From 41b88950ef97305cb53ac47e7d31367f53ed1632 Mon Sep 17 00:00:00 2001 From: Joachim Schoeberl Date: Mon, 13 May 2024 17:38:40 +0200 Subject: [PATCH] Update documentation --- MPIparallel/paralleliteration.html | 260 +++++++++--------- _sources/MPIparallel/paralleliteration.ipynb | 8 + _sources/aposteriori/equilibrated.ipynb | 16 +- _sources/secondorder/erroranalysis.ipynb | 57 ++-- _sources/secondorder/finiteelements.ipynb | 1 + aposteriori/equilibrated.html | 33 ++- .../timedependent/waves/ringresonator.err.log | 44 --- .../waves/wave-leapfrogDG.err.log | 44 --- searchindex.js | 2 +- secondorder/erroranalysis.html | 57 ++-- secondorder/finiteelements.html | 17 +- timedependent/waves/wave-leapfrogDG.html | 2 +- 12 files changed, 232 insertions(+), 309 deletions(-) delete mode 100644 reports/timedependent/waves/ringresonator.err.log delete mode 100644 reports/timedependent/waves/wave-leapfrogDG.err.log diff --git a/MPIparallel/paralleliteration.html b/MPIparallel/paralleliteration.html index 56327e3f..004747a2 100644 --- a/MPIparallel/paralleliteration.html +++ b/MPIparallel/paralleliteration.html @@ -57,7 +57,7 @@ - + @@ -504,7 +504,7 @@

86. Iteration methods in parallel
Starting 4 engines with <class 'ipyparallel.cluster.launcher.MPIEngineSetLauncher'>
-
[0, 1, 2, 3]
+
[0, 1, 2, 3]
@@ -566,104 +566,104 @@

86.1. Richardson iteration 
[stdout:0] 0.002131411608940556
 0.0016785598117133912
-0.0014383176229297977
-0.0012601136081994502
+0.0014383176229297973
+0.0012601136081994504
 0.0011163969057570045
-0.0009957540133089201
-0.000892013039583751
+0.0009957540133089203
+0.0008920130395837511
 0.0008013860460216729
-0.0007213664248458749
-0.0006501978632935307
-0.0005865868529358019
+0.000721366424845875
+0.0006501978632935306
+0.0005865868529358018
 0.0005295374859014327
 0.00047825313447910644
-0.0004320763401729044
-0.0003904510870202349
-0.00035289842915932834
-0.00031900021889388745
-0.0002883878366131279
-0.0002607340736402399
-0.00023574705190481803
-0.00021316549799011987
-0.00019275494792186845
-0.00017430461490466548
-0.00015762474697784634
-0.00014254435985880777
-0.00012890926660311873
-0.00011658034877248673
-0.00010543202872418053
-9.53509125215299e-05
-8.623457969303072e-05
-7.79905007837357e-05
-7.053506705294771e-05
-6.3792719216495e-05
-5.769516408736931e-05
-5.218066951299397e-05
-4.71934292567164e-05
-4.268299050241807e-05
+0.0004320763401729045
+0.00039045108702023503
+0.0003528984291593283
+0.00031900021889388756
+0.00028838783661312803
+0.00026073407364023984
+0.00023574705190481792
+0.0002131654979901198
+0.0001927549479218684
+0.00017430461490466542
+0.0001576247469778465
+0.00014254435985880764
+0.00012890926660311867
+0.00011658034877248662
+0.00010543202872418047
+9.535091252152984e-05
+8.623457969303073e-05
+7.799050078373569e-05
+7.053506705294776e-05
+6.379271921649497e-05
+5.7695164087369264e-05
+5.218066951299396e-05
+4.719342925671642e-05
+4.268299050241808e-05
 3.860373752689044e-05
-3.4914425821440545e-05
-3.1577761578216526e-05
-2.856002200721243e-05
-2.5830712431211527e-05
-2.336225652920634e-05
+3.4914425821440525e-05
+3.15777615782165e-05
+2.8560022007212456e-05
+2.583071243121153e-05
+2.336225652920626e-05
 2.112971647279069e-05
-1.9110540031668575e-05
-1.7284332019465414e-05
-1.563264771416978e-05
-1.4138806122733465e-05
-1.2787721169956143e-05
-1.1565749080625242e-05
-1.0460550393450774e-05
-9.460965197737418e-06
-8.556900320854897e-06
-7.739227318013283e-06
-6.999690227069895e-06
-6.330822151353702e-06
-5.7258698239108366e-06
-5.178725388153858e-06
-4.6838647035290175e-06
-4.236291551289987e-06
-3.831487175498001e-06
-3.4653646485897776e-06
-3.1342275998439424e-06
-2.8347328893352102e-06
-2.5638568499710354e-06
-2.318864756351698e-06
-2.097283211876788e-06
-1.896875175059282e-06
-1.715617372711863e-06
-1.551679871817e-06
-1.4034076037199845e-06
-1.2693036540219163e-06
-1.148014149399825e-06
-1.0383145887165753e-06
-9.390974803808237e-07
-8.493611611120727e-07
-7.681996832028874e-07
-6.947936681618619e-07
-6.284020343837914e-07
-5.683545153190265e-07
-5.140448925990265e-07
-4.649248757945639e-07
-4.204985670127124e-07
-3.8031745444511146e-07
-3.439758843202849e-07
-3.111069655442327e-07
-2.8137886568143383e-07
-2.54491460880751e-07
-2.3017330592352966e-07
-2.0817889380370092e-07
-1.8828617717300159e-07
-1.7029432662799943e-07
-1.540217032071367e-07
-1.3930402462835032e-07
-1.259927067541289e-07
-1.1395336353993158e-07
-1.0306445032160044e-07
-9.321603674500005e-08
-8.430869694970844e-08
-7.625250580230147e-08
+1.911054003166856e-05
+1.7284332019465428e-05
+1.5632647714169753e-05
+1.4138806122733448e-05
+1.2787721169956141e-05
+1.156574908062523e-05
+1.0460550393450818e-05
+9.460965197737416e-06
+8.556900320854911e-06
+7.739227318013271e-06
+6.999690227069886e-06
+6.330822151353692e-06
+5.725869823910824e-06
+5.1787253881538545e-06
+4.683864703529036e-06
+4.236291551289979e-06
+3.831487175497996e-06
+3.4653646485897827e-06
+3.1342275998439534e-06
+2.8347328893352373e-06
+2.563856849971057e-06
+2.318864756351678e-06
+2.0972832118768083e-06
+1.8968751750592811e-06
+1.7156173727118508e-06
+1.551679871817023e-06
+1.4034076037199853e-06
+1.269303654021906e-06
+1.1480141493998358e-06
+1.0383145887165588e-06
+9.390974803808109e-07
+8.493611611120771e-07
+7.68199683202876e-07
+6.947936681618856e-07
+6.284020343837986e-07
+5.683545153190316e-07
+5.140448925990182e-07
+4.649248757945668e-07
+4.2049856701270923e-07
+3.80317454445105e-07
+3.4397588432028707e-07
+3.1110696554423406e-07
+2.81378865681444e-07
+2.544914608807454e-07
+2.301733059235292e-07
+2.0817889380369626e-07
+1.8828617717299907e-07
+1.7029432662800023e-07
+1.5402170320713728e-07
+1.3930402462834775e-07
+1.2599270675412542e-07
+1.1395336353993284e-07
+1.03064450321599e-07
+9.321603674500055e-08
+8.430869694970864e-08
+7.625250580230092e-08

@@ -678,7 +678,7 @@

86.1. Richardson iteration - +

Very similar for other iteraton methods, such as Conjugate Gradients. The matrix operation goes from consistent to distributed without communication, the preconditioner does the cumulation. The inner products are between different vector types:

@@ -700,40 +700,40 @@

86.1. Richardson iteration86.1. Richardson iteration - +
diff --git a/_sources/MPIparallel/paralleliteration.ipynb b/_sources/MPIparallel/paralleliteration.ipynb index 032fc728..fcee3772 100644 --- a/_sources/MPIparallel/paralleliteration.ipynb +++ b/_sources/MPIparallel/paralleliteration.ipynb @@ -154,6 +154,14 @@ "source": [ "c.shutdown(hub=True)" ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6defb2f8-1a74-4f3e-a2ae-09ef1828406d", + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": { diff --git a/_sources/aposteriori/equilibrated.ipynb b/_sources/aposteriori/equilibrated.ipynb index b484f782..47860db7 100644 --- a/_sources/aposteriori/equilibrated.ipynb +++ b/_sources/aposteriori/equilibrated.ipynb @@ -63,10 +63,12 @@ "The lower bound depends on the shape of elements and the coefficient $\\lambda$, but is robust with respect to the polynomial order $k$.\n", "\n", "The main idea is the following: Instead of calculating the $H^{-1}$-norm of $r$, we compute a lifting $\\sigma^\\Delta$ such that $\\operatorname{div} \\sigma^\\Delta = r$, and calculate the $L_2$-norm of $\\sigma^\\Delta$. Since $r$ is not a regular function, the equation must be posed in distributional form:\n", + "\n", "$$\n", "\\int_\\Omega \\sigma^\\Delta \\cdot \\nabla \\varphi = - r(\\varphi) \\qquad \\forall \\, \\varphi \\in V\n", "$$\n", "Then, the residual can be estimated without envolving any generic constant:\n", + "\n", "\\begin{eqnarray*}\n", "\\| r \\|_{A^\\ast} & = & \\sup_{v \\in V} \\frac{r(v)}{\\| v \\|_A} = \\sup_v \\frac{\\int \\sigma^\\Delta \\cdot \\nabla v }{\\|v \\|_A} \\\\\n", "&= & \\sup_v \\frac{\\int \\lambda^{-1/2} \\sigma^\\Delta \\cdot \\lambda^{1/2} \\nabla v }{\\|v \\|_A}\n", @@ -215,11 +217,10 @@ "r2 = MoveTo(0.6,0.6).Rectangle(0.2,0.2).Face()\n", "r2.faces.name='source'\n", "\n", - "# r1 -= r2\n", - "# shape = Glue( [r1,r2] )\n", - "shape = r1\n", - "mesh = Mesh(OCCGeometry(shape,dim=2).GenerateMesh(maxh=0.1))\n", + "r1 -= r2\n", + "shape = Glue( [r1,r2] )\n", "\n", + "mesh = Mesh(OCCGeometry(shape,dim=2).GenerateMesh(maxh=0.1))\n", "Draw (mesh);" ] }, @@ -237,8 +238,7 @@ "u,v = fes.TnT()\n", "\n", "a = BilinearForm(grad(u)*grad(v)*dx).Assemble()\n", - "# source = mesh.MaterialCF( { \"source\" : 1 }, default=1 )\n", - "source = x*y\n", + "source = mesh.MaterialCF( { \"source\" : x*y }, default=0 )\n", "f = LinearForm(source*v*dx).Assemble()\n", "\n", "gfu = GridFunction(fes, name=\"solution\")\n", @@ -285,8 +285,8 @@ "n = specialcf.normal(mesh.dim)\n", "bfequ = (tau*sigma + w*div(tau) + v*div(sigma) + w*mu + lam*v) * dx \n", "bfequ += (-sigma*n*vf-tau*n*wf)*dx(element_boundary=True) \n", - "bfequ += (-1e10*wf.Trace()*vf.Trace())*ds(definedon=dirichlet) # penalty for Dirichlet\n", - "bfequ += (-1e10*mu*lam)*ds(definedon=dirichlet) # no equilibration condition at Dir bnd\n", + "bfequ += (-1e10*wf.Trace()*vf.Trace())*ds(definedon=mesh.Boundaries(dirichlet)) # penalty for Dirichlet\n", + "bfequ += (-1e10*mu*lam)*ds(definedon=mesh.Boundaries(dirichlet)) # no constraint at Dir bnd\n", "\n", "# the residual: element-term + edge term:\n", "lfequ = (source+Trace(gfu.Operator(\"hesse\")))*v*dx \n", diff --git a/_sources/secondorder/erroranalysis.ipynb b/_sources/secondorder/erroranalysis.ipynb index 3cc29095..2fb6653a 100644 --- a/_sources/secondorder/erroranalysis.ipynb +++ b/_sources/secondorder/erroranalysis.ipynb @@ -55,12 +55,13 @@ "\n", "$$\n", "\\int (\\sigma - I_h \\sigma) \\tau_h + \n", - "\\int \\underbrace{(I - P_h) \\opdiv \\sigma}_{\\in V_h^\\ast} \\;\\underbrace{ q_h}_{\\in V_h} + \\int \\underbrace{ \\opdiv \\tau_h}_{\\in V_h} \\, \\underbrace{ (u-P_h u) }_{\\in V_h^\\ast}\n", + "\\int \\underbrace{(I - P_h) \\opdiv \\sigma}_{\\in V_h^\\bot} \\;\\underbrace{ q_h}_{\\in V_h} + \\int \\underbrace{ \\opdiv \\tau_h}_{\\in V_h} \\, \\underbrace{ (u-P_h u) }_{\\in V_h^\\bot}\n", "$$\n", "\n", "Thanks to orthogonality, the second and third term vanish !\n", "\n", "Thus, we get the error estimate \n", + "\n", "$$\n", "\\| \\sigma_h - I_h \\sigma \\|_{H(\\opdiv)} + \\| u_h - P_h u \\|_{L_2}\n", "\\preceq \\sup_{\\tau_h} \\frac{ \\int (\\sigma - I_h \\sigma) \\tau_h} { \\| \\tau_h \\|_{H(\\opdiv)}} \\leq \\| \\sigma - I_h \\sigma \\|_{L_2}\n", @@ -75,32 +76,6 @@ "The flux error is as good as we can interpolate into the flux space. Since the finite element space for $u_h$ is of lower order, the error $u - u_h$ is in general of lower order. But, the filtered error $\\| u_h - P_h u_h \\|$ has the better order." ] }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Local post-processing\n", - "\n", - "Since $\\nabla u = \\lambda^{-1} \\sigma$, we can reconstruct a better approximation $\\widetilde u$ by small, element-wise problems:\n", - "\n", - "$$\n", - "\\widetilde u = \\operatorname{arg}\\min_{v_h \\in P^{k+1} \\atop \\int_T v_h = \\int_T u_h} \\| \\lambda \\nabla v_h -  \\sigma \\|_{L_2, \\lambda^{-1}}^2\n", - "$$\n", - "\n", - "This optimization problems can be written as a mixed variational problem:\n", - "\n", - "Find: $\\widetilde u \\in P^{k+1,dc}$ and $p_h \\in P^0$:\n", - "\n", - "$$\n", - "\\begin{array}{ccccll}\n", - "\\sum_T \\int_T \\lambda \\nabla \\widetilde u \\nabla \\widetilde v \n", - "& + & \\int_{\\Omega} \\widetilde v_h p_h & = & \\sum_T \\int_T \\sigma_h \\nabla \\widetilde v_h & \\forall \\, \\widetilde v_h \\\\\n", - "\\int_{\\Omega} \\widetilde u_h q_h & & & = & \\int_{\\Omega} u_h q_h & \n", - "\\forall q_h\n", - "\\end{array}\n", - "$$" - ] - }, { "cell_type": "code", "execution_count": null, @@ -161,7 +136,33 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Now do the postprocessing. This requires to solve decoupled problems on every element, what is cheap." + "## Local post-processing\n", + "\n", + "Since $\\nabla u = \\lambda^{-1} \\sigma$, we can reconstruct a better approximation $\\widetilde u$ by small, element-wise problems:\n", + "\n", + "$$\n", + "\\widetilde u = \\operatorname{arg}\\min_{v_h \\in P^{k+1} \\atop \\int_T v_h = \\int_T u_h} \\| \\lambda \\nabla v_h -  \\sigma \\|_{L_2, \\lambda^{-1}}^2\n", + "$$\n", + "\n", + "This optimization problems can be written as a mixed variational problem:\n", + "\n", + "Find: $\\widetilde u \\in P^{k+1,dc}$ and $p_h \\in P^0$:\n", + "\n", + "$$\n", + "\\begin{array}{ccccll}\n", + "\\sum_T \\int_T \\lambda \\nabla \\widetilde u \\nabla \\widetilde v \n", + "& + & \\int_{\\Omega} \\widetilde v_h p_h & = & \\sum_T \\int_T \\sigma_h \\nabla \\widetilde v_h & \\forall \\, \\widetilde v_h \\\\\n", + "\\int_{\\Omega} \\widetilde u_h q_h & & & = & \\int_{\\Omega} u_h q_h & \n", + "\\forall q_h\n", + "\\end{array}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This requires to solve decoupled problems on every element, what is cheap." ] }, { diff --git a/_sources/secondorder/finiteelements.ipynb b/_sources/secondorder/finiteelements.ipynb index 25697b89..0ed6b64e 100644 --- a/_sources/secondorder/finiteelements.ipynb +++ b/_sources/secondorder/finiteelements.ipynb @@ -36,6 +36,7 @@ "It has dimension 6. We need two functionals per edge to define the normal component.\n", "\n", "The $BDM_k$ elements are defined as\n", + "\n", "$$\n", "V_T = [P^k]^2\n", "$$\n", diff --git a/aposteriori/equilibrated.html b/aposteriori/equilibrated.html index 4e9b2a09..8a890a29 100644 --- a/aposteriori/equilibrated.html +++ b/aposteriori/equilibrated.html @@ -59,7 +59,7 @@ - + @@ -547,11 +547,12 @@

29.1. General framework\(\lambda\), but is robust with respect to the polynomial order \(k\).

-

The main idea is the following: Instead of calculating the \(H^{-1}\)-norm of \(r\), we compute a lifting \(\sigma^\Delta\) such that \(\operatorname{div} \sigma^\Delta = r\), and calculate the \(L_2\)-norm of \(\sigma^\Delta\). Since \(r\) is not a regular function, the equation must be posed in distributional form: -$\( +

The main idea is the following: Instead of calculating the \(H^{-1}\)-norm of \(r\), we compute a lifting \(\sigma^\Delta\) such that \(\operatorname{div} \sigma^\Delta = r\), and calculate the \(L_2\)-norm of \(\sigma^\Delta\). Since \(r\) is not a regular function, the equation must be posed in distributional form:

+
+\[ \int_\Omega \sigma^\Delta \cdot \nabla \varphi = - r(\varphi) \qquad \forall \, \varphi \in V -\)$ -Then, the residual can be estimated without envolving any generic constant:

+\]
+

Then, the residual can be estimated without envolving any generic constant:

\[\begin{eqnarray*} \| r \|_{A^\ast} & = & \sup_{v \in V} \frac{r(v)}{\| v \|_A} = \sup_v \frac{\int \sigma^\Delta \cdot \nabla v }{\|v \|_A} \\ @@ -676,17 +677,16 @@

29.3. Equilibration in NGSolver2 = MoveTo(0.6,0.6).Rectangle(0.2,0.2).Face() r2.faces.name='source' -# r1 -= r2 -# shape = Glue( [r1,r2] ) -shape = r1 -mesh = Mesh(OCCGeometry(shape,dim=2).GenerateMesh(maxh=0.1)) +r1 -= r2 +shape = Glue( [r1,r2] ) +mesh = Mesh(OCCGeometry(shape,dim=2).GenerateMesh(maxh=0.1)) Draw (mesh);

-
+
@@ -697,8 +697,7 @@

29.3. Equilibration in NGSolveu,v = fes.TnT() a = BilinearForm(grad(u)*grad(v)*dx).Assemble() -# source = mesh.MaterialCF( { "source" : 1 }, default=1 ) -source = x*y +source = mesh.MaterialCF( { "source" : x*y }, default=0 ) f = LinearForm(source*v*dx).Assemble() gfu = GridFunction(fes, name="solution") @@ -708,7 +707,7 @@

29.3. Equilibration in NGSolve -

+

construct \(\sigma\) such that \(\operatorname{div} \sigma = f + \Delta u_h\) from local contributions \(\sigma = \sum \sigma^V\):

@@ -736,8 +735,8 @@

29.3. Equilibration in NGSolven = specialcf.normal(mesh.dim) bfequ = (tau*sigma + w*div(tau) + v*div(sigma) + w*mu + lam*v) * dx bfequ += (-sigma*n*vf-tau*n*wf)*dx(element_boundary=True) -bfequ += (-1e10*wf.Trace()*vf.Trace())*ds(definedon=dirichlet) # penalty for Dirichlet -bfequ += (-1e10*mu*lam)*ds(definedon=dirichlet) # no equilibration condition at Dir bnd +bfequ += (-1e10*wf.Trace()*vf.Trace())*ds(definedon=mesh.Boundaries(dirichlet)) # penalty for Dirichlet +bfequ += (-1e10*mu*lam)*ds(definedon=mesh.Boundaries(dirichlet)) # no constraint at Dir bnd # the residual: element-term + edge term: lfequ = (source+Trace(gfu.Operator("hesse")))*v*dx @@ -762,7 +761,7 @@

29.3. Equilibration in NGSolve -

+

Check equilibrium:

@@ -772,7 +771,7 @@

29.3. Equilibration in NGSolve -

+ diff --git a/reports/timedependent/waves/ringresonator.err.log b/reports/timedependent/waves/ringresonator.err.log deleted file mode 100644 index 784c9dfe..00000000 --- a/reports/timedependent/waves/ringresonator.err.log +++ /dev/null @@ -1,44 +0,0 @@ -Traceback (most recent call last): - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 782, in _async_poll_for_reply - msg = await ensure_async(self.kc.shell_channel.get_msg(timeout=new_timeout)) - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_core/utils/__init__.py", line 198, in ensure_async - result = await obj - ^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_client/channels.py", line 315, in get_msg - raise Empty -_queue.Empty - -During handling of the above exception, another exception occurred: - -Traceback (most recent call last): - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_cache/executors/utils.py", line 58, in single_nb_execution - executenb( - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 1314, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_core/utils/__init__.py", line 165, in wrapped - return loop.run_until_complete(inner) - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/asyncio/base_events.py", line 687, in run_until_complete - return future.result() - ^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 709, in async_execute - await self.async_execute_cell( - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 1005, in async_execute_cell - exec_reply = await self.task_poll_for_reply - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 806, in _async_poll_for_reply - error_on_timeout_execute_reply = await self._async_handle_timeout(timeout, cell) - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 856, in _async_handle_timeout - raise CellTimeoutError.error_from_timeout_and_cell( -nbclient.exceptions.CellTimeoutError: A cell timed out while it was being executed, after 30 seconds. -The message was: Cell execution timed out. -Here is a preview of the cell contents: -------------------- -['scene = Draw (gfu.components[0], order=3, min=-0.05, max=0.05, autoscale=False)', '', 'from time import time', 'ts = time()', 'with TaskManager(): '] -... -[' if i%20 == 0:', ' scene.Redraw()', '', '', 'print ("total time", time()-ts)'] -------------------- - diff --git a/reports/timedependent/waves/wave-leapfrogDG.err.log b/reports/timedependent/waves/wave-leapfrogDG.err.log deleted file mode 100644 index 19281971..00000000 --- a/reports/timedependent/waves/wave-leapfrogDG.err.log +++ /dev/null @@ -1,44 +0,0 @@ -Traceback (most recent call last): - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 782, in _async_poll_for_reply - msg = await ensure_async(self.kc.shell_channel.get_msg(timeout=new_timeout)) - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_core/utils/__init__.py", line 198, in ensure_async - result = await obj - ^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_client/channels.py", line 315, in get_msg - raise Empty -_queue.Empty - -During handling of the above exception, another exception occurred: - -Traceback (most recent call last): - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_cache/executors/utils.py", line 58, in single_nb_execution - executenb( - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 1314, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/jupyter_core/utils/__init__.py", line 165, in wrapped - return loop.run_until_complete(inner) - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/asyncio/base_events.py", line 687, in run_until_complete - return future.result() - ^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 709, in async_execute - await self.async_execute_cell( - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 1005, in async_execute_cell - exec_reply = await self.task_poll_for_reply - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 806, in _async_poll_for_reply - error_on_timeout_execute_reply = await self._async_handle_timeout(timeout, cell) - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - File "/Library/Frameworks/Python.framework/Versions/3.12/lib/python3.12/site-packages/nbclient/client.py", line 856, in _async_handle_timeout - raise CellTimeoutError.error_from_timeout_and_cell( -nbclient.exceptions.CellTimeoutError: A cell timed out while it was being executed, after 30 seconds. -The message was: Cell execution timed out. -Here is a preview of the cell contents: -------------------- -['gfp.Interpolate( exp(-100*(x**2+y**2+z**2)))', 'gfu.vec[:] = 0', '', 'if dim == 2:', ' scene = Draw (gfp, order=3, deformation=True);'] -... -[' cnt = cnt+1', ' if cnt%10 == 0:', ' if dim == 3:', ' gftr.vec.data = traceop * gfp.vec', ' scene.Redraw()'] -------------------- - diff --git a/searchindex.js b/searchindex.js index 4a56b3f9..d80ccaec 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Search.setIndex({"alltitles": {"3D Solid Mechanics": [[74, "d-solid-mechanics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[71, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]], "A Small Number of Constraints": [[82, "a-small-number-of-constraints"]], "A posteriori error estimates": [[33, "a-posteriori-error-estimates"], [46, null]], "A practical introduction": [[45, "a-practical-introduction"]], "Abstract Theory": [[45, "abstract-theory"], [46, null], [53, "abstract-theory"]], "Abstract theory for mixed finite element methods": [[54, "abstract-theory-for-mixed-finite-element-methods"]], "Adding a coarse grid space": [[108, "adding-a-coarse-grid-space"]], "Additional methods": [[45, "additional-methods"]], "Additive Schwarz Methods": [[105, "additive-schwarz-methods"]], "Algorithm": [[66, "algorithm"]], "An Interactive Introduction to the Finite Element Method": [[45, "an-interactive-introduction-to-the-finite-element-method"], [46, "an-interactive-introduction-to-the-finite-element-method"]], "Analysis of the 2-level method:": [[109, "analysis-of-the-2-level-method"]], "Analysis of the DD preconditioner": [[109, "analysis-of-the-dd-preconditioner"]], "Analysis of the Multigrid Iteration": [[64, "analysis-of-the-multigrid-iteration"]], "Analysis of the method": [[108, "analysis-of-the-method"]], "Analysis of the multi-level preconditioner": [[65, "analysis-of-the-multi-level-preconditioner"]], "Appendix": [[45, "appendix"]], "Application of the abstract theory": [[89, "application-of-the-abstract-theory"]], "Approximation of Dirichlet boundary conditions": [[8, "approximation-of-dirichlet-boundary-conditions"]], "Approximation of coercive variational problems": [[28, "approximation-of-coercive-variational-problems"]], "Approximation of inf-sup stable variational problems": [[31, "approximation-of-inf-sup-stable-variational-problems"]], "BDDC - Preconditioner": [[37, "bddc-preconditioner"]], "Basic Iterative Methods": [[52, "basic-iterative-methods"]], "Basic properties": [[27, "basic-properties"]], "Basis functions for the segment:": [[80, "basis-functions-for-the-segment"]], "Basis functions for the triangle:": [[80, "basis-functions-for-the-triangle"]], "Block-Jacobi and general additive Schwarz preconditioners": [[105, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Block-preconditioning": [[84, "block-preconditioning"]], "Boundary Conditions": [[55, "boundary-conditions"], [73, "boundary-conditions"]], "Boundary conditions:": [[57, "boundary-conditions"]], "Bramble-Hilbert Lemma": [[102, "bramble-hilbert-lemma"]], "Building systems from building-blocks": [[30, "building-systems-from-building-blocks"]], "Butcher tableaus of simple methods": [[111, "butcher-tableaus-of-simple-methods"]], "Chebyshev polynomials": [[47, "chebyshev-polynomials"]], "Coercive examples": [[30, "coercive-examples"]], "Coercive variational problems and their approximation": [[28, "coercive-variational-problems-and-their-approximation"]], "Commuting diagram for H^1 - H(\\opcurl)": [[68, "commuting-diagram-for-h-1-h-opcurl"]], "Comparison to DD with minimal overlap": [[109, "comparison-to-dd-with-minimal-overlap"]], "Computation of element-vectors and element-matrices": [[13, "computation-of-element-vectors-and-element-matrices"]], "Computation of the lifting \\| \\sigma^\\Delta \\|": [[34, "computation-of-the-lifting-sigma-delta"]], "Computing dual norms": [[75, "computing-dual-norms"]], "Conjugate Gradients": [[49, "conjugate-gradients"]], "Consistent and Distributed Vectors": [[24, "consistent-and-distributed-vectors"]], "Consistent vectors:": [[24, "consistent-vectors"]], "Constrained minimization problem": [[53, "constrained-minimization-problem"]], "Constrained minimization problems": [[84, "constrained-minimization-problems"]], "Continuity and discrete coercivity of the HDG bilinear-form": [[1, "continuity-and-discrete-coercivity-of-the-hdg-bilinear-form"]], "Convergence of Runge Kutta methods": [[111, "convergence-of-runge-kutta-methods"]], "DG - Methods for elliptic problems": [[2, "dg-methods-for-elliptic-problems"]], "Diagonal preconditioner for L_2-norm": [[106, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[106, "diagonal-preconditioner-for-the-h-1-norm"]], "Diagonally implicit Runge-Kutta methods:": [[111, "diagonally-implicit-runge-kutta-methods"]], "Dirichlet boundary conditions": [[2, "dirichlet-boundary-conditions"]], "Dirichlet boundary conditions as mixed system": [[53, "dirichlet-boundary-conditions-as-mixed-system"]], "Dirichlet boundary conditions by penalty:": [[83, "dirichlet-boundary-conditions-by-penalty"]], "Discontinuous Galerkin Methods": [[45, "discontinuous-galerkin-methods"], [46, null]], "Discontinuous Galerkin for the Wave Equation": [[121, "discontinuous-galerkin-for-the-wave-equation"]], "Discontinuous Galerkin method": [[6, "discontinuous-galerkin-method"]], "Distributed Meshes and Spaces": [[15, "distributed-meshes-and-spaces"]], "Distributed finite element spaces": [[15, "distributed-finite-element-spaces"]], "Distributed vectors and matrices": [[24, "distributed-vectors-and-matrices"]], "Domain Decomposition with Lagrange parameters": [[40, "domain-decomposition-with-lagrange-parameters"]], "Domain Decomposition with minimal overlap": [[108, "domain-decomposition-with-minimal-overlap"]], "Dual mixed formulation": [[89, "dual-mixed-formulation"]], "Dual mixed method": [[60, "dual-mixed-method"]], "Efficiency of the residual error estimator": [[36, "efficiency-of-the-residual-error-estimator"]], "Efficient implementation:": [[121, "efficient-implementation"]], "Efficiently computable multi-level decomposition": [[66, "efficiently-computable-multi-level-decomposition"]], "Eigenvalues of the discretized Laplace-operator": [[121, "eigenvalues-of-the-discretized-laplace-operator"]], "Eqivalent versions of the Poincar\u00e9 inequality": [[102, "eqivalent-versions-of-the-poincare-inequality"]], "Equilibrated Residual Error Estimates": [[34, "equilibrated-residual-error-estimates"]], "Equilibration in NGSolve": [[34, "equilibration-in-ngsolve"]], "Equivalent norms on H^1 and on sub-spaces": [[101, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Error Analysis in L_2 \\times H^1": [[87, "error-analysis-in-l-2-times-h-1"]], "Error estimate of the L_2 projection": [[79, "error-estimate-of-the-l-2-projection"]], "Error estimates in L_2-norm": [[8, "error-estimates-in-l-2-norm"]], "Error estimates:": [[62, "error-estimates"]], "Essential boundary conditions": [[73, "essential-boundary-conditions"]], "Example": [[35, "example"]], "Example: Dirichlet boundary condition by penalty": [[56, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Finite elements for Stokes": [[54, "example-finite-elements-for-stokes"]], "Example: Nearly incompressible materials": [[56, "example-nearly-incompressible-materials"]], "Examples": [[84, "examples"], [111, "examples"]], "Exercise A:": [[107, "exercise-a"]], "Exercise B:": [[107, "exercise-b"]], "Exercise:": [[11, "exercise"], [111, "exercise"], [121, "exercise"]], "Exercise: Robust preconditioners": [[107, "exercise-robust-preconditioners"]], "Exercises": [[30, "exercises"], [75, "exercises"], [102, "exercises"]], "Exercises:": [[112, "exercises"]], "Expanding the Krylov-space": [[49, "expanding-the-krylov-space"]], "Experiment with CoefficientFunctions": [[75, "experiment-with-coefficientfunctions"]], "Experiments with BilienarForms and LinearForms": [[75, "experiments-with-bilienarforms-and-linearforms"]], "Experiments with norms": [[103, "experiments-with-norms"]], "Experiments with overlapping DD": [[109, "experiments-with-overlapping-dd"]], "Experiments with the Richardson iteration": [[48, "experiments-with-the-richardson-iteration"]], "Explicit Euler method (EE)": [[112, "explicit-euler-method-ee"]], "Explicit methods:": [[111, "explicit-methods"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[112, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Exponential Integrators for Parabolic Equations": [[115, "exponential-integrators-for-parabolic-equations"]], "Extending boundary data": [[66, "extending-boundary-data"]], "Extension operators": [[100, "extension-operators"]], "FETI methods": [[38, "feti-methods"]], "FETI-DP": [[39, "feti-dp"]], "Finite Element Error Analysis": [[86, "finite-element-error-analysis"]], "Finite Element Method": [[9, "finite-element-method"], [46, null]], "Finite Element Spaces": [[58, "finite-element-spaces"]], "Finite Elements in H(\\operatorname{div})": [[88, "finite-elements-in-h-operatorname-div"]], "Finite element error analysis": [[8, "finite-element-error-analysis"]], "Finite element error estimates": [[54, "finite-element-error-estimates"]], "Finite element system assembling": [[13, "finite-element-system-assembling"]], "Flux recovery in H(\\operatorname{div})": [[33, "flux-recovery-in-h-operatorname-div"]], "Flux-recovery error estimates with NGSolve": [[33, "flux-recovery-error-estimates-with-ngsolve"]], "Fourth Order Equation": [[3, "fourth-order-equation"]], "Fractional Sobolev spaces": [[102, "fractional-sobolev-spaces"]], "Friedrichs\u2019 Inequality": [[92, "friedrichs-inequality"], [95, "friedrichs-inequality"]], "Friedrichs\u2019 inequality": [[102, "friedrichs-inequality"], [103, "friedrichs-inequality"]], "Friedrichs\u2019 inequality in 1D": [[102, "friedrichs-inequality-in-1d"]], "Friedrichs\u2019 inequality on the square": [[102, "friedrichs-inequality-on-the-square"]], "Galerkin method in space": [[113, "galerkin-method-in-space"]], "General definition:": [[100, "general-definition"]], "General framework": [[34, "general-framework"]], "Generalized derivatives": [[98, "generalized-derivatives"]], "Geometric mesh refinement": [[8, "geometric-mesh-refinement"]], "Geometry with local details:": [[118, "geometry-with-local-details"]], "Get started with Netgen-Opencascade geometric modeling": [[75, "get-started-with-netgen-opencascade-geometric-modeling"]], "Goal driven error estimates": [[35, "goal-driven-error-estimates"]], "Graded meshes around vertex singularities": [[8, "graded-meshes-around-vertex-singularities"]], "Graph-based mesh partitioning": [[108, "graph-based-mesh-partitioning"]], "Grating": [[44, "grating"]], "H(\\operatorname{div}) on sub-domains": [[90, "h-operatorname-div-on-sub-domains"]], "H(div)-conforming Stokes": [[7, "h-div-conforming-stokes"]], "H^1-norm with small L_2-term": [[106, "h-1-norm-with-small-l-2-term"]], "Heat Equation": [[113, "heat-equation"]], "Hellan-Herrmann-Johnson method": [[67, "hellan-herrmann-johnson-method"]], "Helmholtz Equation": [[43, "helmholtz-equation"]], "High Order Finite Elements": [[46, null]], "High order elements": [[8, "high-order-elements"]], "Hybrid DG for elliptic equations": [[1, "hybrid-dg-for-elliptic-equations"]], "Hybrid Interfaces": [[0, "hybrid-interfaces"]], "Hybridization Techniques": [[91, "hybridization-techniques"]], "Hybridized C^0-continuous interior penalty method:": [[3, "hybridized-c-0-continuous-interior-penalty-method"]], "Implement a parallel BDDC preconditioner": [[42, "implement-a-parallel-bddc-preconditioner"]], "Implement our own system assembling": [[11, "implement-our-own-system-assembling"]], "Implementation of Finite Elements": [[10, "implementation-of-finite-elements"]], "Implementation of High Order Finite Elements": [[80, "implementation-of-high-order-finite-elements"]], "Implicit Euler method (IE)": [[112, "implicit-euler-method-ie"]], "Implicit Euler time-stepping": [[113, "implicit-euler-time-stepping"]], "Inf-sup stable variational problems": [[31, "inf-sup-stable-variational-problems"]], "Inner products:": [[24, "inner-products"]], "Installing MPI and PETSc without conda": [[20, "installing-mpi-and-petsc-without-conda"]], "Installing NGSolve": [[46, "installing-ngsolve"]], "Installing conda-packages using pip": [[20, "installing-conda-packages-using-pip"]], "Installing with conda": [[20, "installing-with-conda"]], "Instationary Transport Equation": [[4, "instationary-transport-equation"]], "Integration by parts": [[100, "integration-by-parts"]], "Interface conditions": [[55, "interface-conditions"]], "Interfaces": [[0, "interfaces"]], "Interpolation space H^s": [[41, "interpolation-space-h-s"]], "Interpolation spaces": [[100, "interpolation-spaces"]], "Introduction to MPI with mpi4py": [[20, "introduction-to-mpi-with-mpi4py"]], "Introduction to Non-overlapping Domain Decomposition": [[40, "introduction-to-non-overlapping-domain-decomposition"]], "Inverse estimates": [[103, "inverse-estimates"]], "Iteration Methods": [[45, "iteration-methods"], [46, null]], "Iteration methods in parallel": [[21, "iteration-methods-in-parallel"]], "Iterative Solvers": [[45, "iterative-solvers"], [77, "iterative-solvers"]], "Jacobi and Gauss Seidel Preconditioners": [[51, "jacobi-and-gauss-seidel-preconditioners"]], "Kirchhoff Plate equation": [[67, "kirchhoff-plate-equation"]], "Korn\u2019s inequality": [[103, "korn-s-inequality"]], "Legendre Polynomials": [[79, "legendre-polynomials"]], "Linear Algebra": [[75, "linear-algebra"]], "Linear elasticity": [[74, "linear-elasticity"]], "Linear stability classification": [[112, "linear-stability-classification"]], "Literature": [[46, "literature"]], "Local post-processing": [[86, "local-post-processing"]], "Marked edge bisection": [[33, "marked-edge-bisection"]], "Mass-lumping and Local time-stepping": [[118, "mass-lumping-and-local-time-stepping"]], "Matrix vector multiplication:": [[24, "matrix-vector-multiplication"]], "Maxwell equations:": [[83, "maxwell-equations"]], "Mean-value interpolation": [[102, "mean-value-interpolation"]], "Mesh refinement algorithms": [[33, "mesh-refinement-algorithms"]], "Minimization problem": [[30, "minimization-problem"]], "Mixed Finite Element Methods": [[45, "mixed-finite-element-methods"], [46, null]], "Mixed Methods for Elasticity": [[45, "mixed-methods-for-elasticity"]], "Mixed Methods for Plates and Shells": [[45, "mixed-methods-for-plates-and-shells"]], "Mixed Methods for Second Order Equations": [[45, "mixed-methods-for-second-order-equations"], [46, null]], "Mixed Methods for second order equations": [[57, "mixed-methods-for-second-order-equations"]], "Mixed method for second order equation": [[53, "mixed-method-for-second-order-equation"]], "Multi-level Extension": [[66, "multi-level-extension"]], "Multigrid Methods": [[45, "multigrid-methods"], [46, null]], "Multigrid Preconditioning": [[63, "multigrid-preconditioning"]], "Multigrid and Multilevel Methods": [[63, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[63, "multilevel-preconditioner"]], "NGSolve - PETSc interface": [[14, "ngsolve-petsc-interface"]], "Nano-optics: A ring-resonator": [[119, "nano-optics-a-ring-resonator"], [120, "nano-optics-a-ring-resonator"]], "Natural boundary conditions": [[73, "natural-boundary-conditions"]], "Natural trace space": [[41, "natural-trace-space"]], "Nearly optimal analysis of the ML - preconditioner": [[65, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Newmark time-stepping method": [[114, "newmark-time-stepping-method"]], "Nitsche\u2019s Method for boundary and interface conditions": [[0, "nitsche-s-method-for-boundary-and-interface-conditions"]], "Nitsche\u2019s method:": [[0, "nitsche-s-method"]], "Non-conforming Finite Element Methods": [[12, "non-conforming-finite-element-methods"]], "Non-linear dynamics": [[59, "non-linear-dynamics"]], "Non-overlapping Domain Decomposition Methods": [[45, "non-overlapping-domain-decomposition-methods"], [46, null]], "Nonlinear Shells": [[70, "nonlinear-shells"]], "Normal-trace of functions in H(\\operatorname{div})": [[90, "normal-trace-of-functions-in-h-operatorname-div"]], "Numerical analysis of Parabolic Equations": [[45, "numerical-analysis-of-parabolic-equations"]], "Numerical analysis of Wave Equations": [[45, "numerical-analysis-of-wave-equations"]], "Numerical verification": [[97, "numerical-verification"]], "One sup is enough": [[30, "one-sup-is-enough"]], "Optimal analysis of the multi-level preconditioner": [[65, "optimal-analysis-of-the-multi-level-preconditioner"]], "Optimal convergence of the V-cycle": [[64, "optimal-convergence-of-the-v-cycle"]], "Optimizing the relaxation parameter \\alpha": [[48, "optimizing-the-relaxation-parameter-alpha"]], "Ordinary differential equations": [[45, "ordinary-differential-equations"]], "Orthogonal polynomials on triangles": [[79, "orthogonal-polynomials-on-triangles"]], "Overlapping DD Methods with coarse grid": [[109, "overlapping-dd-methods-with-coarse-grid"]], "Overlapping Domain Decomposition Methods": [[109, "overlapping-domain-decomposition-methods"]], "Overlapping blocks": [[110, "overlapping-blocks"]], "PETSc preconditioner for NGSolve": [[14, "petsc-preconditioner-for-ngsolve"]], "Parallel Solvers": [[45, "parallel-solvers"], [46, null]], "Parameter Dependent Problems": [[56, "parameter-dependent-problems"], [83, "parameter-dependent-problems"]], "Penalty formulation for the Stokes equation:": [[83, "penalty-formulation-for-the-stokes-equation"]], "Penalty formulation for the flux:": [[83, "penalty-formulation-for-the-flux"]], "Piola Transformation": [[88, "piola-transformation"]], "Poincar\u00e9 inequality": [[102, "poincare-inequality"], [103, "poincare-inequality"]], "Poincar\u00e9 inequality in 1D": [[102, "poincare-inequality-in-1d"]], "Point evaluation functional": [[102, "point-evaluation-functional"]], "Poisson equation in NGSolve:": [[76, "poisson-equation-in-ngsolve"]], "Preamble": [[72, "preamble"]], "Preconditioner for S": [[38, "preconditioner-for-s"]], "Preconditioning": [[51, "preconditioning"]], "Primal mixed formulation": [[89, "primal-mixed-formulation"]], "Primal mixed method:": [[60, "primal-mixed-method"]], "Projected preconditioner": [[82, "projected-preconditioner"]], "Projection based interpolation": [[79, "projection-based-interpolation"]], "Projection based interpolation on triangles": [[79, "projection-based-interpolation-on-triangles"]], "Projection matrices from the finest level": [[63, "projection-matrices-from-the-finest-level"]], "Projection onto subspaces": [[32, "projection-onto-subspaces"]], "Proof of the trace inequality:": [[97, "proof-of-the-trace-inequality"]], "Proving discrete LBB by the Fortin operator": [[54, "proving-discrete-lbb-by-the-fortin-operator"]], "Red-Green Refinement": [[33, "red-green-refinement"]], "Regularity-free estimate": [[65, "regularity-free-estimate"]], "Reissner Mindlin Plates": [[69, "reissner-mindlin-plates"]], "Relationship between HHJ and TDNNS": [[68, "relationship-between-hhj-and-tdnns"]], "Reliability of the residual error estimator": [[36, "reliability-of-the-residual-error-estimator"]], "Repeat the exercise in 2D.": [[30, "repeat-the-exercise-in-2d"]], "Richardson iteration": [[21, "richardson-iteration"]], "Riesz representation theorem and symmetric variational problems": [[29, "riesz-representation-theorem-and-symmetric-variational-problems"]], "Robust coarse-grid correction": [[83, "robust-coarse-grid-correction"]], "Robust smoothers": [[83, "robust-smoothers"]], "Robust two-level methods for parameter dependent problems": [[83, "robust-two-level-methods-for-parameter-dependent-problems"]], "Runge Kutta Methods": [[111, "runge-kutta-methods"]], "Saddle-point Problems": [[45, "saddle-point-problems"], [46, null]], "Scaled domain": [[102, "scaled-domain"]], "Schur complement iteration": [[84, "schur-complement-iteration"]], "Schwarz preconditioners for high order finite elements": [[110, "schwarz-preconditioners-for-high-order-finite-elements"]], "Second inf-sup condition means onto": [[30, "second-inf-sup-condition-means-onto"]], "Shift theorems": [[104, "shift-theorems"]], "Simple methods based on the integral equation": [[112, "simple-methods-based-on-the-integral-equation"]], "Single step methods and hyperbolic equations": [[112, "single-step-methods-and-hyperbolic-equations"]], "Single step methods and parabolic equations": [[112, "single-step-methods-and-parabolic-equations"]], "Single-step methods": [[112, "single-step-methods"]], "Sobolev Spaces": [[46, null]], "Sobolev spaces": [[99, "sobolev-spaces"]], "Sobolev spaces over sub-domains": [[100, "sobolev-spaces-over-sub-domains"]], "Solving Stokes in parallel": [[22, "solving-stokes-in-parallel"]], "Solving Stokes\u2019 equation with the non-conforming P^1-triangle": [[12, "solving-stokes-equation-with-the-non-conforming-p-1-triangle"]], "Solving the Poisson Equation": [[76, "solving-the-poisson-equation"]], "Solving the minimization problem": [[49, "solving-the-minimization-problem"]], "Solving the wave equation:": [[121, "solving-the-wave-equation"]], "Some Examples of ASM preconditioners": [[106, "some-examples-of-asm-preconditioners"]], "Some inequalities in Sobolev spaces": [[45, "some-inequalities-in-sobolev-spaces"]], "Splitting Methods for the time-dependent convection diffusion equation": [[5, "splitting-methods-for-the-time-dependent-convection-diffusion-equation"]], "Stability conditions": [[112, "stability-conditions"]], "Stability function": [[112, "stability-function"]], "Stability of the method": [[6, "stability-of-the-method"]], "Stationary Transport Equation": [[6, "stationary-transport-equation"]], "Step function": [[102, "step-function"]], "Stokes Equation": [[58, "stokes-equation"]], "Stokes equation within the abstract theory": [[53, "stokes-equation-within-the-abstract-theory"]], "Structure of Saddle-point Problems": [[84, "structure-of-saddle-point-problems"]], "Sub-space Correction Methods": [[45, "sub-space-correction-methods"], [46, null]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[105, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "Symmetric variational problems": [[29, "symmetric-variational-problems"]], "TDNNS Variational formulation:": [[62, "tdnns-variational-formulation"]], "Tangential displacement normal normal stress continuous finite elements": [[62, "tangential-displacement-normal-normal-stress-continuous-finite-elements"]], "Testing the differential operators": [[121, "testing-the-differential-operators"]], "The 1D case": [[79, "the-1d-case"]], "The Additive Schwarz Lemma": [[105, "the-additive-schwarz-lemma"]], "The Algorithm": [[64, "the-algorithm"]], "The Approximation Property": [[64, "the-approximation-property"]], "The Bramble-Pasciak Transformation": [[81, "the-bramble-pasciak-transformation"]], "The Chebyshev Method": [[47, "the-chebyshev-method"]], "The Chebyshev iteration": [[47, "the-chebyshev-iteration"]], "The Cl\u00e9ment- operator": [[36, "the-clement-operator"]], "The Finite Element Method": [[76, "the-finite-element-method"]], "The First Lemma of Strang": [[12, "the-first-lemma-of-strang"]], "The Galerkin Method": [[45, "the-galerkin-method"], [46, null]], "The Gradient Method": [[50, "the-gradient-method"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[67, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "The Helmholtz equation": [[45, "the-helmholtz-equation"]], "The MPI library": [[20, "the-mpi-library"]], "The ParallelDofs class": [[15, "the-paralleldofs-class"]], "The Poincar\u00e9 inequality": [[96, "the-poincare-inequality"]], "The Richardson Iteration": [[48, "the-richardson-iteration"]], "The Second Lemma of Strang": [[12, "the-second-lemma-of-strang"]], "The Smothing Property": [[64, "the-smothing-property"]], "The Sobolev space H^1, linear and bilinear forms": [[76, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Trace Inequality": [[97, "the-trace-inequality"]], "The Verlet method:": [[118, "the-verlet-method"]], "The Zienkiewicz Zhu error estimator": [[33, "the-zienkiewicz-zhu-error-estimator"]], "The divergence of nn-continuous piece-wise smooth functions:": [[62, "the-divergence-of-nn-continuous-piece-wise-smooth-functions"]], "The function space H(\\operatorname{div})": [[90, "the-function-space-h-operatorname-div"]], "The installation happens in three steps:": [[20, "the-installation-happens-in-three-steps"]], "The preconditioned Richardson iteration": [[51, "the-preconditioned-richardson-iteration"]], "The preconditioned gradient method": [[51, "the-preconditioned-gradient-method"]], "The residual error estimator": [[36, "the-residual-error-estimator"]], "The trace space H^{1/2}": [[100, "the-trace-space-h-1-2"]], "The trace space on one edge of the square": [[100, "the-trace-space-on-one-edge-of-the-square"]], "The upper-bound by the overlap": [[105, "the-upper-bound-by-the-overlap"]], "The weak formulation of the Poisson equation": [[104, "the-weak-formulation-of-the-poisson-equation"]], "This repository contains an interactive introduction to the Finite Element Method": [[25, "this-repository-contains-an-interactive-introduction-to-the-finite-element-method"]], "Time-dependent Problems": [[45, "time-dependent-problems"]], "Time-dependent problems": [[46, null]], "Trace inequality": [[103, "trace-inequality"]], "Trace norm on bottom edge": [[41, "trace-norm-on-bottom-edge"]], "Trace norm on boundary sub-domains": [[41, "trace-norm-on-boundary-sub-domains"]], "Trace theorems and their applications": [[100, "trace-theorems-and-their-applications"]], "Traces spaces": [[41, "traces-spaces"]], "Trapezoidal method": [[112, "trapezoidal-method"]], "Two-level analysis for Maxwell equations": [[83, "two-level-analysis-for-maxwell-equations"]], "Using PETSc": [[23, "using-petsc"]], "Using ipyparallel": [[20, "using-ipyparallel"]], "Variable Coefficients": [[78, "variable-coefficients"]], "Variable coefficients": [[33, "variable-coefficients"]], "Variational Formulation": [[58, "variational-formulation"]], "Variational formulation in space": [[113, "variational-formulation-in-space"]], "Variational formulation:": [[74, "variational-formulation"]], "Vector operations:": [[24, "vector-operations"]], "Visualizing the basis functions": [[76, "visualizing-the-basis-functions"]], "Wave Equation": [[114, "wave-equation"]], "Weak formulation": [[76, "weak-formulation"]], "With the coarse space": [[108, "with-the-coarse-space"]], "Work with GridFunctions": [[75, "work-with-gridfunctions"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[60, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "\\DeclareMathOperator{\\opdiv}{div}\n\\DeclareMathOperator{\\opcurl}{curl}\n\\DeclareMathOperator{\\eps}{\\varepsilon}\nReduced symmetry methods": [[61, "declaremathoperator-opdiv-div-declaremathoperator-opcurl-curl-declaremathoperator-eps-varepsilon-reduced-symmetry-methods"]], "complex-valued problem as real system": [[30, "complex-valued-problem-as-real-system"]], "hp - Finite Elements": [[79, "hp-finite-elements"]], "inf-sup condition": [[30, "inf-sup-condition"]], "inf-sup condition of the first-order derivative bilinear-form": [[30, "inf-sup-condition-of-the-first-order-derivative-bilinear-form"]], "use our own integrators for element matrix calculation:": [[11, "use-our-own-integrators-for-element-matrix-calculation"]], "use our own matrix assembling function:": [[11, "use-our-own-matrix-assembling-function"]], "we can inspect the element matrix:": [[11, "we-can-inspect-the-element-matrix"]]}, "docnames": ["DG/Nitsche", "DG/elliptic", "DG/elliptic_stdDG", "DG/fourthorder", "DG/instationary", "DG/splitting", "DG/stationary", "DG/stokes", "FEM/erroranalysis", "FEM/finiteelements", "FEM/myFiniteElement", "FEM/myassembling", "FEM/nonconforming", "FEM/systemassembling", "MPIparallel/PETSc_interface", "MPIparallel/distmesh", "MPIparallel/hdivnstsimple/NavierStokesMT", "MPIparallel/hdivnstsimple/TestConvert", "MPIparallel/hdivnstsimple/veclaplace", "MPIparallel/hdivnstsimple/veclaplace-auxiliary", "MPIparallel/intro", "MPIparallel/paralleliteration", "MPIparallel/parallelstokes", "MPIparallel/petsc", "MPIparallel/vectypes", "README", "Untitled", "abstracttheory/BasicProperties", "abstracttheory/Coercive", "abstracttheory/RieszRepresentation", "abstracttheory/exercises", "abstracttheory/infsup", "abstracttheory/subspaceprojection", "aposteriori/aposteriori", "aposteriori/equilibrated", "aposteriori/goaldriven", "aposteriori/residualEE", "domaindecomposition/BDDC", "domaindecomposition/FETI", "domaindecomposition/FETI-DP", "domaindecomposition/introduction", "domaindecomposition/traces", "ex20210609", "helmholtz/absorbing", "helmholtz/grating", "iFEM", "intro", "iterative/Chebyshev", "iterative/Richardson", "iterative/conjugategradients", "iterative/gradientmethod", "iterative/preconditioning", "iterative/simple", "mixed/abstract", "mixed/abstractfem", "mixed/boundary", "mixed/pardep", "mixed/secondorder", "mixed/stokes", "mixedelasticity/dynamics", "mixedelasticity/hellingerreissner", "mixedelasticity/reducedsymmetry", "mixedelasticity/tdnns", "multigrid/algorithms", "multigrid/analysisMG", "multigrid/analysisML", "multigrid/multilevel-extension", "plates/hhj", "plates/hhj-tdnns", "plates/reissnermindlin", "plates/shells", "plates/tdnnsplate", "preamble", "primal/boundary_conditions", "primal/elasticity3D", "primal/exercises", "primal/first_example", "primal/solvers", "primal/subdomains", "pversion/hpfem", "pversion/myhighorder", "saddlepoint/bramblepasciak", "saddlepoint/fewconstraints", "saddlepoint/parameterdependent", "saddlepoint/structure", "saddlepoint/useBP", "secondorder/erroranalysis", "secondorder/erroranalysisl2h1", "secondorder/finiteelements", "secondorder/framework", "secondorder/hdiv", "secondorder/hybridization", "sobolev/Friedrichs", "sobolev/Untitled", "sobolev/Untitled1", "sobolev/Untitled2", "sobolev/poincare", "sobolev/trace", "sobolevspaces/GeneralizedDerivatives", "sobolevspaces/SobolevSpaces", "sobolevspaces/Traces", "sobolevspaces/equivalentnorms", "sobolevspaces/exercises", "sobolevspaces/experiments", "sobolevspaces/preciseweak", "subspacecorrection/asm", "subspacecorrection/examples", "subspacecorrection/exercise", "subspacecorrection/minimaldd", "subspacecorrection/overlapping", "subspacecorrection/pversion", "timedependent/ODEs/runge_kutta", "timedependent/ODEs/singlestep", "timedependent/intro/heatequation", "timedependent/intro/waveequation", "timedependent/parabolic/exponential", "timedependent/waves/Maxwell-DG", "timedependent/waves/Untitled", "timedependent/waves/lts", "timedependent/waves/ringresonator", "timedependent/waves/testpml/ringresonator", "timedependent/waves/wave-leapfrogDG"], "envversion": {"sphinx": 61, "sphinx.domains.c": 3, "sphinx.domains.changeset": 1, "sphinx.domains.citation": 1, "sphinx.domains.cpp": 9, "sphinx.domains.index": 1, "sphinx.domains.javascript": 3, "sphinx.domains.math": 2, "sphinx.domains.python": 4, "sphinx.domains.rst": 2, "sphinx.domains.std": 2, "sphinx.ext.intersphinx": 1}, "filenames": ["DG/Nitsche.ipynb", "DG/elliptic.ipynb", "DG/elliptic_stdDG.ipynb", "DG/fourthorder.ipynb", "DG/instationary.ipynb", "DG/splitting.ipynb", "DG/stationary.ipynb", "DG/stokes.ipynb", "FEM/erroranalysis.ipynb", "FEM/finiteelements.ipynb", "FEM/myFiniteElement.ipynb", "FEM/myassembling.ipynb", "FEM/nonconforming.ipynb", "FEM/systemassembling.ipynb", "MPIparallel/PETSc_interface.ipynb", "MPIparallel/distmesh.ipynb", "MPIparallel/hdivnstsimple/NavierStokesMT.ipynb", "MPIparallel/hdivnstsimple/TestConvert.ipynb", "MPIparallel/hdivnstsimple/veclaplace.ipynb", "MPIparallel/hdivnstsimple/veclaplace-auxiliary.ipynb", "MPIparallel/intro.ipynb", "MPIparallel/paralleliteration.ipynb", "MPIparallel/parallelstokes.ipynb", "MPIparallel/petsc.ipynb", "MPIparallel/vectypes.ipynb", "README.md", "Untitled.ipynb", "abstracttheory/BasicProperties.ipynb", "abstracttheory/Coercive.ipynb", "abstracttheory/RieszRepresentation.ipynb", "abstracttheory/exercises.ipynb", "abstracttheory/infsup.ipynb", "abstracttheory/subspaceprojection.ipynb", "aposteriori/aposteriori.ipynb", "aposteriori/equilibrated.ipynb", "aposteriori/goaldriven.ipynb", "aposteriori/residualEE.ipynb", "domaindecomposition/BDDC.ipynb", "domaindecomposition/FETI.ipynb", "domaindecomposition/FETI-DP.ipynb", "domaindecomposition/introduction.ipynb", "domaindecomposition/traces.ipynb", "ex20210609.ipynb", "helmholtz/absorbing.ipynb", "helmholtz/grating.ipynb", "iFEM.ipynb", "intro.md", "iterative/Chebyshev.ipynb", "iterative/Richardson.ipynb", "iterative/conjugategradients.ipynb", "iterative/gradientmethod.ipynb", "iterative/preconditioning.ipynb", "iterative/simple.ipynb", "mixed/abstract.ipynb", "mixed/abstractfem.ipynb", "mixed/boundary.ipynb", "mixed/pardep.ipynb", "mixed/secondorder.ipynb", "mixed/stokes.ipynb", "mixedelasticity/dynamics.ipynb", "mixedelasticity/hellingerreissner.ipynb", "mixedelasticity/reducedsymmetry.ipynb", "mixedelasticity/tdnns.ipynb", "multigrid/algorithms.ipynb", "multigrid/analysisMG.ipynb", "multigrid/analysisML.ipynb", "multigrid/multilevel-extension.ipynb", "plates/hhj.ipynb", "plates/hhj-tdnns.ipynb", "plates/reissnermindlin.ipynb", "plates/shells.ipynb", "plates/tdnnsplate.ipynb", "preamble.ipynb", "primal/boundary_conditions.ipynb", "primal/elasticity3D.ipynb", "primal/exercises.ipynb", "primal/first_example.ipynb", "primal/solvers.ipynb", "primal/subdomains.ipynb", "pversion/hpfem.ipynb", "pversion/myhighorder.ipynb", "saddlepoint/bramblepasciak.ipynb", "saddlepoint/fewconstraints.ipynb", "saddlepoint/parameterdependent.ipynb", "saddlepoint/structure.ipynb", "saddlepoint/useBP.ipynb", "secondorder/erroranalysis.ipynb", "secondorder/erroranalysisl2h1.ipynb", "secondorder/finiteelements.ipynb", "secondorder/framework.ipynb", "secondorder/hdiv.ipynb", "secondorder/hybridization.ipynb", "sobolev/Friedrichs.ipynb", "sobolev/Untitled.ipynb", "sobolev/Untitled1.ipynb", "sobolev/Untitled2.ipynb", "sobolev/poincare.ipynb", "sobolev/trace.ipynb", "sobolevspaces/GeneralizedDerivatives.ipynb", "sobolevspaces/SobolevSpaces.ipynb", "sobolevspaces/Traces.ipynb", "sobolevspaces/equivalentnorms.ipynb", "sobolevspaces/exercises.ipynb", "sobolevspaces/experiments.ipynb", "sobolevspaces/preciseweak.ipynb", "subspacecorrection/asm.ipynb", "subspacecorrection/examples.ipynb", "subspacecorrection/exercise.ipynb", "subspacecorrection/minimaldd.ipynb", "subspacecorrection/overlapping.ipynb", "subspacecorrection/pversion.ipynb", "timedependent/ODEs/runge_kutta.ipynb", "timedependent/ODEs/singlestep.ipynb", "timedependent/intro/heatequation.ipynb", "timedependent/intro/waveequation.ipynb", "timedependent/parabolic/exponential.ipynb", "timedependent/waves/Maxwell-DG.ipynb", "timedependent/waves/Untitled.ipynb", "timedependent/waves/lts.ipynb", "timedependent/waves/ringresonator.ipynb", "timedependent/waves/testpml/ringresonator.ipynb", "timedependent/waves/wave-leapfrogDG.ipynb"], "indexentries": {}, "objects": {}, "objnames": {}, "objtypes": {}, "terms": {"": [8, 9, 12, 18, 19, 25, 27, 28, 29, 30, 32, 34, 39, 45, 46, 53, 54, 64, 65, 70, 71, 72, 74, 75, 76, 77, 80, 81, 82, 84, 92, 100, 101, 102, 104, 105, 111, 112, 115, 118, 121], "0": [0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "00": [53, 76], "0000000000000007": 96, "0000000001010011": 39, "00000000e": 76, "00000001011111010111110": 119, "0000001001000110": 39, "0000010011010000": 39, "0000010110100000": 39, "0000101000001100": 39, "0000110000": 0, "00010095216118821378": 83, "0001011001000000": 39, "00010127534375465337": 115, "00010129028658340466": 48, "00010270136193037442": 48, "00010317733972589769": 84, "00010321518710120788": 84, "00010396456155120892": 84, "00010413209609269955": 48, "00010508880471851492": 85, "00010543202872418053": 21, "00010558276299811141": 48, "000107053640393076": 48, "00010854500989614187": 48, "00010870660188901043": 21, "00010949506857410921": 76, "00010989556910275802": 38, "00011005715705197291": 48, "00011159037138615583": 48, "00011314494646081535": 48, "00011375493321065046": 110, "00011455858334166559": 38, "00011472117993089234": 48, "00011598796595447701": 77, "00011616801306622024": 80, "00011631937360145951": 48, "00011658034877248673": 21, "00011793983348546948": 48, "00011958286986273097": 48, "00012008137501908114": 85, "00012124879733930423": 48, "0001229056447701414": 14, "00012293793490808928": 48, "00012465060600999609": 48, "00012635246751698076": 83, "0001263871385961654": 48, "00012814786519089072": 48, "00012890926660311873": 21, "00012993312295558898": 48, "00013119349751905874": 39, "00013174325375354127": 48, "00013357860421564497": 48, "0001338648614332045": 77, "00013426732432066454": 39, "0001354395258069982": 48, "00013732637489447186": 48, "00013923951281523983": 48, "00014029678008718827": 84, "00014117930594623312": 48, "00014254435985880777": 21, "00014314612577460858": 48, "00014514034896921082": 48, "00014716235745298453": 48, "00014753949588550174": 85, "0001487969376679355": 81, "00014885168670214898": 39, "0001492125384764743": 48, "0001498362645389152": 77, "00015014932580892883": 83, "00015055318197175534": 39, "00015104356593841114": 38, "0001512912846923413": 48, "000153398994230856": 48, "0001555360707766003": 48, "00015762474697784634": 21, "00015770292364614483": 48, "00015989996786680568": 48, "00016212762425659415": 48, "00016400656081982313": 115, "00016421847717886735": 84, "00016434676866042806": 84, "00016438631950522982": 48, "00016491550344460967": 77, "00016602077647740788": 84, "00016667648625635052": 48, "00016757613019838458": 83, "000168998563190804": 48, "00017135299511121462": 48, "00017374023302765528": 48, "00017430461490466548": 21, "00017616073424457884": 48, "00017718883102318885": 110, "0001774124343507029": 63, "00017741243435071894": 63, "00017853792896837772": 77, "0001786149624489968": 48, "00018078405517481033": 91, "0001811033877998466": 48, "00018362648701864706": 48, "0001852656638868365": 83, "00018618474348148243": 48, "00018877864731223603": 48, "0001914086954771383": 48, "00019208909401800605": 77, "00019275494792186845": 21, "00019297576676091736": 84, "0001940753918806743": 48, "0001958658710676972": 85, "00019677924746292868": 48, "00019849103261233912": 38, "00019952078029812246": 48, "00020230051569474947": 48, "00020285820875469017": 21, "00020511898629702412": 48, "00020797673218777952": 48, "00020810720521774676": 77, "00021087430099283637": 48, "00021211111708816004": 38, "00021280037638335268": 81, "00021316549799011987": 21, "0002135880486764573": 84, "00021381224798692054": 48, "00021606301815433576": 83, "00021679113620092907": 48, "00021981153653095173": 48, "0002207732294693555": 115, "00022281477996417238": 84, "00022287402784859796": 48, "00022597919711308994": 48, "00022912763948485545": 48, "00023057965122062087": 77, "00023231995844078663": 48, "00023555676589107158": 48, "00023574705190481803": 21, "00023883868229783445": 48, "00024216633679531238": 48, "0002455403673118469": 48, "0002489614206936301": 48, "0002524301528301413": 48, "0002544824048447953": 84, "00025594722878148363": 48, "0002561028796712345": 85, "0002595133229074687": 48, "0002607340736402399": 21, "0002631291189986039": 48, "00026555485540291635": 77, "00026679531040890817": 48, "00027051260019073385": 48, "00027428170123139963": 48, "00027461820027847346": 83, "0002781033363919083": 48, "000281978238647573": 48, "0002859071512307771": 48, "00028670864286594716": 84, "0002867283203485827": 84, "0002883878366131279": 21, "0002898908277756472": 48, "00029393003246500445": 48, "0002980255401792292": 48, "00030217813664752045": 48, "0003026291735590164": 84, "0003063886186010778": 48, "00031065779392879563": 48, "0003112297948991817": 77, "000314986481834931": 48, "00031900021889388745": 21, "00031937551299935366": 48, "00032163144527917325": 38, "00032219893579614505": 85, "00032382572973983466": 48, "00032833798617694766": 48, "00033291314840126285": 48, "0003375520946429079": 48, "00034225571544383636": 48, "0003437652718687495": 110, "0003470249138323727": 48, "0003518606055004487": 48, "00035289842915932834": 21, "0003567637189834118": 48, "00036173519584250553": 48, "0003641682638948373": 84, "0003658814417513913": 83, "00036677599084996784": 48, "00036683687047396396": 77, "000371887072176918": 48, "00037706942158399916": 48, "00037798057948934126": 76, "0003823240346148249": 48, "00038765192079231786": 48, "00038943081125074715": 21, "0003902376619710067": 38, "0003904510870202349": 21, "00039291080381269393": 85, "0003930541038178998": 48, "0003985316217737422": 48, "00040408552732787813": 48, "0004097168879425399": 48, "0004154267860854548": 48, "0004212163194443691": 48, "00042485165355397067": 81, "0004253862507363e": 48, "00042708660114481857": 48, "00042780257753367363": 77, "0004320763401729044": 21, "0004330387599710549": 48, "00043907394059037843": 48, "0004407752179327705": 84, "000443926110546035": 84, "00044476274680689597": 84, "0004451933037808829": 48, "0004467008934476962": 76, "00045139802666244205": 48, "00045310590428783134": 76, "0004576893029314665": 48, "00046060801777727964": 83, "0004640683430989626": 48, "000470536374732366": 48, "0004761165147485691": 84, "0004770946427009313": 48, "0004778431788232897": 84, "00047825313447910644": 21, "00048374440942494824": 48, "00048575900897660407": 39, "0004904869551287794": 48, "0004933831969703863": 83, "0004942564003676569": 77, "0004973235780976221": 48, "0004994633778866232": 85, "0005": 121, "0005002942167523764": 84, "0005042555949383612": 48, "0005112843408443743": 48, "0005184111698643815": 48, "0005256374551755312": 48, "0005295374859014327": 21, "0005329645893605536": 48, "0005403939846894724": 48, "0005434954855783924": 110, "0005479270734053628": 48, "0005555653080149196": 48, "000563310161583251": 48, "0005645223644237875": 77, "0005668987307889644": 84, "0005669357974305203": 84, "0005711631280335377": 48, "0005791257224512458": 48, "0005865868529358019": 21, "0005871994813932063": 48, "0005953859632015506": 48, "000603686748322705": 48, "0006042081173435654": 83, "0006070238087351294": 76, "00061210343963129": 48, "0006150757230139204": 85, "000620637662759349": 48, "0006292910664307944": 48, "0006352763333349191": 77, "0006380653228011725": 48, "0006469621278029652": 48, "0006501978632935307": 21, "0006559832014964314": 48, "0006568584384695376": 84, "0006602691617452845": 91, "0006651302884261447": 48, "0006744051579833437": 48, "0006815596893261466": 81, "000683642014896617": 83, "0006838096047742041": 48, "0006933454489940485": 48, "000703014536807873": 48, "0007041293195152594": 39, "0007128187407370412": 48, "0007213664248458749": 21, "0007227599600523393": 48, "000725752752707929": 39, "0007272861126943784": 77, "0007281408027218528": 115, "0007328401211737334": 48, "0007372917166384283": 39, "0007430611780766196": 48, "0007534251127049187": 48, "0007639339353912823": 48, "0007745896852841808": 48, "000778573195676947": 85, "0007815351507657772": 83, "000785394430782293": 48, "0007957605690849792": 38, "0007963502699764032": 48, "0007967222328500014": 84, "0007970718790778335": 84, "0007978295322313944": 21, "0008013860460216729": 21, "0008074593310986968": 48, "0008155136133605154": 14, "0008187237729799078": 48, "0008301457855142871": 48, "0008329679456139172": 77, "0008417275901325724": 48, "0008534714402833544": 48, "0008634299729498884": 84, "0008653796219227064": 48, "0008774544540124627": 48, "0008896982890273583": 48, "000892013039583751": 21, "0008992713275613841": 38, "0009021135134710553": 48, "0009022414011419461": 84, "0009147025484014691": 48, "000927467849965456": 48, "0009404119099430902": 48, "000953537256301915": 48, "0009668464537611202": 48, "0009803421043661597": 48, "0009850043882244734": 76, "0009893154642094026": 77, "0009922984038274098": 83, "000994026848073895": 48, "0009957540133089201": 21, "001": [4, 16, 18, 19, 42, 108, 118], "0010069624297562696": 39, "0010079033633484853": 48, "0010219743677684654": 48, "0010257501072204825": 84, "0010362426186451216": 48, "0010507109136523886": 48, "0010534825624365811": 85, "001065382091468879": 48, "0010802590324318184": 48, "0010818082293433016": 63, "0010818082293433783": 63, "0010953446592037578": 48, "001095452880817425": 84, "00110000000001111111111111111111111111111111111111": 108, "0011010100000000": 39, "0011106419374519126": 48, "0011163969057570045": 21, "00112175": 82, "0011261538765406645": 48, "0011307102211392996": 84, "0011334165967012847": 84, "00114188353023771": 48, "0011428466557880732": 110, "0011578339974337177": 48, "0011740084228764713": 48, "001190409997919305": 48, "0011912144262735045": 77, "0012070419612846086": 48, "0012135653881944293": 81, "0012239075998425912": 48, "0012410102494057379": 48, "0012583532955394456": 48, "0012601136081994502": 21, "0012759401743892452": 48, "0012809618903024942": 84, "0012937743735248989": 48, "0013118594328021408": 48, "0013223771120878588": 76, "0013301989452423582": 48, "0013487965579305366": 48, "0013676559729323798": 48, "0013787077033072886": 85, "0013867809482308554": 48, "0013982543263219135": 83, "001406175298682715": 48, "0014258428969956122": 48, "0014383176229297977": 21, "0014397425850613988": 77, "0014457876747266": 48, "0014660136233021089": 48, "0014865247950606303": 48, "001486650169981125": 38, "001495847866702568": 21, "0015073253043182886": 48, "0015284193284582851": 48, "0015498111090446118": 48, "0015715049529611042": 48, "0015837080803695188": 91, "0015921163492495247": 38, "001593505233576355": 48, "0016028677378609755": 80, "0016158163919351772": 48, "0016384429379776749": 48, "0016389793743943e": 48, "0016613894517866516": 48, "0016652310329207363": 85, "0016785598117133912": 21, "0016846605848640193": 48, "0017082610614374385": 48, "0017321956797978454": 48, "0017484379251981065": 77, "0017564693136689225": 48, "0017754830163756964": 76, "001781086913609521": 48, "0018060535084499963": 48, "001815036053916696": 84, "0018222848385735526": 110, "0018231395803895414": 84, "0018313742067635904": 48, "0018570541983739639": 48, "0018830987558998633": 48, "001909513236338424": 48, "0019135138262121013": 84, "0019363030826879543": 48, "00195038440947e": 48, "0019634738256118496": 48, "0019910310851446664": 48, "001996326380935e": 48, "002000033787596275": 83, "002018980572441813": 48, "0020473280915745116": 48, "0020542580850501455": 85, "0020760795413710255": 48, "00209015543474033": 38, "0021046499274098534": 77, "0021052409173061668": 48, "002131411608940556": 21, "002134818313440375": 48, "0021648179244101325": 48, "0021952460474713676": 48, "002217939823003e": 48, "002226109084597644": 48, "0022470117450723555": 81, "0022574135446350964": 48, "0022787217503585614": 110, "002289166045515667": 48, "0023213733165311098": 48, "002354042200669435": 48, "0023859519286668924": 21, "0023871796570160172": 48, "0023928342549886e": 84, "0024127234859912e": 38, "0024207927632217833": 48, "0024548887180404605": 48, "0024894748439375424": 48, "0025083332705242": 77, "002524558589773273": 48, "002560147533562283": 48, "0025621895472479556": 85, "0025962493853125836": 48, "002632871989946613": 48, "0026700233303073207": 48, "002707711530251991": 48, "0027214089686471436": 81, "0027459448578374505": 48, "00274976526375405": 38, "0027644465369766743": 84, "0027847317285992117": 48, "0028240807089284594": 48, "0028410571431563554": 76, "002864000519549906": 48, "002904500039104571": 48, "002938211266571423": 83, "0029400048222059856": 77, "002945588307840723": 48, "002987274531417232": 48, "003029568084823295": 48, "003072478516418594": 48, "003116015552098299": 48, "003160189099587527": 48, "0031661513061312222": 85, "0032050092528697147": 48, "0032504862967541446": 48, "003295602986137172": 76, "0032966307115871057": 48, "0033429816790807e": 48, "003343453178112618": 48, "003390964582487553": 48, "0034196513449910823": 77, "0034274924070966e": 38, "0034391760214572452": 48, "003488098807697223": 48, "0035184047718742643": 84, "003533916534866e": 48, "0035377444753272353": 48, "0035742193069537506": 110, "0035849111823151934": 115, "003588124785603924": 48, "0036392517327986876": 48, "00369054926696621": 21, "003691137550267643": 48, "0037437947167205878": 48, "003797235962696573": 48, "003828543006036e": 48, "0038514742772535683": 48, "003906522914880215": 48, "003962395402637919": 48, "004019105547541827": 48, "004022073286663706": 77, "004044215842409425": 76, "00406449190545115": 103, "004076667444189602": 48, "004135095482647261": 48, "004194404356601538": 48, "004200374856019856": 76, "004204370475404719": 84, "004254609071788941": 48, "004300349628057608": 91, "0043157249547117315": 48, "0043653384802368215": 85, "00436698": 24, "004377767661651467": 48, "004440753187991515": 48, "00444576": 24, "004463903340199571": 83, "004504697877859927": 48, "004569618434104862": 48, "004623704351628802": 84, "004635531928614874": 48, "004702455812997286": 48, "004766219873666417": 77, "0047704079296280195": 48, "004812013162339133": 84, "004823102886311118": 84, "004839406523086855": 48, "0048515408982188635": 14, "0048687652674056035": 21, "0049094702519929204": 48, "0049385274075988955": 38, "004980618201255226": 48, "005": [44, 113], "005050120921221e": 48, "00505286989475429": 48, "005055833317030188": 76, "005065786308053268": 38, "0051166744123425645": 76, "005126245308471082": 48, "005200764884080136": 48, "0052764495430249175": 48, "00532115": 24, "0053533207010933665": 48, "0054314002835130675": 48, "00544510021893": 38, "005481794529587": 84, "005486673124053477": 84, "005510710740585726": 48, "005512321246644729": 110, "005591275063881841": 48, "005596514201958335": 77, "005673116803016881": 48, "005756260083031563": 48, "00579803836835753": 81, "005840729622399467": 48, "00587075e": 76, "0058884106331422": 85, "005926550751686317": 48, "006013749432886288": 48, "006047521099574e": 48, "006052884938201": 84, "006102352279461678": 48, "006183878700617e": 48, "006192386577113453": 48, "006250616150810586": 76, "006268014060966591": 81, "0062838803053114995": 48, "0063391913992392e": 38, "0063768621596142555": 48, "00641114": 24, "0064397891421690305": 77, "006471361574809355": 48, "006567408748907632": 48, "006601850082293865": 83, "006665034668024815": 48, "00666667": 24, "00666667e": 76, "00668231e": 76, "00674705": 24, "0067642711321864776": 48, "0068651507820937825": 48, "006967707126888932": 48, "007071974572961421": 48, "007175095266953569": 39, "007177988453838204": 48, "00720533141359291": 84, "0072391428746526e": 48, "007285785061202624": 48, "007352872683189109": 77, "007363468914282e": 83, "007395401677089779": 48, "007506876607308029": 48, "007538948850570068": 85, "007620249216139159": 48, "007735559962372405": 48, "007767462483043e": 38, "00785285043673079": 48, "00794492344759875": 84, "007972163400751166": 48, "008093542827183131": 48, "008167506192283442": 76, "008217033941975723": 48, "008250496171208304": 77, "008342683267924778": 48, "00838466": 24, "008470538670058644": 48, "008600649402844286": 48, "008733066159301757": 48, "008770264329065e": 48, "008858810330184645": 110, "008867841122120142": 48, "008983784519869005": 77, "009005028016874708": 48, "0091029727105706e": 81, "00914468216745215": 48, "0091636403030867e": 84, "009243880281609152": 85, "00924986": 11, "009286860553797623": 48, "00929141": 24, "00941187124058379": 63, "009411871240584515": 63, "00943162187210645": 48, "0094491081610007e": 48, "009547208619791e": 83, "009567470112709246": 83, "00957902659759177": 48, "009639280718573347": 77, "009656245121461408": 76, "009729137049970023": 48, "00976904": 24, "009775262366193539": 76, "009882017461817013": 48, "00998139": 11, "01": [46, 47, 64, 69, 76, 106, 114, 118], "010000000000006035": 106, "010037734049959545": 48, "0101": 78, "0101101000000000": 39, "010139292127639758": 76, "010196355090081538": 48, "0101979369744123e": 81, "010205668737527e": 77, "010357950994738508": 48, "010359419640115403": 77, "0105225943949912": 48, "01059709e": 76, "010690360225888162": 48, "010835322288439926": 85, "01086132581604804": 48, "01103557098161611": 48, "011161378149839e": 48, "011166401565346e": 48, "011192460122905604": 84, "011213178124896332": 48, "011394232337990156": 48, "011407488793323194": 115, "01148717272069801": 77, "01157882151180759": 48, "0115905": 24, "011767036450854718": 48, "011768153572898": 48, "011843671131637522": 110, "011958970994245996": 48, "0121547221434405": 48, "012187751064729127": 21, "01229555425486588": 38, "01231228e": 76, "0123374": 24, "012354390197259415": 48, "012363": 24, "012479782578389e": 84, "012556324138378401": 81, "012558078894809354": 48, "012765895567014494": 48, "0128688782012293e": 84, "012900680539926e": 77, "012920478670266024": 85, "012977951297551641": 48, "013060236654630365": 77, "013194361094089788": 48, "0133493": 24, "013415244070862429": 48, "0134339": 24, "0134663": 24, "013640723643751601": 48, "013738193462247741": 91, "013747268116727e": 48, "013811543530733e": 84, "013867502737470573": 83, "013870927739243541": 48, "0139698": 24, "014105989018833337": 48, "01413108426448e": 48, "0143129": 24, "01434604512072099": 48, "01450892008875497": 77, "014591238920965869": 48, "014841718816667147": 48, "0148526": 24, "015097639034237173": 48, "0151684": 24, "0152942": 24, "015336450193677e": 48, "015359159966462567": 48, "015471598348598e": 83, "01554149e": 76, "0155923933842377e": 48, "015626448542843247": 48, "0156347": 24, "01570390414750895": 77, "0157114": 24, "01574233336548143": 81, "015899678638718388": 48, "0159095": 24, "016023404407948e": 48, "016142253725977114": 83, "01617903153000453": 48, "0162840947818825e": 48, "016328325811686722": 85, "016464696402088225": 48, "016756870923679827": 48, "017055761899451082": 48, "0173201345490836e": 48, "01736158601934315": 48, "0174519": 24, "01767457072797152": 48, "017677856690926363": 77, "017768": 11, "017929160922748995": 110, "017994955245204867": 48, "018322991779705176": 48, "0183463": 24, "018470559015394374": 80, "0185708": 24, "0185867": 24, "01865894699243924": 48, "0187142": 24, "0189391": 82, "019003103789169966": 48, "0190510611836723e": 48, "01918863e": 76, "01919012190952205": 39, "019355763553309795": 48, "0194531211449646e": 48, "0194674123506714e": 48, "019659": 24, "01971724897913322": 48, "019749": 24, "0197734469007467e": 39, "01983965339095311": 83, "02": [37, 44, 64, 76, 85, 115, 118], "02005492668175809": 38, "020087907739980797": 48, "020188714252706304": 77, "020264232604018606": 103, "020264234576601235": 103, "020347788031209e": 83, "02046811734348254": 48, "02065791386774517": 85, "020831301863592395": 106, "020858291715411852": 48, "021258890368765174": 48, "021302779591812e": 39, "02167043155414069": 48, "021680172914178e": 39, "02209351174170106": 48, "02252883553082078": 48, "0228366": 24, "022852060313368e": 83, "022977263391911276": 48, "023059227507159427": 77, "0231229": 24, "0231253": 24, "0232373": 24, "0232447712545553e": 85, "0232821": 24, "0233311": 24, "0233791": 24, "023439891145440477": 48, "0235113218041585e": 48, "023620592653494303": 21, "0238686459851815e": 48, "023918188335835474": 48, "023921286926102014": 47, "023978513225437e": 48, "0241589": 24, "024273223974312075": 77, "0243206": 24, "02435467077440151": 39, "024384741600227022": 106, "024414250705698613": 48, "024547976218031113": 106, "024931283632543395": 48, "024986973253222e": 14, "02504840e": 76, "0252525392177282e": 48, "025330291735910584": 103, "02538052e": 76, "025474574374679673": 48, "025607813233479115": 77, "0257310986172217e": 77, "025774846052082172": 83, "02605354778036957": 48, "0260991546942733e": 39, "026305373010949e": 48, "0263138": 24, "02642613e": 76, "02651352e": 76, "026686352908694286": 48, "026909264295318674": 85, "026991393488043674": 77, "027036794325895e": 110, "02717043861464373": 110, "027410842175647522": 48, "027487700817419e": 48, "0275036694747827e": 83, "02753241632854": 108, "0278865": 24, "0282924756612993e": 48, "028315523074329833": 48, "02840051e": 76, "028786277945169303": 110, "02881189922538989": 77, "02914557e": 76, "029378085186036483": 77, "0294866473975043e": 48, "02952541052556933": 83, "0297401890608049e": 48, "02974211765794692": 48, "03": [0, 55, 74, 76, 85, 89, 103], "030065910434030483": 83, "03019290019802654": 14, "03049776255121646": 115, "0305053745835908e": 39, "0306445032160044e": 21, "030805": 24, "031006371846374843": 77, "0314919971368875e": 48, "0317430429749966e": 48, "032001079597646e": 39, "0320692": 24, "0321664556812178e": 38, "0322338191445864e": 77, "03235807e": 76, "032415672332935e": 48, "0327422": 24, "032920325740202e": 85, "03296601056961356": 85, "0330722476742483e": 77, "03307384e": 76, "033155171543706e": 48, "0332470429487766e": 48, "03394806e": 76, "0339567": 24, "03430579733377514": 83, "0344221": 24, "0344815": 24, "0346404566701807e": 48, "0349543337339e": 39, "03503614777663428": 77, "035057863467162e": 77, "0352011973243236e": 48, "03576187036516944": 85, "03591803e": 76, "036006096856494e": 84, "0360268233935707": 21, "036081925033399936": 76, "03616817538237669": 110, "036419664573553744": 76, "036882711834327e": 48, "0369993": 11, "0369995": 11, "037204475895342": 77, "037214875409793e": 77, "03732654e": 76, "0377694309296993e": 48, "03798027450044472": 84, "0382184": 24, "0383145887165753e": 21, "038578886687444205": 110, "03867743230879999": 81, "0391573266438795e": 83, "0395349049971244e": 48, "03971919917200752": 38, "0398083805786505e": 48, "039923161314892e": 38, "0399256": 11, "04": [20, 38, 63, 76], "0401019232202664e": 39, "0401499370989996e": 84, "04060539621517081": 83, "04091961212851145": 84, "04114423163307e": 48, "04148923094504725": 38, "04165481393936405": 85, "04182651492449851": 84, "0421362": 24, "042529997182085555": 106, "042700555926476e": 39, "0427258429588364": 109, "04295345e": 76, "04304343e": 76, "04322115e": 76, "04333400e": 76, "043410523067387e": 39, "04378948e": 76, "0440850635700838e": 48, "0441077852041042e": 48, "04445533e": 76, "04452565e": 76, "0448318988346594e": 48, "0456629339641523e": 77, "0458612819916141e": 48, "0460550393450774e": 21, "046167213571327394": 21, "04632471e": 76, "04640769538476629": 70, "0464076953847663": 70, "046559202537099e": 21, "047585281273978e": 48, "0476407832691725e": 48, "047831452027208586": 77, "0479234984947036e": 48, "04829608173279241": 77, "0483439025780243e": 80, "04844263e": 76, "04897732520653299": 106, "04901124e": 76, "0493228166340138e": 38, "05": [0, 6, 7, 14, 16, 18, 19, 21, 22, 38, 39, 42, 43, 48, 55, 58, 63, 76, 77, 78, 81, 83, 84, 85, 91, 106, 108, 110, 113, 114, 115, 119, 120, 121], "05017114e": 76, "0502118520135045e": 38, "050345607732123e": 48, "05056242146491291": 83, "05066057588433867": 103, "05070470124514234": 21, "05072273e": 76, "050808588781358e": 48, "051314861515644065": 85, "051348606027109": 48, "05147539e": 76, "0518600005494": 103, "0522261642833315e": 48, "05225642481371e": 48, "05261663e": 76, "05275907536407711": 0, "053091225238836e": 77, "053108611128666e": 48, "053506705294771e": 21, "0540162324290298e": 48, "0543115072670755e": 83, "054442809432531e": 48, "05473882e": 76, "054825222580238e": 48, "055292332179194e": 39, "05615795e": 76, "056463652129439e": 63, "056463652131211e": 63, "0565477676177174e": 48, "056816693425752e": 48, "056875365473615e": 80, "057699886783593e": 48, "05785815e": 76, "0583026929967562e": 48, "058629781795617e": 48, "05886485e": 76, "0596323": 24, "05969811055414346": 70, "059962390744391e": 77, "06": [14, 21, 38, 39, 48, 63, 76, 77, 80, 81, 83, 84, 85, 91, 110, 115], "060046408245776e": 48, "060188892523224485": 70, "060197008583245344": 70, "060197013034633824": 70, "060197013034635836": 70, "06041231e": 76, "060421857389568e": 77, "0604307395852179e": 48, "060638566853438e": 48, "0612004296881115e": 77, "06173136e": 76, "0622350370452333e": 48, "063041512342942e": 48, "0635527812072218e": 48, "06372227e": 76, "06389531595609149": 83, "064378099172241e": 38, "064615076190458e": 48, "06479309e": 76, "064853508518673e": 84, "06562033841012835": 76, "0658429693010248e": 38, "0665306914429312e": 39, "0666980034078006e": 83, "06672284005037e": 38, "0667455150851594": 85, "06687894741407885": 109, "0668842887956969e": 48, "067774904721001e": 48, "06815699e": 76, "06822155257283025": 81, "06852126e": 76, "0686992932246637e": 48, "06871088406211634": 76, "068880327604704e": 48, "06924526e": 76, "069312434365388e": 83, "06934594552348519": 110, "06941104395496575": 76, "069583126200502e": 83, "06967160e": 76, "06978025266715498": 115, "07": [7, 14, 16, 18, 19, 21, 38, 39, 48, 62, 63, 76, 77, 80, 81, 83, 84, 85, 91, 110, 115], "07053899603094559": 21, "070630129019807e": 77, "0708562851366104e": 38, "0708899746727645": 63, "07092390115693398": 38, "070986974525415e": 48, "0710721": 11, "07109762e": 76, "071677623287515e": 48, "071689163842817e": 48, "0718173613822063e": 38, "07213824e": 76, "072583399247498e": 48, "0728344173676783e": 38, "0733771059172907e": 48, "0736797747190897e": 83, "0752031579870268e": 48, "0754609239472857e": 48, "07563602425424365": 91, "0757704058947594e": 83, "0761093388261588e": 48, "07623135e": 76, "076773468362999e": 110, "07679038814089593": 83, "0769147677204536e": 48, "076925": 11, "0770325975566385e": 48, "07795637134606523": 23, "078183206613188e": 39, "07867885e": 76, "079255401192082e": 77, "0793232627093197e": 76, "0798810040492257e": 77, "0799504864837223e": 77, "08": [14, 21, 38, 39, 48, 63, 77, 80, 81, 83, 84, 85, 91, 110, 115], "0800838951175999e": 77, "080363248841906e": 84, "0816681711721685e": 76, "0817466086216224e": 48, "0817889380370092e": 21, "082154822634119e": 48, "082394154349409e": 48, "08243": 108, "0825055582772711e": 21, "082967519865576": 48, "08333333333333291": 76, "08333333333333307": 76, "08335855188353485": 84, "0835868976279029e": 48, "0837253706493335e": 77, "0838424427178355e": 110, "08403599336703109": 38, "08513052e": 76, "0851966894625983e": 48, "085203085594984e": 48, "0855068742571685e": 38, "085538139091289e": 83, "0861189697629896e": 39, "08658007e": 76, "08686469e": 76, "0871232244493624e": 39, "087166664294984e": 48, "08733": [119, 120], "08754364e": 76, "087705232362404e": 48, "088329890520765e": 48, "08848867205207e": 83, "088744055878357e": 48, "08916336e": 76, "089339379343828e": 48, "08966319416902137": 85, "09": [14, 21, 22, 38, 39, 48, 63, 77, 83, 84, 85, 91, 110, 115], "090080875328715e": 48, "090181364978928e": 48, "090382846710801e": 48, "0905496474263337e": 83, "091879177514704e": 48, "0920362969957407e": 48, "092098918926114e": 83, "0922993209640892e": 48, "09282291331667633": 83, "0938448611287085e": 39, "09384602e": 76, "093847080520956e": 38, "093854065721601e": 48, "09425754e": 76, "09561861e": 76, "0956401951870296e": 48, "095680466304": 103, "0956820207492046e": 84, "09574110e": 76, "095939553147835e": 84, "09680027e": 76, "0968159694944652e": 48, "096870591581053e": 39, "0971473988404078e": 84, "0972441079209224e": 84, "097283211876788e": 21, "097469463440972e": 39, "097522274092465e": 48, "0985340239060944e": 84, "09862199e": 76, "0986818950368167e": 48, "09976291818609e": 48, "0_i": 47, "0_j": 48, "0m": 23, "0x10627a8b0": 76, "0x1098e6250": 23, "0x10e19adf0": 39, "0x10e1f1570": 38, "0x1105e7790": 23, "0x113fac400": 23, "0x11e514450": 23, "1": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 95, 96, 97, 98, 99, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "10": [0, 1, 2, 4, 5, 12, 14, 18, 19, 21, 38, 39, 40, 42, 43, 44, 47, 48, 49, 50, 51, 55, 58, 62, 63, 64, 65, 66, 73, 77, 80, 81, 82, 83, 84, 85, 86, 91, 92, 96, 97, 105, 110, 115, 118, 120, 121], "100": [0, 16, 21, 35, 38, 39, 48, 64, 77, 83, 84, 85, 89, 108, 110, 113, 114, 115, 116, 118, 121], "1000": [18, 19, 48, 64, 81, 82, 85, 112], "10000": [33, 35, 48, 50, 51], "100000": 11, "1001": 48, "100139362740519e": 77, "1002": 48, "1003": 48, "1004": 48, "1005": 48, "1006": 48, "10065": 118, "10066207816841645": 103, "1006946297625608e": 77, "1007": 48, "1008": 48, "1009": 48, "1009066348664202e": 48, "101": [38, 39, 48, 77, 83, 84, 85, 110], "1010": 48, "101080": 22, "1011": 48, "1012": 48, "1013": 48, "10132037983620129": 103, "10132093820158772": 103, "10132105302197744": 103, "1013211807015426": 103, "10132118307694862": 103, "1014": 48, "1014556933272555e": 48, "1015": 48, "1016": 48, "10163889987329": 108, "101670": 121, "1017": 48, "1018": 48, "1019": 48, "102": [38, 39, 48, 77, 83, 84, 85, 110], "1020": 48, "1021": 48, "1022": 48, "1023": 48, "10238050892831017": 85, "1024": 48, "1025": 48, "102520165565222e": 48, "1026": 48, "1027": 48, "1028": 48, "1029": 48, "103": [38, 39, 48, 77, 79, 83, 84, 85, 110], "1030": 48, "1031": 48, "1032": 48, "1033": 48, "1034": 48, "1034116": [65, 105], "1034909470615006e": 48, "1035": 48, "1036": 48, "1037": 48, "1038": 48, "103843864311542e": 48, "1039": 48, "104": [38, 39, 48, 77, 83, 84, 85, 110], "1040": 48, "1041": 48, "1042": 48, "1043": 48, "1044": 48, "1045": 48, "10455003e": 76, "10459024e": 76, "1046": 48, "1047": 48, "1048": 48, "1049": 48, "105": [38, 39, 48, 77, 83, 84, 85, 110], "1050": 48, "105030753170029e": 48, "1051": 48, "1052": 48, "1053": 48, "1053682271319834e": 48, "1054": 48, "1055": 48, "1056": 48, "1057": 48, "1058": 48, "1059": 48, "105x35": [], "106": [38, 39, 48, 77, 83, 84, 85, 110], "1060": 48, "1061": 48, "1062": 48, "1063": 48, "1064": 48, "1065": 48, "1065178148932502e": 77, "1066": 48, "1067": 48, "10673815e": 76, "1068": 48, "1069": 48, "107": [38, 39, 48, 77, 83, 84, 85, 110], "1070": 48, "1071": 48, "1072": 48, "1072387109440784e": 48, "1072490070102246e": 48, "107279980834582e": 84, "1073": 48, "1073268104192383e": 77, "107346442453208e": 48, "1074": 48, "1075": 48, "1076": 48, "1077": 48, "1078": 48, "107896060669359e": 48, "1079": 48, "108": [38, 39, 48, 77, 83, 84, 85, 110], "1080": 48, "108072": 11, "1080931524797886e": 39, "1081": 48, "1082": 48, "1083": 48, "10835203e": 76, "1083784321502765": 48, "1084": 48, "1085": 48, "1085370483308783e": 38, "1086": 48, "1087": 48, "1088": 48, "1089": 48, "109": [38, 39, 48, 77, 83, 84, 85, 110], "1090": 48, "1091": 48, "1092": 48, "1093": 48, "1094": 48, "1095": 48, "10956896174344156": 81, "1096": [48, 67], "1097": 48, "1098": 48, "1099": 48, "109953797594372e": 91, "10999455e": 76, "11": [14, 15, 21, 38, 39, 48, 63, 77, 80, 81, 83, 84, 85, 91, 105, 108, 110, 111, 120], "110": [38, 39, 48, 77, 83, 84, 85, 110], "1100": 48, "1101": 48, "1102": 48, "1103": 48, "1104": 48, "1105": 48, "11052790e": 76, "1106": 48, "1107": 48, "1108": 48, "1109": 48, "110998": 11, "111": [38, 39, 48, 77, 83, 84, 85, 110], "1110": [39, 48], "111069655442327e": 21, "1111": 48, "1111000000000000": 39, "1111111111111111111111111111111111111": 108, "11111111111111111111111111111111111111111111111111": 108, "11111111e": 76, "1112": 48, "1113": 48, "1114": 48, "1114129704008765e": 85, "1115": [15, 48], "1116": [15, 48], "1117": 48, "11175628e": 76, "1118": 48, "111842042028687e": 48, "1119": 48, "112": [38, 48, 77, 83, 84, 85, 110], "1120": 48, "11208822466509e": 115, "1120952556356904e": 48, "1121": 48, "1121913244649083e": 84, "1122": 48, "1123": 48, "1124": 48, "1125": [15, 48], "1126": [15, 48], "1127": 48, "11274286e": 76, "1128": 48, "1129": 48, "112971647279069e": 21, "113": [38, 48, 77, 83, 84, 85, 110], "1130": 48, "1131": 48, "1132": 48, "11326140e": 76, "1133": 48, "1134": 48, "1134450354834574e": 83, "1135": 48, "1136": 48, "1137": [48, 65, 105], "1138": 48, "1139": 48, "1139871745692467e": 48, "114": [38, 48, 77, 83, 84, 85, 110], "1140": 48, "1141": 48, "1142": 48, "114226012216223e": 48, "1142446998583604e": 48, "1143": 48, "114305718318": 103, "1143891383548138e": 83, "1144": 48, "1144795853898352e": 48, "1145": [15, 48], "1146": [15, 48], "1147": 48, "1148": 48, "1149": 48, "115": [38, 48, 77, 83, 84, 85, 110], "1150": 48, "1151": 48, "11519878e": 76, "1152": 48, "1153": 48, "1153690820937868e": 83, "1154": 48, "1155": 48, "1156": 48, "11562856971462293": 85, "1157": 48, "115776481808797e": 77, "1158": 48, "1159": 48, "116": [38, 48, 77, 83, 84, 85, 110], "1160": 48, "1160794010521349e": 38, "116080611146976e": 38, "1161": 48, "1162": 48, "1163": 48, "1164": 48, "1165": 48, "1166": 48, "1166378431252478e": 38, "1167": 48, "1168": 48, "1168648242240292e": 39, "1169": 48, "117": [38, 48, 77, 83, 84, 85, 110], "1170": 48, "1171": 48, "1172": 48, "1173": 48, "1174": 48, "1175": 48, "11756578349743e": 83, "1176": 48, "1177": 48, "11771686788897e": 48, "1178": 48, "117841478485599e": 48, "117889181764347": 63, "1179": 48, "118": [38, 48, 77, 83, 84, 85, 110], "1180": 48, "1181": 48, "1181654480683108e": 76, "1182": 48, "11825842e": 76, "1183": 48, "1184": [15, 48], "1185": 48, "1186": 48, "1187": 48, "1188": 48, "1188632171403412e": 48, "1189": [34, 48], "119": [15, 38, 48, 76, 77, 83, 84, 85, 110], "1190": 48, "1191": 48, "1192": 48, "1192648791598475e": 85, "1193": 48, "1194": 48, "11949520952077154": 83, "1195": 48, "1196": 48, "1197": [34, 48], "119777984490796e": 48, "1198": 48, "1199": 48, "12": [14, 15, 21, 38, 39, 47, 48, 58, 63, 71, 77, 80, 81, 83, 84, 85, 91, 110, 111, 118, 121], "120": [15, 38, 48, 76, 77, 83, 84, 85, 110], "1200": 48, "12007716e": 76, "1201": 48, "1202": 48, "120263469848235e": 83, "1203": 48, "1204": 48, "1205": 48, "1206": 48, "1207": 48, "1207666510181047e": 48, "1208": 48, "1209": 48, "121": [15, 38, 48, 77, 83, 84, 85, 110], "1210": 48, "1211": 48, "1212": 48, "121237851055954e": 48, "1213": 48, "1214": 48, "1214463186327964e": 48, "1215": 48, "1216": 48, "1217": 48, "1218": 48, "1219": 48, "12198974e": 76, "122": [38, 48, 77, 83, 84, 85, 110], "1220": 48, "1221": 48, "1222": 48, "1222150109879732e": 38, "1223": 48, "1224": 48, "1225": 48, "1226": 48, "1226736392497831e": 48, "1227": 48, "1227484703156794": 81, "1228": 48, "122841804400887e": 48, "1229": 48, "123": [38, 48, 77, 83, 84, 85, 110], "1230": 48, "1231": 48, "1232": 48, "1233": 48, "1234": [48, 51], "1235": 48, "1236": 48, "1236004679313227e": 85, "1237": 48, "1237307982241775e": 39, "1238": 48, "1239": 48, "124": [38, 48, 77, 83, 84, 85, 110], "1240": 48, "1241": 48, "1242": 48, "12424681e": 76, "1243": 48, "1244": 48, "1245": 48, "1246": 48, "1247": 48, "1248": [15, 48], "12481": 63, "124854140373132e": 48, "1249": 48, "125": [38, 48, 76, 77, 83, 84, 85, 108, 110], "1250": 48, "12502359e": 76, "125091180878627e": 48, "1251": 48, "12517048e": 76, "1252": 48, "1253": 48, "1254": 48, "1255": 48, "12559322630402628": 84, "1256": 48, "1257": 48, "1258": 48, "1259": 48, "126": [38, 48, 77, 83, 84, 85, 88, 110], "1260": 48, "12604564e": 76, "1261": 48, "12612009e": 76, "1262": 48, "1263": 48, "1264": 48, "1265": 48, "1266": 48, "1267": 48, "1268": 48, "1269": 48, "127": [38, 48, 77, 83, 84, 85, 88, 110], "1270": 48, "1271": 48, "1272": 48, "1273": 48, "1274": 48, "1275": 48, "1275873912643786e": 48, "1276": 48, "1277": 48, "12771016328862472": 85, "1278": 48, "127897790833774e": 14, "1279": 48, "128": [38, 48, 77, 83, 84, 85, 88, 110], "1280": [15, 48], "1281": 48, "1282": 48, "1282174616192325e": 84, "1283": 48, "128385842495598e": 38, "1284": 48, "1284839854964043e": 83, "1285": 48, "1286": 48, "1286760270120043e": 80, "1287": 48, "1288": 48, "1289": 48, "128937157167869": 48, "129": [38, 48, 77, 83, 84, 85, 110], "1290": 48, "1291": 48, "1292": 48, "1293": 48, "1294": 48, "1295": 48, "1295056655675621e": 48, "1296": 48, "1297": 48, "1298": 48, "1299": 48, "13": [14, 15, 21, 38, 39, 48, 58, 63, 77, 80, 81, 83, 84, 85, 91, 110], "130": [38, 48, 77, 83, 84, 85, 110], "1300": 48, "130024942605001e": 38, "1301": 48, "1302": 48, "13029395438108e": 39, "1303": [15, 48], "1304": [15, 48], "1305": 48, "1306": 48, "1307301970457093e": 48, "1308682699054299e": 38, "131": [38, 48, 77, 83, 84, 85, 110], "131205540534497e": 85, "1313": 15, "1313848999463623e": 77, "1314": 15, "1315250560178633e": 39, "1317": 15, "1318": 15, "132": [38, 48, 77, 83, 84, 85, 110], "132234434912973e": 48, "13234921846973": 108, "1324039739542678e": 77, "132574138753976e": 39, "133": [10, 38, 48, 77, 83, 84, 85], "13317477e": 76, "133217721853538e": 83, "133433727673969e": 48, "133572632812229e": 48, "133659": 103, "134": [38, 48, 77, 83, 84, 85], "1342275998439424e": 21, "134355059697924e": 48, "1344496302303633e": 48, "134630462323265e": 48, "135": [38, 48, 77, 83, 84], "1354489295710067e": 38, "136": [38, 48, 77, 83, 84], "1363485282619605e": 39, "136379583878647e": 48, "13658800e": 76, "137": [38, 48, 77, 83, 84, 108], "13724910e": 76, "1372528539117695e": 85, "138": [38, 48, 77, 83, 84], "138056451376175e": 38, "1383131459279391e": 48, "138764267205407e": 48, "138838983489785e": 77, "139": [38, 48, 77, 83, 84], "13912": 108, "13920": 120, "139264286603403e": 48, "139315271444288e": 48, "1393692959144535e": 84, "13946661e": 76, "1395336353993158e": 21, "13978020e": 76, "14": [14, 15, 21, 38, 39, 48, 58, 63, 77, 81, 83, 84, 85, 91, 110], "140": [38, 48, 77, 83], "140103064211771e": 83, "140448925990265e": 21, "1407461783284897e": 84, "141": [48, 77, 83], "1410181950377555e": 77, "14103409e": 76, "1415927925658518": 103, "1415940039476355": 103, "1415964586986087": 103, "141601546084023": 103, "142": [48, 77, 83], "14225686e": 76, "142329644220869e": 48, "142460317688723e": 84, "14247326e": 76, "1427436944464753": 103, "142946942564037e": 83, "142970521525175e": 48, "143": [48, 77, 83], "143020583810261e": 115, "143292625079821e": 63, "143292625081005e": 63, "1432953401769533e": 48, "14351": 118, "1436471197986192e": 21, "14367": 85, "1436973564022294e": 48, "143924722657": 103, "144": [2, 15, 48, 77, 83], "14401498e": 76, "1441040745721575e": 48, "1449": 15, "145": [48, 77, 83], "1450": 15, "145066796269033e": 48, "145240338203072e": 48, "145481": 55, "145700503692558": 48, "1457055730195295e": 38, "1459": 15, "146": [48, 77, 83, 108], "1460": 15, "146247392809796e": 63, "146247392809993e": 63, "14630626612767e": 48, "14643913718708476": 85, "1466": 62, "14677104e": 76, "147": [48, 77, 83], "14706588838844e": 77, "14717576e": 76, "1473442451657755e": 48, "148": [48, 77, 83], "14800181052391703": 83, "148014149399825e": 21, "1487": 15, "1488": 15, "149": [15, 48, 77, 83], "14929094e": 76, "1495557599322867e": 38, "14967093e": 76, "15": [12, 14, 15, 21, 38, 39, 48, 63, 77, 81, 83, 84, 85, 91, 110, 118, 121], "150": [48, 77, 83], "1501": 15, "150172892273618": 63, "1502": 15, "1502531949919336e": 48, "150513": 82, "1509993528977585e": 48, "151": [48, 77, 83], "151239212626374e": 77, "1517867052582708e": 85, "152": [48, 77, 83], "152210014075458e": 48, "1523": 15, "1524": 15, "152829147699458e": 77, "153": [15, 48, 77, 83], "153162058902952e": 48, "15375584e": 76, "154": [48, 77, 83], "1541705203835396e": 48, "154452942554955e": 85, "15456810e": 76, "154591039558322e": 84, "15479975e": 76, "155": [48, 77, 83], "1550389688295956e": 77, "155122265013862e": 48, "15531330e": 76, "155560354442374e": 48, "1557028423569548e": 83, "156": [48, 77, 83], "156042934268715e": 48, "1565545505886938e": 84, "1565749080625242e": 21, "15669640292831505": 14, "15674039e": 76, "157": [15, 48, 77, 83], "157232828249118e": 39, "1577761578216526e": 21, "1578660947274213e": 48, "158": [48, 77, 83], "158325024488876e": 21, "15852596e": 76, "15866512e": 76, "1587": 15, "15873168e": 76, "1588": 15, "159": [48, 77, 83], "1592221117566454e": 48, "159226802859874e": 84, "159386196627027": 63, "159492305023506": 48, "15x15": 120, "16": [14, 21, 38, 39, 48, 63, 67, 76, 77, 81, 83, 84, 85, 88, 91, 103, 108, 110, 121], "160": [48, 77], "1601092281831917e": 77, "1604125116859914e": 48, "161": [48, 77], "1611": 15, "161194204007716e": 48, "1612": 15, "1614959668268976e": 110, "16192264e": 76, "162": [48, 77], "16212877624059613": 70, "16227787e": 76, "1627": 15, "162771755192398e": 48, "1628": 15, "163": [48, 77], "1631690327272827e": 38, "1632383109280923e": 48, "1637": 15, "1638": 15, "164": [48, 77], "164002978391902e": 38, "164087870949272e": 48, "164454944361612e": 48, "164814423789034e": 48, "165": [48, 77], "1652": 55, "1653": 15, "1654": 15, "1655": 108, "16551145145630625": 84, "166": [15, 48, 77], "1662768813921747e": 48, "1663": 15, "1664": 15, "166607912750655e": 77, "16666666666666605": 76, "167": [22, 48, 77], "16703366696607e": 48, "167355481226384e": 48, "167788478028774e": 84, "1678144767741274e": 110, "168": [48, 77], "1682609714160612e": 48, "168625673411682e": 48, "169": [48, 77], "16902662e": 76, "1690509397482924e": 39, "1695613375814276e": 77, "16992": 103, "17": [11, 14, 21, 38, 39, 48, 62, 63, 67, 76, 77, 81, 83, 84, 85, 91, 110, 121], "170": [48, 77], "17018031e": 76, "1702487976526392e": 48, "17034605e": 76, "170953469301783": 48, "171": [48, 77], "17104988e": 76, "171242169527921e": 38, "1715084968823221e": 110, "171539579468978e": 48, "171964483263876e": 48, "172": [15, 48, 77], "17266415e": 76, "1727565": [], "173": [48, 77], "173145243523698e": 38, "1731567271102746e": 38, "173560312218078e": 48, "1736012556074996e": 48, "17363736e": 76, "174": [48, 77], "17444576e": 76, "17444909e": 76, "17484926e": 76, "175": [15, 48, 77], "1750190919030915e": 39, "17521646837299754": 63, "17521646837299884": 63, "175370751686568e": 48, "175796464246952e": 83, "176": [48, 77], "177": [15, 48, 77], "177084140486483e": 48, "1772580024815873e": 48, "1773703164820084e": 48, "177430": 77, "17787146650364064": 84, "177975481974726e": 77, "17798587e": 76, "178": [15, 48, 77], "178359635677189e": 48, "1786334543121044": 63, "17864942689424e": 48, "178725388153858e": 21, "1789330005993": 103, "1789412583431215e": 77, "179": [15, 48, 77], "179038940459572e": 83, "179130114255554e": 91, "179276503633465e": 77, "1797414167033695e": 85, "18": [21, 38, 39, 48, 63, 77, 81, 83, 84, 85, 91, 110, 121], "180": [15, 48, 77], "180098524265756e": 77, "1801844070219922e": 83, "1803": 118, "18047030359955413": 70, "180523233901374e": 85, "180581597168563": 48, "1809640801639115e": 48, "181": [15, 48, 77], "181275397026587e": 84, "181384164542866e": 84, "18166648405462782": 85, "181919481874446e": 39, "18198359e": 76, "182": [15, 48, 77, 121], "18239326239495e": 38, "182464315033788e": 48, "18248908455579e": 48, "1825238009470132e": 48, "1829807261609314e": 83, "183": [15, 48, 77, 121], "183205809357812": 63, "18378034e": 76, "184": [15, 48, 77], "184518646616133e": 38, "1845355279110761e": 48, "1848020578536776e": 85, "185": [15, 48, 77], "18560": 120, "186": [15, 48, 77], "18615733922752714": 83, "186441139990981e": 48, "186544775446272e": 77, "1865510550568645e": 48, "1869": 108, "187": [15, 48, 76, 77], "18746770e": 76, "18773436e": 76, "1879032798831167e": 48, "188": [15, 48, 76, 77], "188084777154004e": 110, "1881411344828965": 70, "18829278797325397": 70, "18829323589939168": 70, "1882932359053115": 70, "18844850e": 76, "18854148259682096": 38, "18856044e": 76, "1886623726682327e": 84, "188762568457179": 48, "189": [15, 48, 77], "189004510910192e": 38, "1897989532745137e": 48, "19": [21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 91, 96, 110, 121], "190": [15, 48, 77], "190039": 82, "1905083170126303e": 48, "191": [15, 48, 76, 77], "19131": 108, "191412740756462e": 84, "1915768422357135e": 91, "19168273e": 76, "1917443509672249e": 48, "192": [15, 48, 76, 77], "192982629062653e": 48, "193": [15, 48, 77], "193017889368154e": 84, "1932267987396277e": 77, "1937717716297164e": 48, "194": [15, 48, 77], "19409740372493e": 48, "19414012e": 76, "194234877618424e": 48, "195": [15, 48, 77], "19502289e": 76, "1957737541722982": 76, "195795581626075": 48, "196": [15, 48, 77], "1960973838068986e": 84, "1961228339113691e": 84, "19613792e": 76, "1962463458313255e": 48, "196926650939042e": 48, "197": [15, 48, 77], "1973176149489421e": 77, "197377": 63, "19756385e": 76, "19794827628847217": 76, "198": [15, 34, 48, 77], "1985": [63, 118], "198666754833949e": 48, "1988": 81, "1989970409922486e": 48, "199": [48, 77], "1992": [65, 105], "19954481e": 76, "199662120862354e": 77, "1997": 5, "1_": [84, 109], "1_0": 76, "1_g": 98, "1_t": 9, "1d": [9, 10, 92, 97, 100], "1e": [1, 2, 5, 7, 14, 17, 18, 23, 47, 48, 49, 50, 51, 58, 62, 74, 81, 82, 83, 85, 106, 118], "1e10": 34, "1e16": [14, 23], "1e3": [0, 55], "1e4": 74, "1e5": 56, "1e6": [37, 39, 76], "1h": 31, "1j": [43, 44, 112], "2": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "20": [0, 4, 6, 21, 38, 39, 43, 47, 48, 51, 55, 63, 77, 81, 83, 84, 85, 91, 110, 115, 118, 119, 120, 121], "200": [47, 48, 49, 77, 81, 83, 84, 91, 103], "2000": [79, 83, 118], "20000": 63, "20003175e": 76, "2001": 63, "2002": [83, 119, 120], "2003": [24, 37], "2005": [60, 83], "2008": 34, "2009": [1, 34, 118], "201": [48, 77], "2010": [3, 7, 115], "201036798469105e": 48, "2014": 118, "2014276423085216e": 63, "20142764230892e": 63, "2016": 7, "20161982e": 76, "2017": 72, "20185210357764e": 83, "20185701640442e": 48, "201912825655762": 48, "202": [48, 77], "2021": 63, "2021410480985224e": 84, "2022": 115, "2024": 103, "20246277e": 76, "20278040915028e": 48, "20278558e": 76, "203": [48, 77], "2030804127834024e": 48, "203340764549582e": 38, "203514": 82, "203579372682444e": 38, "20360315e": 76, "2038392742993523e": 48, "204": [48, 77], "20406673503597e": 48, "20416263e": 76, "204170427930421e": 76, "2047": 118, "204721838426908e": 85, "204985670127124e": 21, "205": [48, 77], "205086292997377e": 48, "205791598142954e": 80, "206": [48, 77], "20600328199376758": 115, "207": [48, 77], "20710259e": 76, "20729480460904": 48, "207304064918974e": 48, "2073949988316887e": 110, "207500945135074": 63, "2075884760544393e": 48, "207797463801602e": 39, "2078": 118, "2079866858818379e": 39, "208": [48, 77], "2081699334725462e": 21, "2083460448557326e": 48, "209": [48, 77], "209841802955364e": 38, "20x15": 120, "21": [21, 38, 39, 48, 63, 77, 81, 83, 84, 85, 91, 110], "210": [48, 74, 77], "2104017086174213e": 48, "210516495096637e": 38, "2105692323236415e": 115, "210914171261402e": 48, "211": [48, 77], "2113462356347832e": 48, "212": [48, 77], "212082195607958": 48, "212176823477045e": 83, "21240144e": 76, "2127666328263005e": 48, "2128164846413054e": 77, "213": [48, 77], "2131": 108, "2135": 103, "213965912854191e": 48, "214": [48, 77], "21422613e": 76, "21448524e": 76, "215": [48, 76, 77], "215611127226652e": 85, "2156997683938283e": 48, "2159255177579773": 81, "216": [48, 76], "216735134902866e": 48, "217": [48, 63, 76], "217767941323583e": 48, "218": [48, 76], "218066951299397e": 21, "218534373172311e": 48, "2189016160123873e": 91, "219": 48, "219015968824155e": 84, "21921127e": 76, "2198400344857686e": 48, "22": [15, 21, 38, 39, 48, 63, 77, 81, 83, 84, 85, 88, 91, 105, 108, 110, 121], "220": 48, "2200814982392898e": 84, "220089440885009e": 48, "22097120565400583": 85, "221": [48, 76], "2210233759685583e": 48, "221168193883369e": 38, "221342747583671e": 48, "221997": 11, "222": [48, 76], "22214879e": 76, "222468225080945e": 48, "22267988e": 76, "222905569117103e": 77, "223": [48, 76], "2231484138440663e": 77, "22396274e": 76, "224": [48, 76], "22447412e": 76, "224582426593474e": 84, "2248018497117877e": 48, "225": [48, 76], "2250262778494888e": 83, "22507913e": 76, "2251790106502366e": 48, "226": [48, 76], "22649519e": 76, "2266542848291076e": 21, "226823012568907e": 48, "227": [48, 76], "2272633103295307e": 48, "2273752434466936e": 84, "22749714e": 76, "227869591181543e": 48, "228": [48, 106], "228111988549506e": 38, "2285806360324746e": 110, "22889271e": 76, "229": [15, 48], "22902254e": 76, "229139870798433e": 110, "2296515582177255e": 48, "23": [15, 21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 91, 108, 110, 121], "230": [15, 48], "230062807166462e": 77, "230426840079289e": 83, "23052075e": 76, "230671159880186e": 48, "230803664237456e": 48, "231": 48, "231350929217185e": 84, "23170524e": 76, "232": [15, 48, 121], "23222011e": 76, "2322662719717318e": 39, "2323126337494496e": 48, "2326351878065917e": 48, "23298683e": 76, "233": [48, 76, 108, 121], "23352169e": 76, "2336993139926075e": 84, "23383303211193643": 85, "234": [48, 121], "2340": 2, "234217088721998e": 48, "234404212444103e": 91, "2345400347809227e": 48, "234563447979218e": 84, "2346168202002402e": 38, "23468220e": 76, "234688003055873e": 85, "2347321588196066e": 48, "23493810e": 76, "235": 48, "23554864e": 76, "235558457392675e": 84, "235698848966272e": 38, "23588458e": 76, "236": 48, "236291551289987e": 21, "236763300738726e": 48, "236833127892583e": 48, "2368607926721074e": 48, "237": 48, "23725": 103, "23770460e": 76, "238": 48, "238341470639586e": 48, "239": [48, 121], "239553": 11, "24": [15, 20, 21, 26, 38, 39, 46, 48, 77, 81, 83, 84, 85, 91, 108, 110, 111, 121], "240": [48, 121], "240522273706592e": 115, "241": [15, 48, 121], "24123503e": 76, "241311930663991e": 48, "24131829e": 76, "242": [48, 121], "242147288218785e": 80, "2421516344125203e": 48, "2422464676774915e": 48, "2424762666737146e": 48, "243": [48, 121], "2434571319271414": 85, "2435260263769448": 103, "244": 48, "2441091631346645e": 84, "2443598039825515e": 48, "2444781742724771e": 38, "245": 48, "2452143271619613e": 77, "24523562e": 76, "245276461511493e": 14, "2453548564298967e": 77, "245672547037219e": 48, "246": 48, "24637796e": 76, "2464607544411714e": 48, "246753131763292e": 77, "247": 48, "24729233e": 76, "248": 48, "2481": [15, 23], "2481652589569217e": 48, "2482396528288556e": 38, "24856511e": 76, "249": 48, "2498065056018573e": 48, "249928356147489e": 48, "24999999999999872": 15, "24999999999999895": 15, "25": [5, 8, 15, 21, 33, 35, 38, 39, 48, 74, 77, 81, 83, 84, 85, 91, 108, 110], "250": 48, "251": 48, "25121637e": 76, "25169512e": 76, "2519326977123633e": 48, "2519635254418385e": 48, "2519836807584633e": 48, "252": 48, "252011089157917e": 38, "25270": 22, "252700643269565e": 48, "253": 48, "25322175e": 76, "2533077908298905e": 83, "253381714290263e": 39, "253472013629693e": 14, "254": 48, "2540629454140222e": 48, "25440808e": 76, "25464403586309087": 83, "254797290530035e": 48, "255": 48, "2550708572760314": 63, "2551405187698492e": 76, "255392285937742e": 48, "255794637574766e": 48, "256": 48, "2561165106050925e": 110, "2562185071306156e": 38, "2563691868200587e": 21, "257": 48, "2570709463492e": 38, "2575225077989435e": 48, "2577596081448574e": 63, "2577596081450706e": 63, "257943509407669e": 48, "258": 48, "25805960e": 76, "2582316265611039": 81, "25880": 121, "259": 48, "2593": [15, 23], "2595516847568828e": 48, "259927067541289e": 21, "26": [21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "260": 48, "26051191101564e": 83, "2607019225332258": 38, "26095800e": 76, "261": 48, "261614742574872e": 21, "2616944617779757e": 48, "262": [34, 48], "26215991e": 76, "262409501838687e": 38, "262631841144873e": 48, "263": 48, "2634537302230126e": 83, "2634965543630566": 85, "263818148215092e": 77, "264": 48, "264312876504194e": 83, "26447785e": 76, "2646": 1, "265": 48, "265668479439641e": 48, "2657": [15, 23], "266": 48, "2662179809977506e": 48, "266740360996993e": 48, "267": 48, "267202512577547e": 48, "2672170441103305e": 48, "2676": 82, "268": 48, "268299050241807e": 21, "268337563450049e": 39, "268589647581184e": 48, "268633095463936e": 48, "26887948e": 76, "269": 48, "269058169182027e": 39, "2693036540219163e": 21, "269372850183054e": 48, "2693827572253804": 81, "269432811736806e": 38, "2695228725235466e": 48, "27": [21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "270": [1, 48], "27043380e": 76, "27068308e": 76, "2706849461530112e": 76, "271": 48, "2715327847755757e": 48, "27177453839846e": 48, "272": 48, "273": 48, "2732527515891966e": 84, "273342095791625e": 39, "2733861726088263e": 48, "27373870e": 76, "274": 48, "2741783209975205e": 84, "27470834e": 76, "275": 48, "2755319017225014e": 39, "276": 48, "27688891e": 76, "277": [48, 49], "277097975239419e": 48, "2773406359217477e": 48, "277596180054449e": 48, "2778255546402775e": 39, "278": 48, "278453604751753e": 38, "278591821354165": 116, "2787374496223366e": 77, "2787721169956143e": 21, "279": 48, "27921192240865e": 38, "279254503545001e": 48, "2792706014737654e": 48, "2794268561723303e": 38, "2799027620592605e": 38, "28": [6, 21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 110], "280": [48, 76], "280805241684333e": 48, "281": [48, 76], "281289667448394e": 48, "28179168e": 76, "282": [48, 76], "283": [48, 76], "28331106e": 76, "2833346908563155e": 48, "284": 48, "28400582e": 76, "284020343837914e": 21, "2848701103604469e": 48, "285": 48, "28519735e": 76, "285198733015978e": 39, "285261803064561e": 39, "286": [48, 115], "28614435e": 76, "28636674165708e": 77, "286505224127795e": 38, "286552374709478e": 48, "28660014129065e": 48, "287": 48, "2870559541431299e": 48, "287100776090044e": 77, "287219173082758e": 48, "28786796e": 76, "288": 48, "2885730706727105e": 48, "289": 48, "2892459896362076e": 48, "29": [15, 21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "290": 48, "290001584068349e": 91, "290161202956915e": 84, "290414512693373e": 48, "290854582112953e": 83, "291": 48, "2916858366707634e": 48, "292": 48, "29215552e": 76, "2924811429051319e": 39, "2926088093011324e": 38, "293": [15, 48, 73], "2931264312198394e": 110, "294": 48, "2941827144102711": 24, "2948886942405e": 48, "29497965478": 106, "295": [15, 48], "2950048008498856e": 38, "295673124262948e": 84, "29588309046074e": 48, "296": [15, 48], "2968072136285665e": 84, "297": 48, "297011872199007e": 84, "2970915870460925e": 48, "297230564528598e": 48, "297285709626039e": 48, "2975": 108, "297607436855025e": 84, "2977358264383947e": 81, "298": [15, 48], "2981497810886503e": 38, "298152195730669e": 38, "29849936e": 76, "2985082498568915e": 84, "298948379400162e": 48, "299": [15, 48], "29946292210663e": 84, "29948886e": 76, "2_": [53, 65, 79, 102, 103, 112], "2_g": 98, "2_t": 35, "2d": [1, 2, 6, 9, 10, 54, 61, 75, 76, 78, 108], "2e6": 76, "2i": 79, "2k": 79, "2kcg": [14, 21, 38, 39, 63, 77, 83, 84, 91, 110], "2klinearsolv": 84, "2l": [63, 83, 109], "2m": 64, "2me": 64, "2n": [47, 79], "2t": 65, "2ta": 27, "2y": 79, "3": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 33, 34, 35, 38, 39, 43, 44, 47, 48, 55, 58, 62, 63, 65, 66, 70, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 88, 89, 91, 92, 96, 97, 98, 101, 103, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "30": [0, 8, 15, 21, 38, 39, 48, 55, 77, 78, 81, 83, 84, 85, 88, 106, 110], "300": 48, "3004844587339315": 51, "3005077941994067e": 77, "300696790397088e": 48, "30074832e": 76, "301": [15, 48], "3011386494338866e": 48, "30138177e": 76, "3017330592352966e": 21, "3017815971438115e": 39, "3019964774111864e": 84, "302": 48, "302264315279244e": 91, "3024": 2, "30269301e": 76, "302714596286466e": 91, "3027690999003073e": 48, "30288929e": 76, "302901864714575e": 48, "303": 48, "303008939522798e": 84, "304": 48, "30408705e": 76, "30491321882643e": 84, "3049853940133655e": 48, "305": [8, 48], "30505582847543e": 48, "3053572711634862": 63, "30571429e": 76, "306": 48, "30621953771071e": 48, "3063314760812088": 81, "3063771523894718": 103, "3064582455185055": 85, "3067389091504954e": 48, "307": 48, "3072059502377703e": 48, "307344765907323e": 48, "307914": 108, "308": 48, "30823297e": 76, "30844242e": 76, "308450122441681": 63, "30858827998629": 63, "30871063e": 76, "309": 48, "309714466727463e": 48, "30x15": 121, "31": [21, 38, 39, 46, 48, 77, 81, 83, 84, 85, 110, 118], "310": 48, "3100766606552662e": 84, "31053506e": 76, "310666361752348e": 84, "3106696194022617e": 39, "3106857175187873e": 84, "311": 48, "311366433940005e": 48, "3114977115398425": 81, "3117181936101093e": 48, "312": 48, "31221": 108, "3129272483674086e": 48, "313": 48, "3130323762459306": 103, "3130351788210097": 103, "3130352795102445": 103, "313035285298448": 103, "31340616e": 76, "3136911569268165e": 110, "314": 48, "31462394e": 76, "3146723828127433e": 48, "315": 48, "3151428740681202e": 48, "3151608293416554e": 48, "315483266142386e": 21, "316": 48, "31624": 55, "3165": 67, "31667701963457284": 63, "3168473544001734e": 91, "3169": 63, "317": 48, "317049926499982e": 48, "3173522661039894e": 48, "31746804233173e": 48, "318": 48, "3182030348089053": 63, "31830950064612856": 103, "318629762870249e": 38, "318747833945106e": 48, "318864756351698e": 21, "319": 48, "3190814707331234e": 48, "31923341e": 76, "319937819401064": 63, "31mout": 23, "32": [15, 21, 38, 39, 48, 70, 77, 81, 83, 84, 85, 103, 108, 110], "320": 48, "3206146501632598e": 77, "3209174368972213e": 48, "321": 48, "321603674500005e": 21, "3219928450516111": 63, "322": [48, 121], "32201279355107e": 48, "32202735e": 76, "322179773720817e": 48, "322459830883026e": 77, "32250970878595636": 84, "3229959045015813e": 48, "323": [48, 121], "323104671078037e": 91, "3231615399420306e": 77, "3231646014039378e": 48, "3232297657475674": 83, "323436571077358e": 39, "323767518440678e": 83, "324": [48, 121], "32474227e": 76, "324899": 108, "325": [48, 121], "3254161040503283e": 48, "32578507e": 76, "326": 48, "326212478264851e": 83, "326469974823354e": 39, "327": [48, 121], "328": 48, "328258117750278e": 48, "329": [48, 79], "3292323296073995e": 48, "32947036e": 76, "3294831066099182e": 83, "329876676381426e": 83, "33": [15, 21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "330": 48, "33003422e": 76, "330822151353702e": 21, "331": 48, "331199046890856e": 48, "3312170911824881e": 48, "332": 48, "33243573e": 76, "3329352056374313e": 77, "333": 48, "33316884e": 76, "3331948524618315e": 48, "33327795e": 76, "3334817867318555e": 48, "333629": 108, "334": 48, "335": 48, "3353005724913072e": 38, "3357651872399246e": 39, "336": 48, "336225652920634e": 21, "33643118654821e": 110, "337": [48, 79], "337032": 108, "3371286004187721e": 77, "337166663552109e": 48, "3375409442559155e": 48, "338": 48, "3387850598294901e": 21, "339": 48, "339109257479704e": 77, "3391246325802083": 70, "3393185901943043e": 48, "3399619828811302e": 85, "34": [15, 21, 38, 39, 48, 65, 76, 77, 81, 83, 84, 85, 105, 108, 110], "340": 48, "34032616e": 76, "34050016e": 76, "340930532910021e": 48, "341": 48, "341206724850267": 63, "341597055866288e": 48, "342": 48, "3420282227507415e": 48, "34223115e": 76, "3426066734105022": 24, "3427726344761957e": 83, "34281243904970204": 116, "343": 48, "343218821995261e": 48, "343368177647046e": 48, "343495008483151e": 39, "3438799364311088e": 48, "344": 48, "345": 48, "34503153e": 76, "345120350599629e": 48, "345286858268262e": 39, "34534586041024e": 115, "345465916709958e": 48, "34576114e": 76, "346": 48, "346108176246629e": 77, "34633": 103, "346737344250734e": 83, "34678613e": 76, "346953929262571e": 48, "347": 48, "34701629e": 76, "3473941636760532e": 48, "348": 48, "34808772e": 76, "34826800e": 76, "348315394185244e": 48, "348678469928673e": 84, "348913389301089e": 48, "349": [15, 48], "3491991701423895e": 77, "34920635e": 76, "34923238484178e": 84, "34927074006952e": 14, "34946": 55, "349635933929682e": 83, "3497617218728434e": 48, "35": [21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 88, 110], "350": 48, "3509906761220354e": 38, "351": [15, 48], "351116172274719e": 48, "3513876287934866e": 48, "352": [15, 48], "352057965747212e": 48, "352206186975885e": 84, "35262210e": 76, "353": 48, "3530": 103, "3538480472957442e": 77, "354": [15, 48], "354153128306859e": 38, "3542094017002498": 81, "3543762860934322e": 83, "3546130827285455e": 110, "35472110553465824": 70, "355": 48, "35572": 108, "3557276175693415e": 48, "355860847201666e": 48, "35591475658727e": 48, "356": 48, "3560043386452667": 63, "35640106452074e": 77, "35674978e": 76, "357": 48, "3578525479156935e": 48, "3579760806659418e": 48, "358": [15, 48], "35802220286281283": 70, "35820310983034076": 70, "35823429355522934": 70, "3582355218606291": 70, "35823552663766817": 70, "3582355266377316": 70, "359": 48, "36": [21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 110], "360": [15, 48], "360246232685851e": 48, "3602862850662532e": 48, "3606471893290851e": 38, "361": 48, "3616798896013715e": 48, "361744949075666e": 83, "362": 48, "36233754e": 76, "362346931360255e": 83, "3626009812903334e": 48, "3628551164479e": 48, "363": 48, "36321061217922673": 24, "363532192858452e": 77, "363564916457091e": 48, "36373064e": 76, "364": [15, 48], "3642450612653385": 51, "365": 48, "365027": 108, "3656976172150075e": 48, "3657345829248144e": 38, "366": 48, "366728196158377e": 48, "366915850720791": 85, "367": 48, "367143553464296e": 110, "3673484810309763e": 48, "367665665217973e": 48, "368": [15, 48], "3680394619761025e": 84, "368440535182689e": 48, "3685646891753328e": 48, "369": 48, "3690705057364501e": 84, "36909436e": 76, "3692871776920687e": 48, "3697248238186124e": 48, "3697656755438388e": 77, "37": [15, 21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "370": 48, "370873187419749": 63, "3708929217814924e": 48, "371": 48, "371148838672236e": 83, "3712": 15, "3715758714675145e": 38, "37161190298016": 83, "372": 48, "37266314e": 76, "373": 48, "373443200688844e": 39, "374": 48, "374538642017021e": 48, "374778284382607e": 48, "375": 48, "3759257345796876e": 38, "37594503814550667": 91, "376": [15, 48], "376294787821936e": 38, "3767551669114815e": 38, "376893472830978e": 48, "377": 48, "3777": 118, "378": [15, 48], "37816023e": 76, "3782375941630555e": 39, "37848562e": 76, "378785299480245e": 48, "378919613280206e": 115, "379": 48, "3792358662036097e": 48, "3792719216495e": 21, "37948302e": 76, "37989344758183696": 63, "38": [15, 21, 38, 39, 48, 66, 76, 77, 81, 83, 84, 85, 110, 118], "380": 48, "380094733222787e": 48, "381": 48, "3815828217680814e": 48, "381900873406225e": 48, "382": 48, "382220957458106e": 48, "383": 48, "3834677872461816e": 77, "3834878735036261e": 21, "383681884834191e": 48, "38371554e": 76, "384": 48, "3840348380620775e": 48, "3841438305254625e": 48, "384548734703711e": 77, "385": 48, "385554547969789e": 70, "386": 48, "38610419e": 76, "386817": 108, "387": 48, "3876295931521244e": 48, "38776040e": 76, "38792594e": 76, "38794309e": 76, "388": 48, "38805825978633e": 48, "38832225e": 76, "389": 48, "389007726660379e": 39, "389791809557051e": 48, "3899902597436885e": 48, "39": [10, 15, 21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 96, 110], "390": 48, "39022": 108, "390974803808237e": 21, "391": 48, "391198049839224e": 39, "392": 48, "3925445641690011e": 77, "392817659702821e": 83, "392841102594967e": 48, "393": 48, "3930402462835032e": 21, "39347185e": 76, "39376057770696e": 38, "3938574627111934e": 77, "394": 48, "39435204e": 76, "39436240052696975": 85, "39444956e": 76, "3945794772264324e": 48, "394595459241093e": 48, "39464": 22, "394917627264948e": 77, "395": 48, "395565888293071e": 48, "396": 48, "3960743990071075e": 48, "396161722131952e": 48, "39617577e": 76, "39634548e": 76, "3965604941420359e": 77, "39661832e": 76, "39682540e": 76, "397": 48, "3974281500533848e": 83, "398": [15, 48], "3980831180153135e": 84, "3984494259649153e": 48, "3986531644012435e": 48, "399": [47, 48], "3d": [1, 2, 6, 8, 10, 33, 34, 45, 46, 54, 61, 62, 75, 76, 77, 91, 108], "3ex": [68, 86], "4": [0, 5, 7, 8, 10, 11, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 32, 35, 38, 39, 40, 47, 48, 55, 58, 63, 65, 66, 67, 70, 75, 76, 77, 79, 80, 81, 83, 84, 85, 88, 91, 103, 104, 105, 106, 108, 110, 111, 114, 115, 119, 121], "40": [15, 21, 38, 39, 44, 48, 76, 77, 81, 83, 84, 85, 110], "400": [14, 15, 23, 48, 77, 84, 110, 121], "40022829e": 76, "40034946e": 76, "40049428e": 76, "400626986584763e": 48, "400764857435027e": 83, "4008290909238202e": 48, "400991083489937e": 83, "401": 48, "401402217647644e": 48, "40140778631308777": 85, "4014811948944164e": 38, "4017278282832695": 51, "401844236565597e": 83, "402": 48, "402546355528151e": 110, "402736540876462e": 48, "403": 48, "4032958225323066e": 77, "4034076037199845e": 21, "4036767277543443e": 85, "4038894412150484e": 14, "404": 48, "40405870e": 76, "404563477407528e": 39, "404629236349358e": 48, "405": 48, "405474055921332e": 48, "406": 48, "4060059508048765e": 83, "4069600824566074e": 48, "407": 48, "4074313271603042e": 38, "407631613365212e": 38, "408": [15, 48], "4086948298572175": 85, "40888996300808e": 48, "409": 48, "4093536347508964e": 48, "409720725027058e": 77, "41": [7, 15, 16, 18, 19, 21, 22, 38, 39, 48, 76, 77, 81, 83, 84, 85, 110], "410": 48, "41001347e": 76, "41019727e": 76, "410421337747203e": 48, "4105955877807516e": 77, "410634905428783e": 77, "41063837e": 76, "411": [15, 48], "4110760181683581e": 83, "411348617119802e": 38, "412": 48, "41280497087718e": 85, "413": 48, "413180318215851e": 85, "413218056147077e": 48, "413250840996505e": 48, "41373102e": 76, "4138806122733465e": 21, "414": 48, "4141783443506158e": 83, "415": 48, "4155225382861335e": 48, "416": 48, "4162233154336305e": 48, "416405828703569e": 48, "416415558992872e": 48, "416445795958666e": 48, "4164879047972151": 63, "417": 48, "41712": 22, "417356345279921e": 48, "417417797477789e": 77, "4179306418471782e": 48, "418": [15, 48], "41806640e": 76, "41888": 22, "419": 48, "41938614e": 76, "41983539667056e": 48, "42": [38, 39, 44, 48, 76, 77, 81, 83, 84, 85, 88, 108, 110], "420": [15, 48], "4201528772553445e": 83, "4203434724225752e": 48, "420344679804673e": 110, "420623649790215": 63, "42063694e": 76, "421": 48, "421373232085135e": 38, "422": [15, 48], "42272101e": 76, "423": [15, 48], "424": 48, "425": 48, "42512937e": 76, "42524628e": 76, "425660340878632e": 48, "426": 48, "4261": 67, "42626166e": 76, "426433": 11, "4265598564692116e": 48, "427": [15, 48], "42789858702521705": 83, "427937357377234e": 48, "428": 48, "428188197729902e": 48, "42879081e": 76, "428927912054942": 51, "4289867528748286e": 48, "429": 48, "42984684e": 76, "43": [15, 38, 39, 48, 76, 77, 81, 83, 84, 85, 110], "430": 48, "43052514e": 76, "43069641e": 76, "430869694970844e": 21, "431": [15, 48], "43101": 85, "431176412435023e": 48, "432": [15, 48], "4320678016306087e": 48, "4329112850224e": 48, "433": [15, 48], "433285162208582e": 85, "4333454712960677e": 83, "433399195958605e": 38, "433511536023269e": 48, "43369229e": 76, "434": [15, 48], "434050375655718e": 83, "43405785e": 76, "43407504e": 76, "434177851031543e": 39, "434918283921399e": 48, "435": [15, 48], "4350095376417224": 76, "4352415865436047e": 48, "43566373e": 76, "4357214966138604e": 48, "435860602860818e": 48, "4359368486597825e": 77, "436": 48, "436024997236365e": 38, "43606881e": 76, "4362081331413285e": 48, "437": 48, "4370135861158414e": 48, "437213226838117e": 81, "43752689e": 76, "43762385593524e": 83, "437683242447562e": 48, "438": [15, 48], "438531508912747e": 83, "439": 48, "439758843202849e": 21, "439845270680144e": 83, "44": [38, 39, 48, 76, 77, 83, 84, 85, 109, 110], "440": 48, "4401297012520998e": 48, "440247699033299e": 77, "440663639564687e": 21, "440755426206628e": 83, "441": 48, "4419425838448128": 85, "442": 48, "4428682911205534e": 48, "44287657201762e": 84, "44292495e": 76, "443": 48, "443176683566252e": 77, "443267856418581e": 48, "4438616067029353e": 83, "4439636276027798e": 84, "444": 48, "444456354642378e": 48, "44487840792874156": 22, "445": 48, "4456256298405192e": 106, "445632525789044e": 83, "445665566833718e": 48, "44579395591044385": 85, "446": 48, "4464326683285988e": 48, "446668911117099e": 77, "4468688322695497e": 48, "447": 48, "447257507807357e": 48, "448": 48, "4488933718100165e": 48, "448900761214395": 48, "449": 48, "449409064134838e": 48, "4499449472635008e": 115, "45": [0, 15, 38, 39, 48, 55, 76, 77, 83, 84, 85, 108, 110], "450": 48, "45079365e": 76, "450846077211376e": 38, "451": 48, "4510315297654936e": 48, "451059188051417e": 84, "451197927296262": 51, "451200504182154e": 84, "452": 48, "45204653576357e": 48, "452057792644223e": 48, "4521077551150774": 81, "452500262063084e": 110, "45261752e": 76, "453": 48, "453639114498798e": 80, "454": 48, "455": 48, "4552353471039174e": 48, "4553367053311626e": 83, "455420448937443e": 48, "45569001e": 76, "456": 48, "45688875e": 76, "457": 48, "45725397386741407": 63, "4577110083756291e": 48, "45788548e": 76, "457930495020287e": 48, "458": 48, "45816144e": 76, "4582431551015187e": 84, "458332278094533e": 48, "4586797121211655e": 48, "459": 48, "45942323241555755": 85, "46": [15, 38, 39, 48, 76, 77, 83, 84, 85, 110], "460": 48, "4601915644276485e": 48, "460411365408639e": 21, "460965197737418e": 21, "461": 48, "46135753154810366": 63, "46155": 11, "461725575996149e": 48, "461759935102699e": 48, "46177993e": 76, "462": 48, "4620885627225223e": 83, "46255587e": 76, "462716319985976e": 48, "46279346e": 76, "4628668141158896": 85, "463": 48, "4635327629266928": 83, "463813301050634e": 48, "4639916534842686e": 39, "464": 48, "46499": 108, "465": 48, "46500883e": 76, "4650109967698113e": 91, "46507838e": 76, "46524770740059607": 85, "4653646485897776e": 21, "465366340985294e": 84, "4659479241412505e": 48, "466": 48, "46644119e": 76, "4665823171826354e": 48, "466601479835905": 63, "467": 48, "46723283e": 76, "46730851e": 76, "467336896710443": 85, "46795278e": 76, "467966534081841e": 48, "468": 48, "46811512e": 76, "468997807675418e": 63, "46899780767589e": 63, "469": 48, "4690773015519533e": 48, "469188331783693e": 84, "4695364680635591": 85, "47": [15, 38, 39, 48, 76, 77, 83, 84, 85, 110], "470": 48, "47010385e": 76, "470109807340424e": 38, "4701460070543832e": 48, "4703083710549495e": 48, "4707544746314215e": 84, "4709697187344926": 51, "471": 48, "471088928931207e": 48, "4712663856765383e": 85, "471857535916869e": 21, "472": 48, "472437150273745e": 38, "472796970933967e": 38, "473": 48, "4731343120998787e": 39, "473282009251378e": 39, "4733739358435825e": 84, "4734783166392862": 85, "474": 48, "4743845285220006e": 84, "475": 48, "4755076200566074e": 48, "47570024488714774": 85, "476": 48, "47646191e": 76, "4764670997617955e": 84, "47657413e": 76, "4767521894186756e": 84, "47682560e": 76, "477": 48, "47711133461441e": 77, "477162405364844e": 48, "477239892603522e": 83, "477359372021876e": 77, "477929338164458e": 48, "478": 48, "4780177726832662e": 48, "47830128e": 76, "478486166301863e": 76, "478769914585975": 96, "4789746989912544e": 48, "47899861e": 76, "479": 48, "48": [15, 38, 39, 48, 76, 77, 83, 84, 85, 110], "480": 48, "4801364239010767e": 83, "48026033e": 76, "4805329017218728e": 48, "4807078864792983e": 77, "480955141420751e": 48, "481": 48, "4814120552195498": 63, "48148537704207556": 85, "48165354867617e": 38, "482": 48, "482969009887021e": 77, "483": 48, "48304492e": 76, "48344467893236e": 77, "48356595e": 76, "48361914822910884": 85, "483675900724828e": 48, "484": 48, "484232104216341e": 48, "484543077553714e": 83, "4846707043150605e": 38, "484893189401684e": 48, "485": 48, "485175829259252e": 48, "4856977656683026e": 83, "485707407058169e": 21, "486": 48, "487": 48, "4870126649549754e": 48, "488": [48, 108], "48834566896636394": 85, "489": [48, 80], "4892925374272817": 85, "4893134619348585e": 38, "489456390475471": 51, "4895424052178241e": 48, "4895926840424588e": 38, "489842622889638e": 39, "489875390949823e": 48, "49": [15, 38, 39, 48, 77, 83, 84, 85, 110], "490": 48, "4903482436642743e": 38, "490762189941222e": 110, "490829651617686e": 48, "490978217279541e": 48, "491": 48, "4914425821440545e": 21, "491900037077236e": 48, "492": 48, "492327674466352e": 83, "493": 48, "493285147546725e": 48, "493611611120727e": 21, "494": 48, "494419632350608e": 63, "494419632355397e": 63, "494706044701668e": 48, "4947798856049585": 85, "495": 48, "49537": 63, "49543876866909387": 85, "496": 48, "4960536837939773e": 48, "496062301292538e": 48, "4967019608058028": 106, "497": 48, "49706348e": 76, "497513684323499e": 48, "49754848815741554": 85, "4976441070032594e": 77, "498": 48, "498607421968768e": 48, "498665639019465e": 48, "499": 48, "499121459722843e": 38, "49944425e": 76, "49950661134314e": 39, "4996244938407006e": 21, "4999999999999974": 76, "499999999999999": 76, "5": [0, 2, 3, 4, 5, 6, 7, 8, 12, 14, 16, 18, 19, 21, 22, 33, 34, 35, 37, 38, 39, 43, 44, 47, 48, 55, 58, 62, 63, 70, 74, 76, 77, 78, 79, 80, 81, 83, 84, 85, 91, 92, 96, 97, 103, 108, 110, 112, 113, 114, 115, 116, 118, 119, 121], "50": [4, 15, 38, 39, 43, 47, 48, 76, 77, 78, 83, 84, 85, 108, 110, 121], "500": [22, 38, 39, 47, 48, 67, 69], "50000": 121, "500116704668778e": 38, "500146983254948e": 48, "5001615510129478": 85, "50016337e": 76, "500300016615044e": 48, "501": [48, 103], "5011576064000944e": 48, "501242952996463e": 48, "5017360038189035": 85, "502": 48, "5025194394231773": 85, "503": 48, "503552894229021e": 21, "50358631e": 76, "5036485771930268": 85, "5036598509313203e": 77, "503722035383636e": 48, "504": 48, "5042919233686429e": 83, "50442051e": 76, "504556658304414e": 48, "505": 48, "505165174500594e": 48, "505801129603892e": 48, "506": 48, "506101484240894e": 48, "5061468121661231": 85, "5061729904018873": 85, "5061795188774857e": 84, "506336456463105e": 39, "506645750744541e": 83, "50682327e": 76, "506908218354334e": 77, "507": 48, "507026363501187e": 77, "5072557909082451": 51, "5073268053984769": 85, "50759366e": 76, "5077276185552118e": 48, "508": 48, "50835": 121, "5087570130423114e": 38, "509": 48, "509437297093308e": 48, "51": [15, 38, 39, 48, 77, 83, 84, 85, 110], "510": 48, "510174891172396e": 83, "5102925997151507e": 48, "510630588292769e": 48, "5107434241281112": 85, "511": 48, "5116661890005876e": 77, "5116759654916236e": 77, "512": 48, "5120662430136316e": 48, "512365097809827e": 38, "512762255199913e": 83, "513": 48, "5130": 1, "513125186992284e": 48, "514": 48, "5142198151406168e": 39, "51484360347095e": 77, "5149410265326552e": 77, "514952075563536e": 38, "515": 48, "5155162937637426e": 48, "515875985552747e": 83, "516": 48, "516488531819619e": 48, "5169033203791969e": 48, "517": 48, "517507837203933e": 38, "5175925277867024": 85, "517814835797797": 85, "518": 48, "518172758996038e": 81, "518845029685": 103, "519": 48, "5194838971425951e": 48, "519641137784833e": 77, "519795778795417e": 48, "519927604494129e": 83, "52": [15, 38, 39, 48, 77, 83, 84, 85, 110], "520": 48, "52020762e": 76, "52031126e": 76, "5203140321780024": 85, "5205257369901945e": 85, "5205385000271846": 85, "521": 48, "521850746831488e": 85, "521910": [], "522": 48, "5220696259625265e": 48, "5221995110452216e": 83, "5225189901845548": 39, "5225660928343492": 63, "523": 48, "523134156230471e": 48, "523391567900036e": 85, "524": 48, "52444753e": 76, "5246072026789994": 51, "5248845689998295e": 48, "525": 48, "525126310870192": 85, "52514963956198e": 77, "525322494411694e": 48, "5253742720462283": 85, "526": 48, "52636441e": 76, "527": 48, "5274388649262574e": 48, "52761847e": 76, "528": 48, "5284325663857127e": 38, "5287311423528937e": 48, "528784983278807e": 84, "529": 48, "52915266e": 76, "529219410275793e": 83, "5293620323637416e": 110, "53": [15, 38, 39, 48, 77, 83, 84, 85, 110], "530": 48, "5308251803984455e": 48, "531": 48, "531331856542912e": 48, "53171058e": 76, "532": 48, "532582373313546e": 48, "533": 48, "5330312963314107e": 83, "533439783514639e": 48, "5336307237815367e": 38, "533828877556726e": 39, "534": 48, "534231389431823e": 48, "534722304045493e": 48, "535": 48, "53509125215299e": 21, "535130653762732e": 48, "5355291443173377": 85, "5358648222961435": 85, "536": 48, "537": 48, "537520206911699e": 38, "538": 48, "5380346646190164e": 48, "5387047082349326e": 14, "539": 48, "539446673074001e": 48, "5394591497527506e": 48, "53986083e": 76, "54": [15, 38, 39, 48, 77, 83, 84, 85, 110], "540": 48, "5401318134887053": 85, "540217032071367e": 21, "5403099220737555e": 48, "5404009954080077e": 39, "5406511938359275e": 48, "541": 48, "541248535042474": 66, "5414706478167903": 109, "5415208146951107": 51, "5419012858384775e": 38, "542": 48, "542308870311771e": 110, "54285092e": 76, "54285512e": 76, "543": 48, "5432729629011974e": 48, "543414380099021e": 38, "544": 48, "544138507039576e": 48, "54491460880751e": 21, "545": 48, "545348237201027e": 48, "545934144326212": 70, "546": 48, "5466647999174853e": 14, "547": 48, "548": 48, "54836020e": 76, "5483623051359466e": 77, "548906053040536e": 48, "549": 48, "54910123e": 76, "5491577368906567": 85, "54938804180488e": 48, "5494201485235853e": 77, "5497371221971641": 63, "55": [15, 38, 39, 48, 77, 83, 84, 85, 110], "550": 48, "5500272589058757e": 48, "550103983705584e": 77, "55036412e": 76, "550386658345834e": 48, "5505589020810506e": 48, "5505614653778326e": 83, "551": 48, "551059202512336e": 48, "5515751696702515e": 39, "551679871817e": 21, "552": 48, "5524220444695906e": 48, "5526642024859626e": 48, "553": 48, "5530923154043247e": 39, "5533557874560733": 70, "553474254749655e": 84, "554": 48, "554507451136723e": 110, "554898004471734e": 48, "5549435215560785e": 48, "555": 48, "5558353127250373e": 84, "556": 48, "55632805e": 76, "556456083826408": 81, "556900320854897e": 21, "557": 48, "557134991353894e": 39, "5578609944298918": 51, "558": 48, "5582836045941635e": 84, "559": 48, "559079216644538": 106, "55921383e": 76, "5594603828450653e": 48, "55951239e": 76, "5598812394438284": 70, "56": [15, 38, 39, 48, 76, 77, 83, 84, 85, 108, 110], "560": 48, "560166670756158e": 48, "5602257500879521": 70, "560256843307439": 70, "5602573408039925": 70, "5602573409856751": 70, "56080516e": 76, "56081511e": 76, "5608872971611618e": 83, "560991376723551e": 48, "561": 48, "562": 48, "5621133631968823e": 48, "562143501253274e": 83, "562231195951519e": 48, "562283477553": 103, "56260820e": 76, "562726802531528e": 81, "5628463859901822": 51, "5629754213375459": 85, "563": [48, 118], "563264771416978e": 21, "5638568499710354e": 21, "564": 48, "564004473485101e": 110, "56415614e": 76, "56429751911025e": 14, "564771675466398e": 48, "56493273089252e": 48, "56496750e": 76, "565": 48, "565176484800696e": 38, "566": 48, "56608104233486e": 48, "5661478981187622e": 21, "56649872e": 76, "567": 48, "567307643680844e": 48, "5676517902103517e": 77, "567924758563753e": 48, "568": 48, "568405424836091e": 48, "568521269026107e": 80, "569": 48, "56906916257152e": 39, "56969874e": 76, "57": [15, 38, 39, 48, 77, 83, 84, 85, 110], "570": 48, "57030863e": 76, "5704465018664383e": 48, "571": 48, "5716200429890295e": 48, "571878152242305": 81, "572": 48, "573": 48, "5734139487420609": 51, "573761126328625e": 77, "573997293731076e": 48, "574": 48, "5742754062254667e": 81, "5742937204616135e": 48, "57468": 85, "574788922438835e": 85, "5748227292996332e": 48, "575": 48, "575167343788693e": 48, "575570123894986e": 38, "575855": [], "576": 48, "5765781335362466e": 48, "576679382931486e": 77, "5767269529447143e": 39, "577": 48, "57756990e": 76, "57787847517935e": 48, "578": 48, "578665958559203": 81, "579": 48, "5794": 108, "579604898714938e": 48, "58": [15, 38, 39, 48, 77, 83, 84, 85, 110], "580": 48, "58003557e": 76, "5802375801172456e": 38, "5808008395922967e": 77, "58097206e": 76, "581": [48, 65, 105], "5811845708510086e": 48, "582": 48, "58213525e": 76, "582460201324559e": 38, "582662659968393e": 48, "583": 48, "5830712431211527e": 21, "583408203975017e": 84, "58369941e": 76, "583722552709536e": 48, "5838745132739276e": 48, "58390741e": 76, "584": 48, "585": 48, "585181086951244e": 48, "5856683186987715e": 84, "586": 48, "5860896744937203e": 48, "5865698784456028e": 48, "586717823012357e": 39, "587": 48, "5871503487735405e": 39, "58730159e": 76, "587397638255004e": 48, "58768986e": 76, "587918844820352e": 48, "587947333438726": 51, "588": 48, "5887660933918107e": 48, "589": 48, "5892049430232538": 63, "59": [15, 38, 39, 48, 77, 83, 84, 85, 110], "590": 48, "5904892246529526e": 48, "591": 48, "5917993944175503e": 77, "591872379395178e": 84, "592": 48, "592476359799728e": 110, "59289558e": 76, "593": 48, "593513625303192e": 48, "5937316187541024": 85, "594": 48, "5941864085865704e": 85, "595": 48, "595331929789697e": 48, "59571120e": 76, "5957238886105585e": 48, "596": 48, "596224550241645e": 48, "596431939936081e": 48, "597": 48, "5975760598785288e": 77, "598": 48, "598344404734277e": 48, "598740783185647e": 85, "598901": 108, "599": 48, "5e4": 74, "5em": 67, "6": [6, 14, 20, 21, 23, 34, 35, 38, 39, 40, 44, 47, 48, 58, 63, 70, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 88, 91, 103, 108, 110, 111, 112, 115, 118, 119], "60": [15, 38, 39, 48, 77, 80, 83, 84, 85, 110], "600": 48, "600238226564242e": 84, "601": 48, "6012589050513686": 51, "601489072773148e": 85, "601828051333945e": 48, "602": [48, 51], "60229352735577e": 48, "602422442088333e": 48, "60282600161908": 84, "603": 48, "6032113977710588e": 48, "603559520622855e": 48, "60372397e": 76, "603811278315021e": 83, "604": 48, "60426442e": 76, "6044659570055255e": 48, "605": 48, "6056347314808085e": 83, "605938808689276e": 48, "606": 48, "6060002408897057e": 83, "60625432046131e": 48, "6063431809725343e": 77, "606634889215221e": 84, "607": 48, "608": 48, "608253257484617e": 48, "608585554903282e": 38, "608662509689858e": 63, "608662509690732e": 63, "608671743947409e": 48, "6087661839121814e": 83, "609": 48, "609184832867373e": 48, "61": [15, 38, 39, 48, 63, 76, 77, 83, 84, 85, 110], "610": 48, "6103899203810424e": 77, "6105980661982686e": 48, "6106915979525028e": 48, "611": 48, "6118242125742767e": 48, "612": 48, "612651755831167e": 115, "613": [48, 65, 105], "613209024712011": 51, "61373488e": 76, "614": 48, "614005175441214e": 48, "615": 48, "6157121995544838e": 48, "6158399286683607e": 48, "616": 48, "617": 48, "617057007632026e": 48, "618": [48, 84], "6181368587478409": 81, "6184608892926782e": 48, "6186157638062868e": 77, "619": 48, "619150701221642e": 84, "62": [15, 38, 39, 48, 76, 77, 83, 84, 85, 110], "620": 48, "6209503608706": 106, "621": 48, "621197626469506e": 48, "621371600445931e": 48, "621944158161234e": 85, "622": 48, "6221154116416602e": 48, "623": 48, "623330289029238e": 77, "623457969303072e": 21, "62349341e": 76, "62363235e": 76, "623692451270183e": 48, "6237346368041081": 51, "62395745e": 76, "624": 48, "625": 48, "625231949488012e": 38, "625250580230147e": 21, "625257042884367e": 48, "625532145380201e": 48, "6255450647095954e": 48, "62576805756395e": 77, "625844366509997e": 38, "626": 48, "62624400e": 76, "626339847741722e": 39, "6264019576717673e": 48, "626576251343839e": 48, "627": 48, "627318460818314e": 39, "628": 48, "6283104727068234e": 48, "629": 48, "6298437650060975e": 77, "63": [38, 39, 48, 77, 83, 84, 85, 110], "630": 48, "63003331e": 76, "63058946e": 76, "631": 48, "6310815021997474e": 48, "631558620476784e": 48, "631718022580567e": 48, "632": 48, "632571239832806e": 48, "632783294297951e": 76, "632846787976177": 51, "633": 48, "633510002398147e": 83, "634": 48, "63404990e": 76, "63421433e": 76, "63463979e": 76, "6349639615833913e": 63, "6349639615837404e": 63, "63499185e": 76, "635": 48, "6351941581130006e": 110, "6357941765358154e": 77, "635817941556107e": 83, "636": 48, "63605128e": 76, "63609902e": 76, "6361360075956317e": 84, "6365287726637194e": 48, "636716541085751e": 84, "637": 48, "638": 48, "638073035428225e": 48, "6382200125324833e": 48, "638759920077451e": 48, "639": 48, "6397979567617955": 81, "63998469e": 76, "64": [38, 39, 48, 77, 83, 84, 85, 103, 108, 110], "640": 48, "6406169867162068": 51, "641": 48, "6410069935234305e": 48, "64119171e": 76, "6417376943526984e": 83, "641909355189532e": 48, "642": 48, "642007949458749e": 38, "642173216119092e": 83, "642257486881171": 85, "6425609626973794e": 48, "643": 48, "644": 48, "6441": 82, "6442510876374e": 14, "64428432e": 76, "645": 48, "645976545818384e": 48, "646": 48, "64691905e": 76, "647": 48, "6470601443285914e": 48, "6471585575973842": 51, "647555890574575e": 85, "648": 48, "6480462116": 103, "6481898622687437e": 48, "648411830187808e": 91, "648429": 11, "648471378408556e": 48, "6487932956182e": 91, "64890769583779e": 77, "64893941e": 76, "649": 48, "6491627752406698e": 38, "649248757945639e": 21, "6495382905210624e": 63, "6495382905217427e": 63, "64960": 120, "649936644773816e": 83, "64996884e": 76, "65": [38, 39, 48, 77, 83, 84, 85, 108, 110], "650": 48, "65063196e": 76, "65080144e": 76, "6508451302837636": 85, "6508768563981184e": 81, "650993786938887e": 48, "651": [34, 48], "6518312303347455e": 48, "65192015e": 76, "652": 48, "65287341e": 76, "653": [48, 103], "65359773e": 76, "6538034423658578e": 48, "653829414967276e": 110, "654": 48, "6542": 55, "654555424792147e": 48, "6546782690498294e": 48, "655": 48, "655442290755333e": 83, "655581574754693e": 48, "656": 48, "65603819e": 76, "657": 48, "658": 48, "6581012527089735e": 38, "6586430081313818e": 48, "659": 48, "65974500215008e": 48, "65982942e": 76, "66": [38, 39, 48, 77, 83, 84, 85, 110], "660": 48, "660895805916253e": 48, "661": 48, "661041373862165e": 48, "6612253938470774e": 38, "6612431188635448": 85, "6614054364313376e": 85, "662": 48, "662800535624399e": 76, "6628845003728513e": 39, "663": 48, "6631659914077622e": 48, "66339889e": 76, "663515736510115e": 83, "6638671781442963e": 48, "664": 48, "665": 48, "665234128197446e": 48, "66524538e": 76, "665576913766078e": 48, "665986": 11, "666": 48, "66666667e": 76, "667": 48, "66713703e": 76, "667577486274225e": 48, "6676902403680316e": 48, "668": 48, "66811773135793e": 81, "6687818393381375e": 39, "669": 48, "669498526513673e": 48, "67": [38, 39, 48, 67, 77, 83, 84, 85, 110], "670": 48, "6702589191564442e": 83, "671": 48, "6711501048505505e": 48, "67167662e": 76, "67198008e": 76, "672": [34, 48], "67211217e": 76, "6729157958379345e": 77, "673": 48, "6739930930784257e": 48, "674": 48, "6741148658528377e": 38, "6741498611348335e": 77, "67482292575374e": 48, "675": 48, "675150622392839e": 84, "6757907507089311": 47, "676": [15, 48], "676841913354895e": 48, "676916499495534e": 48, "6769198434232496e": 48, "677": [48, 50], "677150772435917e": 83, "67777778e": 76, "678": 48, "679": 48, "67903582e": 76, "6792916466594385": 85, "679373377825878e": 48, "67975607e": 76, "68": [38, 39, 48, 77, 83, 84, 85, 106, 110], "680": 48, "680141277151451e": 48, "68033669500627e": 48, "680627093063401e": 83, "68063272e": 76, "680695703563715e": 48, "681": 48, "681809447713418e": 84, "681996832028874e": 21, "682": 48, "68249759e": 76, "683": 48, "6831414237428172e": 38, "6831750753998855e": 48, "68347587e": 76, "683545153190265e": 21, "6838647035290175e": 21, "6839353293700188e": 48, "684": 48, "68410527e": 76, "684111020480524e": 48, "684180647878445e": 48, "684247238336984e": 48, "685": 48, "6852922515902963e": 39, "6858896178966197e": 77, "686": 48, "6868": 108, "687": 48, "6870458184868213e": 48, "68751578e": 76, "6878351817818138e": 77, "68795971e": 76, "687990675892528e": 39, "688": 48, "688103183630812e": 48, "688325338782177e": 48, "6886868114230752e": 39, "689": 48, "68927196e": 76, "689323984540329e": 77, "689450198832634e": 48, "69": [38, 39, 48, 77, 83, 84, 85, 110], "690": 48, "690000195677042e": 48, "6904904901851865e": 39, "6906601592346228e": 84, "6907017058419193": 83, "690711212048912e": 21, "690793878626996e": 38, "691": 48, "6916369308059416": 63, "6917198260952784": 106, "6917198310276222": 106, "69173": 106, "691753672693137": 48, "692": [15, 48], "6924136155685763": 84, "69243": 108, "692753635219975e": 39, "692782966558739": 109, "693": 48, "694": 48, "6944302008399361e": 48, "695": 48, "6956794549358718e": 48, "696": 48, "696078939451168e": 48, "69680035e": 76, "6969039222829026e": 84, "697": 48, "6972458570229063e": 77, "6973127929579093e": 48, "697808726680315e": 39, "697838018857469e": 38, "69786564e": 76, "698": 48, "699": 48, "699126451692341e": 48, "69926933e": 76, "699296772001884e": 48, "699680596318089e": 48, "699684507733734e": 77, "7": [14, 21, 35, 38, 39, 46, 48, 63, 70, 76, 77, 78, 80, 81, 83, 84, 85, 89, 91, 103, 108, 110, 115, 118, 119, 120], "70": [38, 39, 48, 77, 83, 84, 85, 110], "700": 48, "7002013246680412e": 48, "7002654479423812e": 48, "700585818353643e": 48, "70080819e": 76, "701": 48, "70113937456317e": 48, "7017039819867693e": 83, "702": [15, 48], "7024528929213845e": 63, "7024528929217936e": 63, "7029432662799943e": 21, "703": 48, "703743410155175": 84, "7037791513458759": 81, "70398951e": 76, "704": [15, 48], "70420964e": 76, "705": 48, "705147728680346e": 83, "705590068837061e": 48, "706": 48, "706142233350342e": 48, "707": 48, "70764226689353e": 48, "708": 48, "70825255e": 76, "709": 48, "70940954e": 76, "71": [7, 38, 39, 48, 77, 83, 84, 85, 110], "710": 48, "71013089e": 76, "710547350811855e": 48, "71076796e": 76, "711": 48, "7110717192708665e": 77, "7111": 108, "7112644172491408e": 39, "7113164141127007e": 84, "71189422260038e": 48, "712": 48, "712084773076043e": 48, "712122195637452e": 21, "713": 48, "7131383862528984e": 110, "714": 48, "714008685346325e": 39, "71424066226727e": 48, "71428571e": 76, "714443543707031e": 85, "715": 48, "7152486742642178e": 85, "715617372711863e": 21, "716": 48, "7165": 120, "7166280513160016": 85, "7166339448727095e": 48, "7166986112072625e": 48, "717": 48, "7171682211344884e": 77, "717233033971179e": 48, "71729962e": 76, "717550736731428e": 48, "7175647832785993": 85, "7178089277347426e": 84, "717901707233367e": 84, "718": 48, "7180346107522614e": 48, "7186175188828105e": 83, "7187250629149218": 110, "719": 48, "71933788e": 76, "71934292567164e": 21, "72": [38, 39, 48, 77, 83, 84, 85, 110], "720": 48, "7200410895389296e": 21, "720277923249615e": 38, "7209573497964506e": 48, "720980706824187e": 84, "721": [15, 48], "7213242109925206e": 48, "7213460179708344e": 38, "722": [15, 48], "72201759e": 76, "72225926431386e": 48, "7228777450440256e": 77, "723": 48, "72312767e": 76, "723697328914656e": 88, "72386981e": 76, "723886147227309e": 48, "724": [15, 48], "724302716978304e": 77, "724587470723463e": 76, "724628290895526e": 48, "7246579670980434e": 48, "724988999951798": 108, "725": 48, "72535447e": 76, "7258698239108366e": 21, "726": 48, "727": 48, "727365380912911e": 48, "72752866e": 76, "728": 48, "728141472198836e": 48, "7284332019465414e": 21, "7287649730899981": 85, "728766308288062e": 77, "728979506876731e": 48, "729": 48, "72977177e": 76, "73": [38, 39, 48, 67, 77, 83, 84, 85, 110], "730": 48, "731": [48, 108], "7312763680539798e": 77, "73128488e": 76, "7314307180782044e": 48, "732": [15, 48], "732708184104283e": 77, "7327143565346456e": 48, "733": [15, 48], "733231842002208e": 48, "734": [15, 48], "734367123883148e": 48, "7343762731976364e": 48, "7344838517945644e": 48, "734723475976807e": 76, "734731997623074e": 110, "73491052e": 76, "735": 48, "736": 48, "737": 48, "737097636041496e": 48, "73742": 108, "7378817215805044e": 48, "738": 48, "73825402e": 76, "738830450395595e": 39, "738961003585408e": 77, "739": 48, "739208989947937": 96, "739227318013283e": 21, "73992768129038e": 84, "74": [38, 39, 48, 77, 83, 84, 85, 110], "740": 48, "740471030598783e": 84, "740608113448077e": 83, "7408466320005056e": 48, "741": 48, "7419678304298498e": 48, "742": 48, "742066724764176e": 84, "74252409e": 76, "7427297251387739e": 81, "743": 48, "743046389129484e": 48, "743140248349613e": 77, "744": 48, "744154957255584e": 48, "744737564650945e": 91, "7449312891786651e": 48, "745": 48, "745338379740247e": 48, "746": 48, "746744809618211e": 39, "746988905891274e": 77, "747": 48, "747900914242783e": 48, "748": 48, "749": 48, "749865690221446e": 48, "75": [4, 5, 38, 39, 48, 77, 83, 84, 85, 110], "750": 48, "75053646e": 76, "7508301157973897e": 48, "751": 48, "751962927012061e": 48, "75197995777629e": 39, "752": 48, "7521849481247322e": 38, "75245664e": 76, "752780375295666e": 39, "753": 48, "75317153e": 76, "753411194334228e": 48, "754": 48, "7545438069154676e": 48, "75456096e": 76, "75494085e": 76, "755": 48, "755550557607619e": 48, "756": 48, "756363093945918e": 39, "757": 48, "75704791e": 76, "7572110993411804e": 48, "7572606786524734": 63, "7576109259346328e": 83, "758": 48, "75810": 22, "7584322804740845e": 48, "7584634540051176e": 39, "7585": [15, 23], "7585371463963765e": 48, "759": 48, "75906396e": 76, "7592339454418208e": 48, "759305": 108, "7594147515889492e": 84, "759467113806551e": 48, "759621420205265e": 48, "759857607170855e": 21, "76": [38, 39, 48, 77, 83, 84, 85, 110], "760": 48, "761": 48, "761187835174138e": 48, "761498064357539e": 48, "76156412171864e": 84, "762": 48, "762472029308668": 106, "7626795389173299": 85, "763": 48, "763102225221426e": 91, "763145703838146e": 84, "7636030771172796e": 48, "764": 48, "76404167e": 76, "7642322774630083e": 38, "7644792731733917e": 85, "76492876e": 76, "765": 48, "7651544666507633": 63, "76577828e": 76, "766": 48, "766234453974245e": 48, "76652503e": 76, "767": 48, "76707234881702e": 14, "767206937178794": 85, "767520055746677e": 85, "768": 48, "7683333378414936e": 39, "7684222523134668e": 38, "76870051e": 76, "769": 48, "769239199303403e": 48, "769265216173688e": 48, "769516408736931e": 21, "769929839694957e": 48, "76999840e": 76, "77": [3, 34, 38, 39, 48, 77, 83, 84, 85, 110], "770": 48, "77092485563709e": 21, "771": 48, "7710075176906277e": 83, "7713073556456228e": 48, "772": 48, "7722502220772948e": 48, "77256": 108, "772794989088839e": 38, "773": 48, "7731": 15, "77344724236405e": 84, "77372": 83, "774": 48, "774391311803948e": 48, "775": 48, "775007175940237e": 77, "77504466331908e": 85, "7753694641455832": 81, "775540546546009e": 38, "776": 48, "7760220119243286e": 48, "776226828362013e": 48, "7766245457196026e": 39, "777": 48, "777798497624973e": 77, "778": 48, "77881017e": 76, "7789183584086107e": 21, "779": 48, "779080482626883e": 48, "7794": 108, "779751500223053e": 77, "77984470e": 76, "78": [38, 39, 48, 77, 83, 84, 85, 110], "780": 48, "780006401287035e": 48, "780076662289142e": 48, "7804": 103, "7805639122786753e": 38, "781": [15, 48], "781165681543078e": 81, "78117344e": 76, "78149891668565e": 48, "78173444e": 76, "782": [15, 48], "782339415582198e": 48, "782735872932991e": 39, "783": 48, "7830345946531105e": 48, "784": 48, "784093837858262e": 14, "7843641797264262e": 83, "784415947420096e": 84, "785": [15, 48], "785161810542102e": 38, "786": [15, 48], "7865073893161035e": 48, "786976191748678e": 38, "787": [15, 48], "787694209726307e": 48, "787969": 63, "788": [15, 48], "7881729236898696e": 48, "788196322974487e": 85, "78832306e": 76, "78842972e": 76, "789": 48, "79": [38, 39, 48, 76, 77, 83, 84, 85, 110, 121], "790": [15, 48], "790269700494604e": 48, "790475207480968e": 48, "7907346494362": 103, "7907524159514695e": 38, "7908391227508154e": 48, "7909557666719956e": 91, "791": 48, "791426428112145e": 84, "7917614043413373": 85, "792": [15, 48], "79291620834115e": 48, "79295905709389e": 48, "793": 48, "793885733041347e": 48, "794": [15, 48], "794336920486616e": 38, "7946": 108, "7948968585386554e": 38, "795": [15, 48], "7956752990680338": 48, "796": [15, 48], "796780788441403e": 84, "7969387311232552e": 48, "797": 48, "79711823e": 76, "7976717886559725e": 48, "7976737044634345e": 110, "7978921512925343e": 39, "798": 48, "7986443153856565e": 48, "799": 48, "79905007837357e": 21, "799562528394244e": 48, "8": [0, 6, 14, 17, 18, 20, 21, 23, 35, 38, 39, 40, 43, 47, 48, 49, 50, 51, 55, 63, 74, 76, 77, 80, 81, 83, 84, 85, 91, 103, 108, 110, 111, 115, 118, 120, 121], "80": [12, 38, 39, 48, 61, 76, 77, 83, 84, 85, 110, 121], "800": 48, "800144210675474e": 21, "80040475e": 76, "801": 48, "801171468526849e": 110, "801601436853941e": 77, "80161548e": 76, "802": 48, "803": 48, "803081364117639e": 48, "8031745444511146e": 21, "803815226253984e": 80, "803880010352867e": 39, "804": 48, "804375697464095e": 48, "804802927823209e": 48, "805": 48, "805223866626383e": 48, "806": 48, "806223719927285e": 39, "807": 48, "807312175357524e": 38, "80767289e": 76, "807873306659926e": 48, "8079": 108, "808": 48, "808004503566659e": 39, "808420681895741e": 48, "808562920476457e": 110, "809": 48, "809551240211115e": 48, "809635276715441e": 39, "8097411151552746e": 85, "809913282837525e": 48, "8099867233648296e": 77, "81": [38, 39, 48, 77, 83, 84, 85, 88, 110, 118, 121], "810": 48, "81073187e": 76, "81090441e": 76, "811": 48, "811130133982537e": 48, "811443703268355e": 48, "81158": 55, "812": 48, "813": 48, "81353155e": 76, "8136342454941936e": 77, "8137155683617994": 63, "8137886568143383e": 21, "814": 48, "814250201672253e": 48, "814352226978437": 48, "814693618811081e": 48, "815": 48, "815059836395663e": 84, "815400446992752e": 77, "815523139527669": 63, "8157865664294941e": 48, "81598785e": 76, "816": 48, "816032266267512e": 48, "816107452862656": 63, "8163916471489756e": 76, "81662709e": 76, "817": [38, 48, 63], "818": 48, "81873156e": 76, "818875607297657e": 48, "819": 48, "819016339413737e": 48, "819628984093517e": 48, "82": [38, 39, 48, 77, 83, 84, 85, 110, 121], "820": 48, "820233339776046e": 83, "820313116279269e": 48, "821": 48, "821704532877173e": 48, "8219274917452472e": 115, "8219711666983736e": 48, "822": 48, "822718134465209e": 48, "82286681e": 76, "823": 48, "824": 48, "8241752321672265e": 48, "82433803e": 76, "825": 48, "825107658370613e": 84, "825268149129328": 63, "826": 48, "826066219922423e": 22, "827": 48, "8270138008684686e": 48, "82734566e": 76, "8274090136404232e": 110, "82746652e": 76, "82748412619732e": 38, "8277013571751647e": 83, "827828516101396e": 48, "828": 48, "8287315681024927": 76, "829": 48, "829576433200149e": 83, "8299448851004696e": 48, "83": [38, 39, 48, 64, 76, 77, 83, 84, 85, 110, 121], "830": 48, "831": [15, 48], "831487175498001e": 21, "831823159500368e": 48, "832": [15, 48], "83208976e": 76, "8322959488698175e": 39, "83273533446297e": 48, "8329546473066384e": 83, "833": 48, "833058036442482e": 48, "833505": 55, "83371241e": 76, "834": 48, "83425690e": 76, "83432663e": 76, "8347": 55, "8347328893352102e": 21, "834859801621774e": 39, "835": 48, "8350121151719e": 48, "835720635727922e": 48, "836": 48, "8366450976608554e": 48, "837": 48, "837272655973157e": 39, "83757108e": 76, "837711441740575e": 80, "838": 48, "838580733733294e": 83, "838939606568788e": 77, "839": 48, "83915107e": 76, "839255645128162e": 48, "839699790942513e": 48, "839719533898004e": 39, "8399139114264356e": 84, "8399291261764208": 81, "84": [38, 39, 48, 76, 77, 83, 84, 85, 110, 121], "840": 48, "8402817281136264e": 83, "84032156624639e": 110, "841": [15, 48], "8410815328296521e": 48, "841635350994417e": 110, "842": [15, 48], "8420679358690194e": 81, "84206913e": 76, "84225488e": 76, "842446272464718e": 83, "842549500246563e": 38, "843": [15, 48], "843210063311627e": 39, "843722051876385e": 38, "84372631e": 76, "84374591e": 76, "843765347749792e": 48, "844": 48, "8442136050473178e": 48, "844363215072037e": 84, "845": 48, "845707984343893e": 38, "8457974486872674e": 48, "846": 48, "847": [15, 48], "8470072131627334e": 39, "8473523199457097e": 48, "848": 48, "848906170432547e": 39, "848960189067868e": 48, "849": [15, 48], "8490570129782678e": 48, "849634445174112e": 48, "8499": 108, "85": [38, 39, 48, 64, 67, 77, 83, 84, 85, 110], "850": 48, "851": [15, 48], "851545498892002e": 77, "852": [15, 48], "852581180555624e": 48, "852603540368492e": 48, "8529520194379847e": 77, "853": 48, "8531733433027572e": 84, "853903943783089e": 48, "854": 48, "854864477206059e": 39, "855": [15, 48], "8554370838921893e": 48, "856": [15, 48], "856002200721243e": 21, "856060475024522e": 48, "857": [15, 48], "8570338555821037e": 77, "85716531130414": 109, "857209320503717e": 48, "85722573273506e": 76, "85774805e": 76, "858": 48, "858427395202262e": 39, "85855089e": 76, "8585936041936275e": 48, "858810500619308e": 48, "859": 48, "859291633883496e": 39, "8595864561205e": 48, "86": [38, 39, 48, 77, 83, 84, 85, 88, 110, 121], "860": [15, 48], "8600961405495873e": 38, "8601107411343858": 38, "860373752689044e": 21, "861": 48, "86183": 108, "862": 48, "862620510921282e": 48, "863": 48, "86300492e": 76, "86388258e": 76, "863948847061986e": 77, "864": 48, "864238910083353e": 48, "86452589e": 76, "865": 48, "8652485005035088e": 85, "86533680e": 76, "865492736595519e": 48, "866": 48, "86639575404848e": 48, "8667288714541835e": 48, "8667683822022966e": 77, "867": 48, "8670169881917116e": 38, "86702751e": 76, "8675": 103, "868": [15, 48], "868009654151403e": 38, "8682450158839147e": 83, "868479538952061e": 77, "869": [15, 48], "86914888e": 76, "86919182473513e": 48, "869305704937035e": 48, "869604494897487": 96, "869604494897613": 96, "8699045759210372e": 48, "87": [15, 38, 39, 48, 77, 83, 84, 85, 88, 110, 121], "870": [15, 48], "871": 48, "8712721072740293e": 48, "8718180412834895e": 83, "871898147569218e": 81, "872": 48, "873": [15, 48], "8730870487562395e": 48, "873362458040147e": 83, "8738829146687e": 48, "874": [15, 48], "874839108748753e": 85, "875": 48, "87581228e": 76, "875922769123227e": 77, "875970509314675e": 48, "876": 48, "87607688173": 103, "8761613319769555e": 48, "877": 48, "878": 48, "8783881773434057e": 77, "878469863474089e": 48, "878772652942522e": 48, "879": 48, "8791455960178543e": 83, "879557505654338e": 39, "88": [38, 39, 48, 77, 83, 84, 85, 110, 121], "880": 48, "8800": 83, "881": 48, "8810774163262462e": 38, "881284405121477e": 48, "88135271e": 76, "8813631598073865e": 84, "881460064130038e": 38, "882": 48, "882693116313131e": 48, "8828617717300159e": 21, "883": [15, 48], "883349334841279e": 77, "8834840815121618e": 77, "883740543600398e": 77, "884": [15, 48], "8842911389966815e": 38, "8844848973340027e": 48, "885": 48, "885234043123831e": 48, "886": 48, "886127569791755e": 48, "88629681e": 76, "886769225627127e": 48, "887": 48, "88721028e": 76, "88735676e": 76, "888": [15, 48], "8887460381335765e": 48, "889": [15, 48], "88970269e": 76, "88978359878323e": 38, "89": [15, 38, 39, 48, 67, 77, 83, 84, 85, 110], "890": 48, "891": 48, "89120579e": 76, "892": [15, 48], "8920270278170713e": 39, "8927334941616805e": 48, "892738714831724e": 48, "89298369e": 76, "893": [15, 48], "8932471540524113e": 84, "8936006388080886e": 84, "89366049152692e": 48, "893754159196256e": 84, "893809231662017e": 84, "894": 48, "89402445e": 76, "894284530766281e": 48, "895": [15, 48], "895082990001085e": 48, "895117949331075e": 38, "895246097319718e": 48, "89537770278461e": 38, "895578637246975e": 48, "8959534368152337e": 48, "896": [15, 48], "89614036e": 76, "896183345282972e": 48, "896875175059282e": 21, "8969487973256553e": 84, "897": [15, 48], "897243155937746e": 83, "89738810e": 76, "897947407783534e": 38, "898": 48, "898245924316241": 48, "89843315e": 76, "89867542e": 76, "89899776e": 76, "899": [15, 48], "8991802786965872e": 48, "8993136308760616e": 83, "8993384818526183e": 83, "8993719203166894e": 48, "8pt": [68, 86], "9": [11, 14, 20, 21, 22, 38, 39, 48, 49, 63, 70, 76, 77, 80, 81, 83, 84, 85, 91, 96, 108, 110, 115, 119, 120], "90": [8, 33, 38, 39, 48, 77, 83, 84, 85, 110], "900": [15, 48], "90065562e": 76, "901": 48, "9014339920530615e": 77, "90154230e": 76, "901698813932705e": 84, "90189805e": 76, "901962432716777e": 84, "902": 48, "902098365795283e": 110, "903": [15, 48], "904": [15, 48], "90411155e": 76, "90491892e": 76, "905": [15, 48], "905310270171566e": 48, "905832846232014e": 38, "906": [48, 62], "906013148755061e": 48, "906326312921551e": 48, "906366472377646e": 84, "907": [15, 48], "9074917935399035e": 48, "908": [15, 48], "9081958235744878e": 76, "908274034770534e": 77, "909": 48, "90956415e": 76, "909777725564041e": 48, "9098392782407787e": 84, "91": [15, 38, 39, 48, 77, 83, 84, 85, 110], "910": [15, 48], "910553350297205e": 84, "910736871305874e": 48, "91093951e": 76, "911": [15, 48], "9110540031668575e": 21, "9111102267874765e": 84, "91127060197297e": 48, "911627307675492e": 83, "912": [15, 48], "912312105164127e": 84, "91241542438913e": 48, "912723274012169e": 48, "91277642e": 76, "913": 48, "91358750e": 76, "91395496e": 76, "914": [15, 48], "9148388881307": 106, "91494610e": 76, "915": [15, 48], "916": [15, 48], "916054099105543": 83, "916228055046904e": 48, "916429066888301e": 48, "916509026727359e": 48, "917": 48, "9171532625864503e": 83, "91715363e": 76, "917196533515586e": 48, "917762596575794e": 48, "918": 48, "919": [15, 48], "9191003622777862e": 48, "91997734e": 76, "92": [38, 39, 48, 77, 83, 84, 85, 110], "920": [15, 48], "921": 48, "921050242517416e": 48, "92176436e": 76, "922": 48, "92210142e": 76, "9223651737713088e": 48, "922398248092175e": 77, "922852287859793e": 91, "923": [15, 48], "923091927017895e": 48, "923210728679797e": 38, "9236962476294173e": 38, "924": [15, 48], "924816628679435e": 39, "924873078433326e": 48, "92491430e": 76, "925": [15, 48], "9256370039507276e": 48, "926": [15, 48], "926136120458067e": 83, "927": [15, 48], "927885769755884e": 48, "928": [15, 48], "928987954782195e": 48, "929": 48, "93": [38, 39, 48, 64, 77, 82, 83, 84, 85, 110], "930": [15, 48], "930665437036461e": 48, "93070738e": 76, "931": [15, 48], "931123263847961e": 48, "931468204779986e": 77, "932": [15, 48], "933": [15, 48], "9332724156658535e": 48, "9335995335618345e": 110, "93362996e": 76, "9339708712085968e": 48, "934": [15, 48], "934064265915551e": 48, "935": [15, 48], "935021255196529e": 85, "9357393093205987e": 0, "93598132e": 76, "936": [15, 48], "9361261198607126e": 85, "93682062e": 76, "936851784211181e": 85, "937": [15, 48], "937354550595616e": 48, "937851126352127e": 85, "938": [15, 48], "938071598197655e": 39, "939": [15, 48], "939175519254065e": 38, "93996993e": 76, "94": [10, 38, 39, 48, 77, 83, 84, 85, 110], "940": [15, 48], "940263595804976e": 48, "94066240e": 76, "941": 48, "941213925298137e": 48, "942": 48, "94214740e": 76, "943": [15, 48], "943188092742385e": 48, "94354799e": 76, "944": [15, 48], "94415843236444e": 48, "944919015348339e": 21, "945": 48, "9458345541265103e": 48, "946": 48, "946429851535188e": 48, "946813128879899e": 48, "946966834883386e": 48, "947": 48, "947450640234518e": 83, "947907556144233e": 77, "947936681618619e": 21, "948": 48, "948041627199521e": 48, "949": 48, "949029909160572e": 76, "949144841480855e": 48, "949180345347565e": 84, "94941910837591e": 63, "9494191083769515e": 63, "94990000871906": 108, "94996": 108, "95": [38, 39, 48, 77, 83, 84, 85, 110], "950": 48, "951": 48, "95165835e": 76, "9518262989727527e": 48, "952": 48, "9521726909671235e": 81, "9523838011088034e": 77, "952462288277464e": 48, "9525636053155235e": 83, "953": [15, 48], "953692726428111e": 48, "954": [15, 48], "9544491028129525e": 38, "954524988040401e": 38, "954697536053879e": 38, "95474670424036e": 91, "9548447702941143e": 48, "955": 48, "95521894798872e": 83, "955252997009489e": 84, "955476591094188e": 77, "95574753e": 76, "955800959468116e": 48, "956": 48, "9564189071730394e": 84, "956788505148804e": 84, "9568529356807312e": 48, "956866708243204e": 84, "957": [15, 48], "95764202e": 76, "958": [15, 48], "9583025195272006e": 38, "959": 48, "959344694700465e": 77, "959623023960013e": 77, "9597946922305316e": 39, "959861760711115e": 48, "96": [38, 39, 48, 77, 83, 84, 85, 110], "960": 48, "961": [15, 48], "9610069089538715e": 48, "961192049799402e": 77, "961884348845643": 48, "962": [15, 48], "96251115e": 76, "962748743407555e": 77, "963": 48, "9632743582485296e": 48, "9634170326035583": 63, "964": 48, "9642431866822456e": 48, "964343029696402e": 48, "9647467894946307e": 48, "965": [15, 48], "966": [15, 48], "966100092903907e": 21, "966964255127543e": 48, "967": 48, "967068": 0, "967218775966888e": 48, "967816862422729e": 39, "968": 48, "968077666128608e": 115, "9685811568329946e": 84, "969": [15, 48], "969780154913643e": 83, "9697904651577707e": 48, "97": [38, 39, 48, 77, 83, 84, 85, 110], "970": [15, 48], "970987207121857e": 48, "971": 48, "971758200881336e": 48, "972": 48, "972737960055163e": 48, "9729411628519188e": 48, "973": [15, 48], "973280254961185e": 83, "973354406340976e": 63, "973354406342218e": 63, "9739": 108, "974": [15, 48], "9748427980784953e": 48, "975": 48, "97567491e": 76, "97584791e": 76, "975969": 0, "976": 48, "976297565594483e": 48, "97669399e": 76, "977": [15, 48], "97722": 0, "977370313033323e": 48, "97741566142421e": 39, "977742889310151e": 48, "978": [15, 48], "97870424274299e": 38, "9788107295271953e": 77, "978929725877198e": 48, "979": 48, "97906": 0, "97926710e": 76, "97947213e": 76, "979591970780287e": 77, "9796612659768636e": 48, "98": [38, 39, 48, 77, 83, 84, 85, 110], "980": 48, "9800158109928926e": 38, "981": 48, "982": 48, "982218896602567e": 48, "98236278e": 76, "982454733891398e": 84, "9824564452862945e": 84, "983": 48, "983324464566756": 66, "983673740203993e": 77, "984": 48, "984159946931181e": 77, "984341809190383e": 39, "984965667907077e": 39, "984998868844853e": 48, "985": 48, "98506472e": 76, "9851038052631544e": 38, "986": 48, "987": 48, "98757560193412e": 48, "98786396035351e": 48, "988": 48, "988324878740525e": 48, "989": 48, "989051753270956e": 38, "98985998902484e": 48, "99": [38, 39, 48, 77, 83, 84, 85, 110], "990": 48, "990325906208014e": 48, "990504227339704e": 85, "990897": 0, "991": 48, "99116470e": 76, "99169": 118, "991707474082117e": 48, "992": 48, "992503628103648e": 48, "99252178e": 76, "99272270e": 76, "99280637e": 76, "992946960452109e": 48, "993": 48, "993479432078307e": 48, "994": 48, "995": 48, "995153767474519e": 48, "996": 48, "9961888045847009": 63, "9962490853348933": 63, "996273588831822e": 84, "9967208414315467": 63, "99673786e": 76, "996951398840777": 63, "997": 48, "9970667770354094": 63, "99724319e": 76, "997501320449961e": 48, "99761394e": 76, "998": 48, "998043749532844e": 48, "9982691913414063": 22, "9987999466057931": 63, "999": 48, "999105875654954": 85, "999690227069895e": 21, "9999999999993516": 109, "999999999999967": 55, "9999999999999964": 76, "9999999999999974": 57, "9999999999999993": [81, 96], "A": [0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 47, 48, 49, 50, 51, 52, 53, 54, 56, 60, 61, 62, 63, 64, 65, 69, 72, 73, 75, 76, 78, 79, 81, 83, 84, 85, 91, 99, 101, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115, 118, 121], "As": [1, 12, 13, 38, 64, 89, 90, 105, 113], "At": 40, "But": [3, 6, 24, 31, 33, 34, 35, 47, 54, 58, 76, 79, 86, 100, 101, 104, 105, 106], "By": [8, 31, 32, 37, 47, 50, 51, 53, 54, 63, 65, 76, 79, 83, 84, 100, 101, 103, 105, 108, 109, 112, 113], "For": [1, 2, 5, 9, 12, 13, 23, 27, 28, 29, 30, 32, 33, 36, 40, 41, 46, 51, 53, 55, 58, 64, 65, 66, 69, 76, 77, 79, 81, 83, 84, 90, 97, 98, 99, 100, 102, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115], "If": [1, 4, 6, 8, 9, 10, 12, 20, 24, 25, 26, 27, 28, 29, 32, 33, 36, 41, 46, 47, 48, 49, 51, 53, 54, 56, 63, 64, 66, 69, 76, 79, 81, 83, 84, 91, 97, 98, 100, 101, 103, 104, 105, 109, 111, 112, 113, 121], "In": [0, 6, 8, 9, 11, 12, 20, 23, 27, 28, 29, 32, 33, 34, 35, 36, 40, 46, 49, 50, 51, 52, 54, 56, 57, 58, 61, 63, 69, 73, 76, 78, 79, 81, 83, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115, 119, 120, 121], "It": [0, 1, 5, 6, 9, 12, 15, 24, 25, 27, 28, 31, 32, 33, 34, 36, 37, 40, 47, 53, 54, 60, 63, 64, 73, 76, 77, 79, 82, 83, 84, 88, 91, 97, 98, 100, 109, 112, 115], "Its": [37, 83, 88, 100, 104, 105], "NOT": 103, "Of": 84, "On": [0, 8, 12, 13, 31, 34, 35, 36, 38, 41, 53, 63, 66, 83, 89, 100, 101, 103, 104, 106, 108, 112, 121], "One": [0, 6, 8, 24, 33, 36, 37, 41, 50, 51, 79, 83, 84, 100, 108, 110, 111, 112], "Such": [33, 40, 46, 109], "That": 12, "The": [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 21, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 39, 40, 41, 43, 49, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 65, 66, 69, 72, 73, 74, 75, 77, 78, 80, 82, 83, 84, 86, 87, 88, 89, 91, 98, 99, 101, 103, 106, 108, 109, 110, 111, 112, 113, 115, 121], "Their": 6, "Then": [0, 1, 8, 12, 13, 15, 20, 24, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 41, 46, 47, 48, 50, 53, 54, 63, 64, 65, 66, 79, 80, 81, 83, 84, 88, 92, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 112, 118], "There": [8, 9, 12, 27, 28, 29, 32, 33, 34, 39, 48, 50, 51, 64, 65, 66, 78, 79, 82, 83, 88, 89, 97, 99, 100, 101, 102, 104, 105, 106, 112], "These": [12, 24, 35, 51, 83, 88, 100, 109, 111], "To": [4, 8, 12, 13, 31, 33, 36, 40, 46, 47, 48, 51, 53, 55, 73, 76, 83, 84, 89, 100, 101, 103, 105, 106, 108, 109, 110, 111], "With": [6, 13, 20, 31, 32, 47, 51, 63, 67, 82, 88, 100, 106], "_": [1, 3, 6, 8, 9, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 41, 48, 51, 53, 55, 56, 61, 62, 64, 65, 66, 67, 69, 71, 75, 76, 79, 83, 84, 86, 87, 90, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 113], "_0": [34, 98, 99], "_1": 79, "_2": 105, "__call__": 76, "__dict__": [10, 76], "__doc__": 11, "__eq__": [10, 76], "__file__": 11, "__flags_doc__": [10, 76], "__getitem__": 76, "__getstate__": [10, 76], "__hash__": [10, 76], "__init__": [10, 42, 63, 66, 76, 109], "__loader__": 11, "__memory__": [10, 76], "__mul__": [10, 76], "__name__": 11, "__new__": [10, 76], "__package__": 11, "__pow__": [10, 76], "__setstate__": [10, 76], "__spec__": 11, "__special_treated_flags__": [10, 76], "__str__": [10, 76], "__timing__": [10, 76], "__version__": 46, "_a": [27, 28, 29, 34, 37, 49, 50, 51, 52, 54, 64, 65, 66, 83, 84, 101, 103, 105, 106, 108, 109], "_b": [47, 101, 103], "_c": [37, 47, 49, 51], "_d": 83, "_e": [34, 79, 87, 100], "_h": [12, 36], "_i": [9, 66, 104, 105, 108], "_k": 68, "_m": 106, "_mesh": 76, "_n": [12, 79, 90], "_q": [53, 54, 84], "_registered_draw_typ": 121, "_t": [9, 13, 34, 112], "_u": 101, "_v": [6, 8, 12, 27, 28, 29, 30, 31, 33, 35, 36, 53, 54, 75, 100, 101, 104], "_w": [6, 27, 30, 31, 41, 100, 101], "_x": 104, "a0": [106, 108], "a0inv": 109, "a2": 86, "a_": [13, 24, 51, 53, 63, 64, 73, 76, 105, 111], "a_0": [63, 64, 65], "a_2": 101, "a_h": [12, 83, 109], "a_i": [24, 105], "a_k": 102, "a_l": [63, 64, 65], "a_n": 79, "a_t": [13, 24], "ab": [112, 119, 120], "abbrevi": 20, "abl": 104, "about": [8, 20, 23, 40, 73, 100], "abov": [8, 9, 12, 28, 30, 31, 35, 36, 51, 53, 65, 75, 76, 86, 103, 105, 108, 109, 112], "absorb": [43, 45], "abstract": [41, 76, 101], "ac": 51, "academ": 63, "access": [2, 6, 40, 76], "accid": 86, "accord": [8, 15, 27, 31, 35, 104], "accordingli": [53, 73], "account": [35, 40], "accur": [0, 10, 76, 112], "accuraci": [12, 13, 33], "achiev": [0, 47], "aconv": 5, "across": [1, 8, 12, 34, 36, 42, 55, 79, 88, 100], "act": 21, "acta": 115, "action": [63, 81, 82, 105, 109], "activ": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 39, 46], "actual": [31, 36, 82, 100], "ad": [1, 2, 24, 40, 49, 58, 61, 64, 83, 88, 103, 106, 109, 118], "adapt": [2, 8, 73], "add": [0, 1, 2, 24, 30, 31, 37, 38, 39, 42, 46, 82, 83, 100, 106, 108, 109], "addcircl": 44, "addintegr": [0, 55], "addit": [1, 3, 37, 38, 39, 41, 46, 47, 63, 65, 108, 109, 110, 111, 113, 121], "addition": [28, 35, 46], "addrectangl": [44, 62], "addsurfac": 70, "adiff": 5, "adjac": [36, 108], "adjoint": [1, 2, 37, 48, 51, 105], "adjust": 111, "administr": 15, "advanc": [20, 26, 58, 65, 79], "advantag": [1, 4, 6, 12, 40, 49, 52, 108], "affin": [8, 9, 60], "aform": [113, 114, 115, 118], "after": [8, 10, 26, 48, 76, 84, 101, 105], "afw": 83, "again": [12, 30, 31, 46, 57, 83, 100, 101, 106, 109, 112], "against": 55, "agreement": 103, "aij": 23, "aim": [34, 40, 46], "ainv": [38, 39, 42, 84], "air": [34, 78, 83, 89], "aka": [31, 98, 110], "al": 64, "alanc": 37, "algebra": [88, 105], "algorithm": [8, 40, 45, 47, 49, 52, 63, 76, 84, 97], "align": [1, 8, 9, 12, 13, 28, 29, 31, 33, 34, 35, 36, 47, 48, 50, 51, 52, 53, 55, 56, 79, 81, 98, 100, 101, 104, 121], "all": [1, 2, 6, 8, 9, 10, 13, 15, 20, 24, 27, 29, 30, 31, 32, 33, 34, 36, 37, 39, 46, 49, 51, 52, 53, 54, 63, 65, 66, 69, 72, 73, 74, 76, 81, 83, 88, 90, 99, 100, 101, 102, 104, 105, 106, 108, 109, 110, 111, 112, 113, 115, 121], "all_dofs_togeth": 121, "allow": [0, 5, 7, 9, 12, 13, 32, 36, 37, 41, 47, 50, 51, 56, 58, 67, 76, 79, 89, 100, 113], "allreduc": [15, 24], "almost": 28, "along": [3, 8, 33, 50, 59], "alpha": [0, 1, 2, 3, 5, 6, 8, 9, 12, 13, 30, 33, 49, 50, 51, 53, 73, 79, 81, 83, 98, 99, 101, 102, 104, 106, 108, 121], "alpha_": [48, 50], "alpha_1": [8, 12, 28, 30, 98], "alpha_2": [8, 12, 28, 30], "alpha_d": 98, "alpha_i": 98, "alpha_j": 49, "alpha_t": 9, "alreadi": [1, 34, 41, 64, 65, 81, 109], "also": [5, 6, 7, 8, 9, 12, 15, 20, 27, 28, 29, 30, 31, 33, 34, 35, 36, 40, 41, 47, 48, 49, 50, 51, 53, 55, 56, 58, 63, 65, 72, 76, 79, 83, 84, 89, 98, 99, 100, 103, 104, 105, 106, 111, 112], "altern": [15, 46, 58, 99], "although": [38, 52, 77, 114], "alwai": [27, 33, 111], "am": 20, "amat": 118, "among": 84, "amount": 40, "amplitud": 112, "an": [0, 3, 4, 5, 8, 9, 10, 12, 13, 14, 27, 28, 29, 30, 31, 32, 33, 34, 37, 40, 41, 42, 47, 48, 49, 51, 54, 55, 56, 60, 61, 62, 64, 65, 67, 72, 75, 76, 79, 80, 81, 82, 83, 84, 87, 88, 91, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 112], "anaconda": 20, "analog": 8, "analyi": 87, "analysi": [6, 33, 46, 51, 63, 71, 79, 91, 103, 105, 106, 110], "analyz": [40, 56, 63, 64, 76, 83, 104, 105], "angl": [0, 8, 55, 104], "ani": [10, 27, 29, 34, 52, 54, 65, 76, 79, 83, 105, 106, 112], "anim": [43, 76, 108, 109], "anisotrop": [8, 71], "anl": 23, "anoth": [0, 49, 52, 62, 78, 83, 91, 112], "ansatz": 41, "anton": 63, "anymor": 121, "anywai": [58, 121], "ap": [49, 81, 118], "apart": 38, "apl": 118, "app": 121, "appear": 79, "append": [33, 35, 47, 48, 49, 50, 51, 80, 103, 108, 110, 115, 121], "appl": 20, "appli": [1, 4, 5, 8, 10, 12, 13, 34, 35, 36, 37, 46, 50, 51, 52, 54, 63, 64, 65, 69, 72, 74, 76, 79, 82, 83, 84, 90, 98, 99, 100, 101, 106, 108, 109, 112, 113, 114], "applic": [8, 12, 33, 35, 45, 46, 49, 63, 72, 81, 82, 121], "applym": [10, 76], "approach": [34, 40, 46], "approx": [4, 12, 13, 41, 47, 48, 51, 53, 54, 64, 65, 83, 87, 90, 106, 111, 112, 121], "approxiamt": 79, "approxim": [0, 12, 27, 33, 36, 40, 46, 47, 48, 51, 54, 62, 63, 65, 67, 76, 77, 79, 81, 83, 86, 97, 109, 111, 112, 113, 114, 115, 118], "apsc_loc": 23, "ar": [0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 15, 20, 21, 23, 24, 27, 28, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 82, 83, 84, 87, 88, 89, 90, 91, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112, 115, 121], "arbitrari": [12, 28, 30, 31, 48, 53, 55, 65, 67, 76, 83, 90, 100, 102, 105, 106], "arbitrarili": [3, 106], "arc": 8, "arcco": 47, "arcross": 89, "area": 8, "aren": 76, "arg": [10, 76, 86, 121], "arg0": [10, 76], "arg1": [10, 76], "arg2": [10, 76], "arg3": [10, 76], "argument": [6, 10, 32, 34, 47, 76, 80, 87, 104], "aris": [28, 52], "arithmet": 77, "arnold": [60, 61, 67, 83], "around": [31, 106], "arrai": [5, 6, 12, 23, 24, 38, 40, 52, 53, 55, 56, 57, 58, 60, 61, 62, 67, 68, 69, 73, 76, 81, 82, 84, 86, 89, 91, 98, 100, 102, 105, 111, 112, 121], "array_y_": 76, "artifici": [37, 64], "arxiv": [118, 119, 120], "asc": [10, 11], "ascher": 5, "ascii": 121, "ask": 15, "asm": [45, 46, 63, 65, 83, 105, 108, 109], "asmal": [115, 118], "assembl": [0, 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "assign": [15, 108, 110], "associ": [12, 34, 76, 110], "assum": [1, 8, 10, 12, 13, 24, 28, 29, 30, 31, 34, 49, 50, 52, 53, 55, 64, 65, 73, 76, 83, 90, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 111, 112], "assumpt": [12, 31, 33, 64, 65, 101], "ast": [27, 28, 29, 30, 31, 34, 41, 47, 48, 50, 51, 53, 75, 86, 104, 105], "astrid": 62, "asvector": 42, "asymptot": [48, 106], "atan2": 0, "atild": 42, "atol": [14, 23], "atop": [31, 34, 36, 37, 41, 47, 65, 84, 86, 100, 104, 105, 108, 109, 112], "attain": 75, "attribut": [10, 76], "aubin": [8, 64, 65, 79], "augment": [45, 83], "author": [46, 72, 105], "automat": [8, 10, 70, 76], "autoscal": [4, 5, 119, 120], "autoupd": [10, 33, 35, 63, 66, 76, 83, 84, 109], "avail": [3, 5, 10, 11, 14, 20, 46, 51, 52, 72, 76, 84, 115], "averag": [10, 33, 34, 35, 42, 66, 76, 82, 109, 121], "averagingt": 42, "avoid": [12, 33, 40, 50, 71, 121], "await": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24], "ax": [8, 47], "axi": 112, "aziz": [31, 53], "b": [1, 4, 5, 6, 8, 13, 20, 22, 27, 28, 30, 31, 35, 37, 38, 39, 42, 47, 48, 49, 50, 51, 52, 53, 54, 56, 62, 68, 70, 73, 76, 78, 81, 82, 83, 84, 85, 86, 88, 89, 101, 102, 103, 108, 111, 112, 116, 119, 120, 121], "b64": 121, "b64encod": 121, "b_": [8, 13, 53, 111], "b_1": [91, 102, 111], "b_2": 91, "b_i": [51, 111], "b_k": 102, "b_l": 111, "b_n": 6, "b_r": 102, "b_t": 8, "b_x": 82, "b_y": 82, "babuska": 53, "babu\u0161ka": 31, "back": [8, 13, 15, 20, 37, 47, 51, 61, 84, 100, 102, 109, 112, 121], "backward": [51, 52, 63, 76, 112, 113], "bad": [6, 58, 69, 71, 83, 106], "badli": 98, "baker": 3, "balanc": [40, 55, 57, 58, 79], "ball": [99, 102], "banach": [27, 28, 99, 100], "banana": 20, "band": 40, "bar": [0, 55, 78, 89], "bare": 64, "barycentr": [9, 80], "base": [3, 15, 37, 40, 45, 63, 64, 65, 68, 76, 80, 100, 121], "base64": 121, "basematrix": [10, 42, 63, 76, 82, 96, 106, 108, 109], "basevector": [10, 76, 108], "basewebguiscen": [73, 96, 116, 121], "basi": [3, 4, 8, 9, 10, 12, 13, 34, 36, 38, 47, 49, 63, 66, 79, 83, 88, 100, 103, 105, 106, 110, 113, 114, 115, 118, 121], "basic": [45, 46, 58, 64, 108, 118], "bbboundari": [10, 76], "bbnd": [10, 37, 39, 76], "bboundari": [10, 37, 39, 76], "bc": [0, 37, 38, 39, 44, 55, 62, 70, 73, 108], "bcast": 20, "bddc": [45, 46, 74, 76, 77, 91], "bdm": [7, 34, 88], "bdm2": 88, "bdm_1": 88, "bdm_k": 88, "beauti": 58, "becom": [5, 12, 40, 51, 89, 103, 118], "been": [79, 100], "befor": [26, 37, 109], "begin": [1, 5, 6, 8, 9, 12, 13, 24, 28, 29, 31, 33, 34, 35, 36, 37, 38, 40, 41, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 73, 78, 79, 81, 82, 83, 84, 86, 87, 89, 90, 91, 92, 97, 98, 99, 100, 101, 102, 104, 105, 106, 109, 111, 112, 113, 114, 115, 121], "behav": [91, 98], "behavior": 112, "behaviour": 109, "behind": [20, 46, 49], "being": [27, 76, 103, 108], "bel": [116, 121], "belong": [73, 100, 104, 108, 109], "below": [28, 33, 51], "bend": [62, 67, 69, 70], "best": [8, 28, 54, 67, 79, 110], "beta": [8, 13, 49, 53, 54, 69, 70, 79, 81, 84, 104], "beta_": 31, "beta_1": [30, 31, 53], "beta_2": [30, 31], "beta_h": 69, "beta_i": 104, "beta_k": 49, "better": [8, 33, 49, 66, 86, 103], "between": [0, 3, 8, 13, 14, 20, 21, 24, 33, 34, 38, 40, 41, 45, 46, 62, 73, 74, 76, 97, 100, 108], "bezier_point": 121, "bezierpnt": 121, "bf": [0, 1, 8, 34, 55], "bfa": 81, "bfa1": [22, 85], "bfb": [22, 81, 85], "bfc": [22, 85], "bfequ": 34, "bfnorm": 0, "bfpre": 121, "bfschur": [22, 81, 85], "bg": 51, "big": [13, 30, 40, 47, 51, 53, 54, 56, 64, 65, 66, 77, 79, 83, 97, 103, 105, 106, 108, 109, 112, 113, 115, 121], "bigcup": 109, "bigcup_": 36, "bilinear": [0, 2, 4, 6, 8, 12, 13, 27, 28, 29, 31, 34, 35, 37, 38, 39, 42, 48, 51, 53, 56, 62, 68, 73, 75, 78, 83, 84, 89, 97, 101, 104, 106, 109, 121], "bilinearform": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 17, 18, 19, 21, 22, 23, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "bin": [20, 46], "binari": [20, 121], "bind": [10, 108], "binder": 25, "bisect": [76, 108], "bit": [38, 109], "bitarrai": [10, 66, 73, 76, 108, 109], "bitmask": 78, "bjac": 105, "block": [4, 20, 39, 40, 53, 56, 64, 70, 81, 83, 89, 91, 103, 121], "blockmatrix": 84, "blockvector": [81, 84], "blow": 6, "blue": 112, "bmat": 82, "bnd": [1, 4, 5, 10, 34, 55, 57, 66, 70, 76, 121], "bnddof": [38, 39], "bndmass": 66, "bndpart": 55, "bodi": [59, 62, 74], "boffi": [46, 72], "bone": [15, 20], "bonu": 26, "book": [25, 46], "bool": [10, 76], "bornemann": 111, "boss": 20, "bot": [27, 29, 31, 32, 37, 38, 39, 49, 53, 83, 100, 101], "bot_": [49, 79], "bot_c": 49, "bot_v": 53, "both": [0, 2, 12, 15, 24, 29, 30, 31, 32, 33, 41, 49, 54, 55, 61, 63, 64, 65, 68, 76, 78, 79, 83, 84, 87, 99, 100, 101, 103, 106, 112, 121], "bottl": 40, "bottom": [1, 2, 3, 10, 14, 34, 37, 38, 39, 44, 56, 62, 66, 73, 80, 82, 91, 100, 108, 109], "bottomo": 107, "bound": [8, 9, 12, 27, 28, 30, 31, 33, 34, 35, 36, 47, 48, 49, 51, 65, 83, 84, 97, 98, 99, 100, 101, 102, 103, 104, 106, 108, 109], "boundari": [1, 3, 4, 6, 7, 10, 12, 13, 18, 19, 22, 28, 30, 31, 33, 34, 36, 37, 38, 39, 40, 43, 45, 46, 51, 58, 65, 67, 70, 74, 75, 76, 78, 79, 80, 81, 82, 84, 85, 88, 89, 90, 91, 92, 95, 97, 99, 100, 101, 103, 104, 106, 108, 110, 113, 121], "boundarycf": [73, 76], "bounded": 109, "box": [8, 12, 22, 27, 28, 29, 31, 32, 53, 65, 74, 77, 78, 79, 81, 83, 85, 97, 99, 100, 101, 104, 105, 108, 121], "bp": 81, "braess": [34, 46, 64, 67, 79], "brambl": [8, 12, 36, 45, 46, 64, 79, 85, 101], "bramblepasciakcg": [22, 81, 85], "break": [38, 39, 46, 47, 48, 49, 50, 51, 81, 91], "brenner": [3, 46], "brezzi": [31, 34, 46, 53, 54, 61, 67, 72, 88], "bring": [37, 102], "broadcast": [15, 20], "browser": 46, "bsize": 23, "bstab": [119, 120], "btr": [116, 121], "bu": [4, 5, 6, 31, 84], "bubbl": [54, 58, 61, 76, 80, 118], "build": [8, 11, 12, 38, 42, 74, 108, 110], "buildrefinementtre": 76, "buildrenderdata": 121, "built": [24, 40, 55, 57, 76, 88], "builtin": [10, 76, 80], "bv": 53, "bvp": [1, 8], "bx": 82, "byte": 20, "c": [5, 8, 9, 10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 34, 37, 38, 42, 46, 47, 48, 49, 51, 53, 54, 56, 63, 64, 65, 67, 73, 77, 79, 81, 82, 83, 84, 85, 90, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 111, 112, 113], "c2l": 109, "c_": [1, 9, 13, 24, 51, 63, 64, 65, 70, 79, 83, 105, 106, 108, 109, 111], "c_0": [64, 90, 98, 100], "c_1": [33, 101, 103, 106, 109, 111], "c_2": [33, 101, 103, 106, 109], "c_3": 101, "c_4": 101, "c_f": [92, 102, 103], "c_i": 111, "c_j": 111, "c_l": [64, 111], "c_m": 111, "c_p": 102, "c_t": [9, 13, 24], "cach": 20, "cal": [8, 9, 12, 13, 33, 35, 36, 98, 99], "calcelementmatrix": 11, "calcul": [21, 34, 48, 50, 51, 63, 76, 88, 121], "calculu": [100, 102], "call": [1, 2, 8, 9, 12, 15, 20, 24, 27, 31, 32, 33, 39, 48, 51, 52, 55, 56, 58, 65, 76, 79, 83, 84, 90, 91, 97, 98, 99, 101, 103, 104, 105, 106, 109, 110, 111, 112, 119, 120, 121], "callback": [47, 49], "can": [0, 1, 2, 3, 6, 8, 10, 12, 13, 14, 15, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 60, 62, 63, 64, 65, 66, 69, 72, 73, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 89, 91, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 112, 113, 115, 121], "cancel": [47, 79, 80, 90, 98, 121], "candid": [6, 30, 53, 54, 87], "cannot": [3, 7, 12, 17, 35, 36, 51, 53, 76, 84, 98, 109], "canon": [27, 41, 62, 75, 83], "canva": 47, "cap": [9, 13, 29, 34, 36, 40, 83, 90, 99, 100, 101, 107, 109], "captur": 51, "care": [8, 33], "carefulli": 64, "cartesian": 76, "case": [2, 8, 9, 23, 28, 29, 30, 31, 35, 36, 51, 54, 58, 60, 67, 69, 73, 76, 78, 83, 99, 100, 101, 104, 105, 106, 115], "cauchi": [12, 27, 31, 32, 36, 64, 66, 97, 99, 100, 101, 102, 105], "cc": [5, 24, 38, 61, 73, 81, 82, 84, 91, 111, 121], "ccc": [24, 53, 68, 81, 84, 86, 91, 111], "cccc": 105, "ccccccl": 91, "ccccccll": 61, "ccccl": [53, 57, 58, 60, 84], "ccccll": [12, 40, 53, 55, 56, 57, 58, 62, 67, 69, 84, 86, 89], "cccll": 60, "cdof": 63, "cdot": [3, 6, 12, 13, 27, 28, 29, 30, 31, 32, 34, 37, 47, 48, 53, 54, 57, 62, 65, 66, 75, 76, 79, 87, 89, 97, 98, 99, 100, 101, 105, 113, 121], "cea": [8, 12, 64], "cell": [10, 20, 46, 58, 76, 88, 110, 119, 120, 121], "center": [0, 8, 55, 75, 99, 102], "cerbsim": 46, "certain": [8, 35, 108, 111, 112], "cf": [0, 4, 6, 10, 11, 18, 19, 33, 37, 38, 39, 55, 74, 76, 88, 121], "cfl": 112, "cfn": 44, "cg": [14, 18, 23, 49, 77, 81, 91], "cgsolver": [14, 18, 19, 21, 22, 38, 39, 42, 63, 74, 77, 82, 83, 84, 108, 110], "ch": [83, 109, 112], "ch_i": 109, "chain": [8, 13, 88, 100, 106], "challeng": 60, "chamfer": 74, "chang": [3, 10, 48, 73, 76, 81, 100], "chapter": [12, 28, 52, 99, 111], "character": [32, 50, 100], "charg": 76, "cheap": [4, 12, 39, 51, 63, 66, 77, 82, 86], "cheaper": [37, 82, 108], "cheapli": [49, 56, 83, 84, 91, 121], "cheat": [38, 62], "chebi": 47, "chebyiter": 47, "chebyshev": [45, 46, 49, 52], "check": [0, 6, 32, 34, 46, 76, 99, 108, 121], "check_unus": [38, 39, 66, 103], "chen": [25, 67], "choic": [6, 8, 64, 66, 75, 100, 104, 112], "choos": [0, 6, 8, 12, 28, 31, 32, 33, 36, 47, 48, 51, 53, 55, 73, 76, 83, 87, 97, 100, 103, 104, 108, 118, 121], "chosen": [3, 10, 28, 40, 50, 57, 83, 97, 106, 112], "christoph": [1, 7, 72], "ciarlet": 9, "circ": [0, 8, 9, 12, 13, 36, 55, 89, 100, 102, 106, 121], "circ1": 78, "circ2": 78, "circl": [0, 7, 16, 18, 19, 55, 78, 81, 82, 85, 89, 118, 121], "circo": [0, 55], "circul": 4, "circumst": 8, "cjac": 106, "cl": [6, 12, 36, 98, 102], "claim": [28, 30, 54, 64, 101, 106], "clamp": 3, "clark": 37, "class": [0, 9, 10, 14, 20, 21, 22, 23, 24, 42, 46, 55, 63, 65, 66, 76, 78, 101, 109, 111], "classic": [36, 45, 98, 111], "clean": 46, "clear": [53, 90, 108, 109], "clearli": [100, 103], "clement": 54, "clementin": 20, "click": 25, "client": [15, 42], "clip": [22, 55, 83, 85], "clock": 34, "close": [8, 9, 27, 28, 29, 31, 32, 33, 48, 53, 64, 76, 81, 89, 100, 101, 104], "closest": [0, 32], "closur": [27, 33, 41, 99, 100, 104], "cluster": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 40], "cl\u00e9ment": [65, 109], "cnt": [4, 5, 16, 108, 121], "co": [0, 32, 41, 47, 55, 76, 100, 102, 104, 121], "coars": [42, 63, 64, 66], "coarsebndmass": 66, "coarsedof": 66, "coarseext": 66, "coarsegrid": 108, "coarsemat": 63, "coarsepr": 63, "coarser": [40, 63, 64, 65, 121], "coarsest": [63, 65], "coasr": [63, 64], "cockburn": 1, "code": [26, 46, 50], "codimens": 76, "coeffici": [4, 24, 28, 34, 46, 51, 63, 64, 73, 75, 76, 79, 89, 100, 102, 103, 105, 106, 108, 109, 111, 112, 113, 121], "coefficientfunct": [5, 7, 10, 16, 44, 70, 76], "coerciv": [0, 2, 8, 12, 31, 36, 45, 46, 53, 101, 104], "coffe": 6, "cohen": 118, "coil": 82, "coincid": [8, 37, 98, 100, 111, 112], "col": [23, 42, 63], "col_pardof": 42, "colleagu": [25, 26], "collect": [15, 20, 53, 99], "color": [47, 112], "coloumb": 83, "column": [13, 49, 106, 112, 115], "com": 46, "combin": [2, 6, 8, 10, 15, 30, 35, 36, 49, 51, 58, 63, 64, 65, 76, 81, 100, 105, 110, 115, 121], "come": [6, 30, 40, 55, 76, 84, 86, 109, 121], "comm": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 42, 76], "comm_self": 23, "comm_world": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 42], "command": 46, "commerci": 46, "common": [0, 8, 9, 33, 34, 55, 100], "commun": [15, 20, 21, 24, 40, 76], "communiactor": 15, "commut": [12, 79, 83, 86, 90, 102], "comodi": 67, "comp": [0, 10, 37, 38, 39, 55, 76, 78], "compact": [27, 90, 98, 99, 100, 101], "compar": [1, 34, 40, 47, 54, 79, 110, 112], "comparison": [1, 50, 121], "compat": [76, 83], "compens": [1, 6, 87], "competit": 40, "compil": [70, 74], "complement": [27, 37, 38, 82, 83, 103], "complet": [27, 29, 31, 64, 65, 73, 76, 83, 99, 111, 121], "complex": [9, 10, 31, 40, 43, 44, 52, 63, 76, 77], "complic": 40, "compon": [0, 1, 3, 5, 7, 10, 12, 17, 18, 19, 33, 34, 37, 38, 39, 47, 51, 54, 55, 57, 58, 62, 64, 65, 67, 69, 70, 73, 74, 76, 83, 84, 86, 88, 89, 91, 108, 119, 120, 121], "composit": [100, 101, 118], "compound": [0, 55], "compoundfespac": [10, 76], "compoundfespaceallsam": [10, 76], "compress": [18, 19, 37, 38, 39, 52], "compressed_row_storag": 52, "compuat": 46, "comput": [0, 4, 23, 24, 25, 26, 28, 33, 35, 36, 40, 41, 46, 47, 49, 51, 55, 63, 70, 72, 73, 76, 77, 82, 84, 88, 91, 97, 98, 102, 103, 108, 110, 111, 113, 121], "computation": [51, 52], "concentr": 6, "concept": 112, "conclud": [31, 53, 100], "condens": [1, 76], "condit": [1, 4, 6, 10, 12, 13, 28, 31, 33, 34, 39, 41, 43, 45, 46, 47, 48, 51, 54, 58, 61, 63, 64, 65, 67, 70, 74, 76, 78, 84, 87, 89, 91, 100, 103, 104, 105, 106, 108, 109, 110, 111, 113, 114, 115], "condition": 65, "condtion": 73, "conduct": [78, 83, 109], "configur": 20, "conflict": 46, "conform": [3, 9, 45, 46, 54, 58, 62, 76, 79, 83, 91, 100, 106], "conjug": [21, 37, 38, 45, 46, 52, 77, 82, 84], "connect": [6, 9, 10, 15, 23, 24, 36, 39, 67, 83, 98, 102, 108, 109], "conserv": [45, 118, 121], "consid": [0, 3, 6, 8, 12, 28, 30, 31, 33, 34, 35, 41, 43, 53, 57, 76, 81, 82, 83, 84, 89, 100, 102, 106, 108, 112, 113, 114, 115], "consider": 51, "consist": [0, 2, 9, 12, 15, 21, 23, 25, 36, 37, 40, 41, 42, 45, 46, 60, 62, 64, 76, 82, 83, 84, 99, 100, 105, 111, 112, 121], "consisteni": 0, "consol": 46, "const": 34, "constant": [8, 9, 10, 12, 15, 28, 29, 30, 33, 34, 36, 40, 42, 51, 54, 56, 58, 59, 63, 64, 66, 67, 76, 78, 79, 88, 92, 98, 100, 101, 102, 103, 106, 108, 109, 111, 112, 121], "constantebematrix": [120, 121], "consti": 108, "constrain": [1, 34, 37, 83, 91], "constrainst": 45, "constraint": [12, 38, 46, 54, 58, 69, 79, 83, 84, 105], "construct": [8, 9, 29, 30, 33, 34, 49, 53, 54, 77, 79, 82, 83, 87, 97, 100, 104, 105], "constructor": 10, "consult": [75, 110], "contact": [0, 55], "contactboundari": [0, 55], "contain": [8, 9, 23, 27, 33, 34, 36, 41, 49, 76, 83, 88, 98, 99, 104, 109, 112], "content": 121, "context": 6, "continu": [0, 2, 6, 7, 8, 12, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 53, 54, 55, 56, 58, 60, 61, 64, 65, 68, 76, 78, 79, 83, 84, 87, 88, 89, 90, 91, 92, 95, 97, 99, 100, 101, 102, 103, 104, 108, 109, 121], "contourf": 112, "contract": 28, "contradict": [29, 31, 32, 101], "contrast": [31, 36, 51, 61, 76, 112, 114], "contribut": [13, 24, 33, 34, 36, 98, 104, 108], "control": [33, 74, 76, 106], "conv": [5, 17], "convect": [28, 45, 46], "conveni": [9, 46, 55, 65, 73, 81], "convent": 31, "converg": [8, 27, 28, 31, 32, 33, 38, 39, 41, 48, 50, 51, 56, 65, 75, 77, 79, 83, 99, 100, 101, 103, 110, 115], "convergenct": 51, "convers": [21, 24, 100, 121], "convert": [20, 23, 76], "convertl2oper": [10, 76], "convertoper": [17, 19, 116], "convex": [8, 32, 50, 65, 84, 104, 105], "convu": [5, 17], "convuhat": 17, "coo": [42, 76], "coordin": [0, 9, 76], "copi": [23, 63, 115], "core": 40, "corneliu": 63, "corner": [8, 33, 35, 104], "corollari": [101, 112], "correct": [34, 35, 38, 51, 63, 64, 65, 73, 82, 105, 106, 108, 109], "correspond": [2, 10, 12, 23, 33, 36, 41, 47, 51, 64, 65, 67, 71, 76, 80, 83, 91, 103, 105, 106, 108, 118], "cosen": 48, "cosh": 41, "cosin": 100, "cost": [34, 40, 82, 108], "could": [34, 51, 54, 82, 91, 112], "count": [15, 34], "counter": [34, 103], "coupl": [1, 10, 42, 53, 76, 111], "coupling_typ": [10, 18, 19, 76], "couplingmat": 42, "couplingtyp": [10, 18, 19, 76], "courant": 112, "cours": [25, 46, 72, 84], "covari": 116, "cover": 100, "cpp": [10, 80], "creat": [10, 14, 21, 23, 33, 46, 75, 76, 78, 80], "createaij": 23, "createblocksmooth": 110, "createcolvector": 42, "createdevicematrix": 120, "createdirectsolverclust": [10, 76], "createfromcoo": 42, "creategener": 23, "createi": 23, "creatematrix": [42, 63, 66], "createpetscmatrix": 14, "createrowvector": [42, 66, 84], "createsmooth": [47, 49, 51, 63, 64, 82, 106, 118], "createsmoothingblock": [10, 76], "createsparsematrix": 118, "createtranspos": [63, 66], "createvec": [14, 23], "createvector": [4, 5, 21, 38, 39, 42, 47, 48, 49, 50, 51, 63, 81, 108, 109, 115, 116, 118], "createvvector": 42, "creation": 80, "crime": 12, "critic": [84, 103], "cross": [70, 116], "crouzeix": [12, 111], "csg": [70, 77, 83], "csg2d": [37, 38, 39, 82], "csgeometri": [70, 83], "csr": [1, 2, 23, 52, 76], "csr_matrix": [1, 2, 52, 76], "ctautau": 70, "cube": 77, "cubic": [54, 58], "cumul": [21, 23, 24, 42, 66], "cup": [1, 9, 41, 55, 73, 90, 91, 100, 104, 106, 107], "cup_": [34, 100], "curl": [17, 34, 59, 62, 68, 69, 70, 83, 88, 116], "current": [0, 83], "curv": [0, 7, 8, 12, 16, 18, 19, 44, 55, 60, 64, 67, 70, 74, 76, 77, 78, 81, 85, 89, 121], "cut": [33, 100], "cycl": 33, "cyl": [7, 16, 22, 70, 74, 77, 81, 85], "cyl1": 8, "cyl2": 8, "cylbox": 74, "cylboxedg": 74, "cylind": [0, 8, 22, 70, 74, 77, 83, 85], "c\u00e9a": 28, "d": [0, 1, 2, 5, 6, 7, 8, 11, 12, 13, 14, 18, 19, 23, 27, 28, 29, 32, 34, 36, 37, 38, 39, 43, 44, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 72, 73, 74, 75, 76, 79, 80, 82, 83, 84, 88, 89, 90, 92, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 111, 112, 113, 115, 121], "d_": 63, "d_g": 98, "d_h": 83, "d_i": 90, "d_l": [63, 64, 65], "d_w": 98, "damp": [47, 48, 51, 64], "dampingp": [119, 120], "dampingu": [119, 120], "dash": 47, "data": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21, 22, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 67, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "date": 51, "dc": [58, 86, 91], "dd": [73, 108], "ddot": [105, 112, 114], "de": [52, 116], "debug": 20, "decad": [10, 40, 76], "decai": [8, 66, 108, 109], "decid": 103, "decis": 6, "declar": 76, "declaremathoper": [1, 4, 5, 6, 53, 62, 67, 68, 69, 78, 86, 87, 88, 89, 90, 91], "decod": 121, "decompos": [32, 38, 40, 53, 65, 66, 83, 101, 104, 105, 106, 108, 109], "decomposit": [37, 63, 65, 83, 90, 105, 106, 115], "decoupl": [62, 86, 111], "decreas": [8, 40, 48, 51], "def": [0, 2, 3, 7, 17, 18, 19, 33, 35, 42, 47, 49, 55, 58, 62, 63, 66, 67, 69, 70, 74, 81, 103, 109, 110, 112, 115, 121], "default": [0, 15, 20, 34, 35, 37, 38, 39, 55, 73, 76, 78, 89, 108], "defin": [0, 4, 8, 9, 10, 12, 13, 15, 20, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 41, 42, 47, 48, 50, 52, 53, 54, 55, 56, 58, 59, 62, 63, 64, 65, 66, 75, 76, 79, 84, 88, 90, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 109, 110, 112, 115, 120, 121], "definedon": [0, 2, 6, 7, 10, 12, 18, 19, 22, 34, 37, 38, 39, 55, 66, 73, 76, 81, 84, 85, 108], "defineproblem": 110, "definit": [0, 8, 9, 12, 27, 29, 30, 31, 32, 36, 37, 50, 51, 52, 56, 63, 65, 67, 77, 79, 81, 84, 90, 91, 98, 99, 103, 104, 105, 106, 109, 110, 112], "deflect": [67, 69], "deform": [0, 2, 12, 55, 56, 59, 66, 74, 76, 82, 103, 113, 114, 118, 121], "degener": 83, "degre": [9, 10, 34, 39, 47, 54, 62, 71, 73, 76, 77, 79, 80, 88, 91], "dehat": 116, "deletezeroel": 118, "deliv": 112, "delta": [1, 2, 12, 28, 31, 33, 36, 41, 43, 47, 53, 55, 57, 58, 64, 65, 69, 70, 76, 84, 98, 100, 104, 113, 114, 118], "delta_": [9, 27, 79], "delta_h": 121, "delta_t": 34, "demand": 98, "demkowicz": 79, "denomin": 80, "denot": [27, 79, 98, 99, 112], "dens": [24, 27, 42, 52, 79, 90, 97, 100, 101, 104, 109], "densiti": [8, 58, 83, 90, 100, 113], "dep": 20, "depend": [0, 1, 4, 8, 12, 20, 31, 33, 34, 51, 54, 62, 65, 75, 77, 79, 97, 102, 103, 106, 110, 112, 113, 119, 120], "deprec": [10, 76], "der": 118, "deriv": [1, 2, 3, 6, 8, 9, 12, 13, 33, 36, 40, 46, 53, 55, 57, 65, 70, 73, 76, 81, 84, 99, 100, 109, 112, 113, 114, 118], "describ": [12, 35, 46, 74, 82], "descriptor": [10, 76], "design": [35, 40, 60, 84, 121], "desir": 35, "dest": 20, "destin": 20, "det": [8, 13, 88, 100, 106, 112, 121], "detail": [6, 72, 79], "deterior": [106, 109], "determin": [13, 48, 81, 111], "detour": 83, "deuflhard": 111, "dev0": 20, "develop": [25, 64, 79, 83, 107], "df": 73, "dg": [3, 4, 5, 6, 10, 45, 46, 54, 67, 76, 121], "dgjump": [2, 6, 10, 76, 121], "dh": 116, "diag": [51, 63, 105, 112], "diagon": [4, 12, 21, 22, 49, 51, 52, 56, 81, 85, 91, 105, 118, 121], "diagonalmatrix": [42, 66, 108], "diagoniz": [48, 112], "diagram": [12, 79, 86], "diam": [8, 109], "diamet": 8, "diaz": 118, "dict": [10, 76], "did": [33, 46, 114], "diff": 5, "differ": [0, 1, 8, 9, 10, 21, 33, 45, 46, 47, 48, 54, 58, 63, 64, 65, 66, 73, 76, 77, 78, 79, 101, 109, 112, 113, 118], "differenti": [3, 27, 28, 46, 52, 65, 70, 76, 79, 84, 90, 98, 99, 100, 103, 105, 111, 112, 113], "differentialsymbol": [37, 39], "difficult": [5, 106], "difficulti": [26, 58], "diffus": [28, 45, 46], "dim": [0, 7, 8, 10, 12, 16, 18, 19, 24, 31, 33, 34, 35, 55, 57, 63, 66, 70, 76, 78, 81, 85, 89, 91, 103, 108, 109, 118, 121], "dimens": [8, 9, 34, 36, 63, 76, 88, 100, 109, 115], "dimension": [8, 10, 13, 27, 28, 29, 30, 31, 49, 52, 76, 79, 84, 87, 100, 101, 103, 106, 115], "dinv": 64, "dir": [11, 34, 42], "direct": [8, 34, 40, 50, 52, 53, 54, 63, 74, 77, 81, 85, 97, 100, 108, 109], "directli": [7, 8, 50, 76, 79, 84, 98], "dirichet": 108, "dirichlet": [0, 1, 3, 5, 7, 10, 11, 12, 13, 14, 18, 19, 21, 22, 28, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 55, 57, 58, 62, 63, 66, 67, 69, 70, 73, 74, 76, 77, 78, 79, 80, 81, 82, 84, 85, 89, 91, 92, 100, 103, 104, 106, 107, 108, 109, 113], "dirichlet_bbbnd": [10, 76], "dirichlet_bbnd": [10, 76], "dirk": 111, "disadvantag": [1, 91], "disc": 104, "discontinu": [1, 2, 3, 10, 18, 19, 34, 55, 56, 58, 72, 76, 78, 83, 89, 91, 119, 120], "discret": [0, 2, 4, 6, 7, 12, 31, 33, 34, 40, 46, 48, 52, 55, 56, 58, 59, 60, 62, 63, 64, 67, 69, 76, 83, 84, 86, 87, 91, 97, 105, 106, 109, 112, 113, 114, 115, 118], "discuss": [12, 28, 64, 73, 83], "disjoint": 41, "disp": [62, 67, 69, 70], "disp_dg": 3, "displac": [59, 60, 70, 74, 82], "dissect": 108, "dissert": [80, 88], "dist": [41, 109], "distanc": [8, 50], "distribut": [10, 21, 34, 42, 45, 46, 62, 76, 78, 98], "div": [1, 4, 5, 6, 12, 18, 19, 22, 27, 28, 34, 45, 46, 53, 54, 56, 57, 58, 62, 67, 68, 69, 70, 74, 76, 78, 81, 82, 83, 84, 85, 86, 87, 89, 91, 100, 121], "diverg": [7, 58, 60, 61, 83, 88, 90, 121], "divid": [8, 12, 24, 28, 64, 108], "divtol": [14, 23], "divvel": 7, "dl": [63, 65], "do": [1, 12, 25, 48, 51, 52, 55, 66, 73, 79, 82, 86, 97, 106, 109, 112, 113], "doc": 52, "docinfo": 10, "docu": 75, "document": [10, 72, 75, 110], "doe": [0, 3, 8, 15, 21, 24, 31, 46, 48, 49, 64, 73, 76, 84, 97, 100, 102, 103, 108, 109, 118], "dof": [1, 10, 15, 21, 23, 24, 34, 37, 38, 39, 42, 62, 66, 76, 83, 88, 108, 109, 110, 118], "dof2proc": 42, "dofnr": [10, 76], "dofrang": [10, 76], "doi": [65, 105], "dom": [37, 38, 39, 76, 108], "dom1": [38, 39], "dom2": [38, 39], "domain": [8, 9, 10, 12, 24, 33, 37, 38, 39, 42, 43, 53, 55, 63, 65, 66, 73, 76, 82, 83, 97, 98, 99, 101, 103, 104, 113, 119, 120], "domaindof": 108, "domdof": 108, "domi": 108, "domin": [64, 65, 83, 101], "domtest": [37, 38, 39], "domtrial": [37, 38, 39], "don": [1, 2, 3, 38, 46, 57, 69, 73, 91, 105], "done": [12, 15, 21, 49, 65, 108, 121], "dormann": 37, "dot": [44, 112, 113, 114, 121], "dotimestep": 16, "doubl": [24, 84], "dougla": [24, 34, 88], "down": [37, 56], "downarrow": [68, 86], "download": 20, "dp": [37, 45, 46], "draft": 46, "draw": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 33, 34, 35, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "draw_surf": [83, 116, 121], "draw_vol": [74, 77, 116, 121], "drawback": [1, 2, 84], "drawgeo": 121, "driven": 46, "ds_i": 100, "dst": 20, "dt": [32, 65, 113, 115, 121], "dtype": [23, 121], "dual": [8, 9, 13, 27, 31, 34, 35, 39, 41, 55, 61, 62, 100, 104, 105], "dualiti": [28, 55], "dubin": 79, "due": [35, 36, 53, 64, 74, 83, 104, 109, 121], "dummyargu": [10, 76], "dummypardof": 42, "durian": 20, "dvert": [37, 39], "dx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 63, 64, 65, 66, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 113, 114, 115, 116, 118, 121], "dy": [79, 97, 102], "dynam": [12, 45], "e": [2, 3, 4, 6, 8, 9, 10, 12, 24, 27, 28, 29, 31, 32, 33, 34, 35, 36, 38, 39, 40, 47, 48, 49, 50, 51, 52, 53, 55, 58, 60, 61, 62, 64, 65, 66, 73, 74, 75, 76, 77, 79, 80, 83, 84, 87, 88, 90, 91, 97, 98, 99, 100, 101, 102, 103, 105, 109, 112, 113, 116, 118, 121], "e0": [106, 108], "e_": [66, 105], "e_0": [65, 100, 106], "e_1": 34, "e_2": 34, "e_3": 34, "e_i": [9, 34, 80, 105], "e_l": [64, 65, 66], "each": [8, 10, 12, 13, 15, 24, 33, 36, 51, 58, 63, 76, 82, 83, 91, 98, 99, 100, 108, 121], "earlier": 76, "eas": 55, "easi": [46, 60], "easili": [5, 24, 34, 41, 47, 48, 49, 64, 65, 79, 112], "easter": 26, "ecomposit": 37, "ed": 110, "edg": [0, 1, 2, 3, 6, 7, 8, 9, 10, 12, 16, 18, 19, 34, 35, 36, 54, 55, 59, 62, 68, 74, 76, 78, 79, 80, 81, 83, 85, 87, 88, 89, 91, 103, 108, 110], "edgedof": 76, "edof": 76, "effect": 83, "effici": [33, 34, 40, 49, 63, 79], "effort": 40, "ehat": 116, "ei": [10, 11, 22, 76, 85], "eid": 76, "eigen": [41, 47, 64, 100, 101, 103, 112], "eigenfunct": [41, 92, 96, 97, 103, 108], "eigenpair": 103, "eigensi": 121, "eigenspac": 103, "eigensystem": [27, 64, 115], "eigenvalu": [0, 41, 47, 48, 51, 64, 84, 92, 96, 97, 100, 103, 106, 108, 109, 110, 112], "eigenvalues_precondition": [42, 47, 63, 81, 83, 106, 108, 109, 110], "eigenvector": [48, 51, 103, 112], "eigh": 115, "either": [9, 20, 24, 35, 46, 76, 80, 87, 89, 121], "el": [8, 15, 37, 39, 108, 109], "elast": [60, 82], "elderberri": 20, "electr": [0, 76], "electromagnet": 61, "electrostat": 76, "element": [0, 1, 2, 3, 4, 6, 7, 24, 26, 27, 29, 33, 34, 36, 37, 38, 39, 40, 48, 52, 55, 56, 60, 61, 63, 64, 65, 67, 68, 69, 72, 73, 74, 75, 83, 84, 87, 91, 100, 101, 103, 105, 106, 108, 109, 112, 113, 118, 121], "element_boundari": [3, 4, 5, 6, 7, 17, 18, 19, 34, 62, 67, 69, 70, 91, 116, 121], "element_typ": [10, 76], "element_vb": [1, 121], "element_wis": [33, 35], "elementari": [100, 102], "elementid": [10, 11, 33, 35, 76], "elementrang": 76, "elements2d": 15, "elementtransform": 76, "elimin": [37, 52, 56, 76, 81], "eliminate_hidden": [18, 19], "eliminate_intern": 91, "ellipt": [24, 28, 37, 41, 45, 46, 53, 54, 64, 65, 76, 81, 91, 113], "els": [1, 12, 15, 18, 19, 20, 22, 27, 37, 38, 39, 41, 42, 63, 85, 108, 109, 119, 121], "elsewher": 76, "em": 79, "emb": [38, 39, 101, 105], "emb_p": [119, 120], "embed": [17, 22, 25, 83, 85, 100, 101, 105], "embeddedmatrix": 120, "embeddedtransposematrix": 120, "embu": 17, "embuhat": 17, "ement": 36, "emg": 83, "empti": [9, 105], "emptyset": [36, 100], "en": [14, 111], "enabl": [10, 46, 76], "encod": 121, "encodedata": 121, "end": [1, 5, 6, 8, 9, 12, 13, 16, 24, 28, 29, 31, 33, 34, 35, 36, 37, 38, 40, 41, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 73, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 90, 91, 92, 97, 98, 99, 100, 101, 102, 104, 105, 106, 109, 111, 112, 113, 114, 115, 121], "energi": [34, 35, 37, 45, 50, 51, 52, 66, 69, 70, 83, 118, 121], "enforc": [38, 40, 55, 61], "engin": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 34, 46], "enlarg": 109, "enough": [54, 61, 69, 76, 83, 106], "enrichr": 0, "ensur": 33, "enter": [25, 57, 76, 78], "entiti": 110, "entri": [1, 12, 13, 24, 51, 52, 61, 66, 76, 105], "enumer": [15, 23, 37, 38, 39, 42, 51, 80, 108], "enumerateglob": 23, "envelop": [119, 120], "environ": [46, 73], "envolv": 34, "ep": [5, 83, 106], "epsseminorm": 103, "epub": [65, 105], "eqnarrai": [8, 12, 24, 34, 36, 37, 41, 47, 50, 51, 53, 54, 58, 60, 62, 63, 64, 65, 66, 67, 78, 79, 83, 84, 86, 87, 89, 90, 92, 97, 99, 100, 105, 106, 109, 111, 112, 113, 114, 115, 121], "equ_clement_bh": 36, "equ_clement_bh2": 36, "equ_leadingcoef": 79, "equ_projbased2": 79, "equ_threeterm": 79, "equal": [33, 34, 40, 78, 79, 101, 105], "equat": [2, 8, 13, 28, 31, 33, 34, 36, 38, 40, 41, 48, 52, 54, 55, 56, 61, 65, 73, 74, 77, 78, 79, 81, 82, 84, 89, 91, 100, 101, 105, 111, 118, 119, 120], "equflux": 34, "equilibr": 46, "equilibrium": [12, 34, 74], "equip": [83, 90], "equival": [9, 13, 27, 28, 31, 33, 34, 45, 46, 50, 51, 60, 65, 84, 87, 88, 100, 102, 103, 104, 106, 112], "er": 34, "ern": 46, "eror": 45, "err": [21, 22, 47, 48, 49, 51, 80, 81, 85, 88, 115], "err0": [47, 48, 49, 50, 51, 81], "err2": [50, 51], "errest": [33, 35], "errestdu": 35, "errhist": [47, 48, 50, 51], "errlist": 80, "errmax": [33, 35], "error": [0, 12, 21, 28, 29, 31, 45, 47, 48, 49, 50, 51, 64, 67, 71, 75, 76, 83, 91, 105, 108, 109, 115], "essenti": [8, 34, 55, 57, 65, 89, 98, 100, 103, 108], "establish": [40, 79], "estim": [12, 28, 31, 37, 47, 49, 50, 61, 64, 66, 67, 81, 83, 86, 97, 100, 102, 105, 106, 108, 109], "et": [10, 64, 76], "et_quad": 10, "et_segm": 10, "et_tet": 10, "eta": [33, 34, 35, 36, 100], "eta_t": [33, 35, 36], "etautau": 70, "eu": 100, "euclidean": 24, "euklidean": [32, 48, 75], "euler": [4, 111, 115], "eval": [0, 92, 96, 97, 103], "evalu": [0, 8, 9, 11, 12, 22, 28, 34, 35, 38, 39, 66, 79, 81, 82, 84, 85, 98, 101, 104, 108, 115, 121], "evec": [0, 92, 96, 97, 103, 115], "evei": 6, "even": [4, 12, 40, 64, 78, 79, 121], "everi": [1, 2, 6, 12, 15, 20, 24, 27, 28, 30, 40, 42, 47, 48, 49, 51, 61, 65, 66, 83, 84, 86, 91, 97, 100, 103, 105, 108, 112, 115], "everyon": 20, "everyth": 100, "everywher": 28, "evp": 97, "exact": [12, 13, 34, 64, 111, 121], "exactli": [7, 12, 40, 41, 48, 55, 56, 79, 105, 106, 115], "exampl": [6, 9, 12, 20, 27, 28, 31, 34, 41, 45, 46, 52, 53, 63, 73, 76, 82, 83, 85, 98, 103, 108, 110, 113, 114], "excel": 25, "except": [37, 88, 121], "excercis": [48, 51], "exchang": [1, 20, 40], "exchangeproc": 42, "execut": [20, 46, 75], "exercis": [0, 3, 4, 10, 12, 26, 28, 31, 34, 41, 45, 46, 51, 73, 74, 77, 78, 79, 80, 81, 90, 100, 106, 108, 110], "exist": [1, 2, 8, 9, 12, 27, 28, 29, 30, 31, 32, 33, 34, 48, 50, 54, 66, 79, 83, 90, 99, 100, 101, 102, 103, 104, 105, 106], "exp": [4, 5, 6, 43, 44, 64, 75, 112, 113, 114, 115, 116, 118, 119, 121], "expand": [41, 47, 48, 64, 76, 81, 100, 102], "expans": [13, 41, 63, 112, 113], "expect": [20, 33, 106], "expens": [63, 109, 112], "experi": [25, 46, 47, 51, 72, 77, 78, 84, 97, 106, 108], "experiment": [25, 48], "explain": [46, 72, 75, 100], "explan": [46, 65], "explicit": [4, 5, 12, 30, 34, 45, 100, 109, 118, 121], "explicitli": [38, 49, 84, 105, 109], "explizit": 5, "explor": 110, "exponenti": [45, 79, 112], "exponentialpropag": 115, "export": 20, "express": [9, 10, 12, 30, 62, 65, 76, 102, 105, 113, 114], "exproc": 42, "ext": 66, "extend": [0, 10, 12, 14, 27, 41, 53, 73, 80, 90, 98, 100, 101, 102, 112], "extendrec": 66, "extens": [6, 12, 23, 27, 37, 45, 46, 63, 79, 91], "extent": 112, "extern": 46, "extra": [31, 55, 61, 79, 91, 110], "extract": [21, 101, 105], "extrem": [51, 110], "f": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 18, 21, 22, 23, 24, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 67, 69, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 100, 102, 104, 108, 110, 111, 112, 113, 114, 121], "f2": 86, "f_": [13, 24, 36, 53, 62, 80], "f_0": 53, "f_1": 31, "f_2": 31, "f_d": 73, "f_e": 62, "f_f": 73, "f_h": 12, "f_i": [13, 24], "f_j": [13, 76], "f_t": [13, 24, 34, 62], "fa": 110, "fac": 47, "face": [0, 1, 6, 7, 8, 9, 10, 12, 16, 18, 19, 22, 33, 34, 35, 54, 55, 62, 74, 76, 77, 78, 80, 81, 85, 88, 89, 91, 103, 108, 110, 118, 121], "facedof": 76, "facet": [1, 2, 3, 6, 62, 76, 108, 121], "facetfespac": [1, 5, 34, 91, 121], "facetspac": 3, "facetvari": 17, "fact": 48, "factor": [8, 28, 31, 48, 58, 63, 64, 66, 70, 76, 79, 80, 83, 84, 97, 98, 100, 101, 108, 109, 121], "fail": 58, "falk": [60, 83], "fall": 112, "fals": [0, 1, 4, 5, 10, 18, 19, 22, 38, 39, 47, 49, 55, 66, 74, 76, 77, 80, 82, 83, 85, 96, 97, 103, 109, 110, 116, 118, 119, 120, 121], "famili": [8, 88, 100], "famou": [84, 86], "far": 77, "fast": [5, 46, 50, 100, 103, 112], "faster": [8, 112, 121], "faustmann": 46, "fb": 51, "fbg": 51, "fc": 20, "fcen": 119, "fd": 73, "fdof": 76, "fe": [0, 1, 2, 4, 5, 6, 10, 11, 12, 14, 15, 21, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 63, 64, 66, 70, 73, 74, 76, 77, 78, 80, 82, 83, 89, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 118, 121], "feasibl": [34, 47, 63, 105, 108], "featur": [48, 55], "feb": 115, "fem": [10, 34, 46, 76, 79, 110], "fes1": [1, 5, 70], "fes2": [1, 5, 70], "fes3": 70, "fes_p": 121, "fes_pf": 121, "fes_pt": 121, "fes_tr": 121, "fes_u": 121, "fesc": 116, "fescurl": 17, "fesdom": 109, "fesflux": [33, 34, 35], "fesh1": 12, "fesi": [37, 38, 39], "fesl2": 12, "feslam": [38, 39, 55, 84], "feslami": [38, 39], "fesp": 56, "fespac": [3, 10, 12, 62, 67, 69, 70, 76, 121], "fespaceelementrang": [10, 76], "fesstress": 74, "festr": 116, "fesu": [55, 84], "fesvertex": [37, 39], "feti": [37, 45, 46], "few": [0, 25, 40, 46, 47, 48, 79, 83, 84, 97, 103], "ff": [73, 80], "fg": 51, "fictiti": [37, 83, 105], "field": [0, 6, 12, 28, 39, 40, 54, 57, 58, 62, 70, 73, 76, 82, 83, 84], "fig": [20, 47], "figsiz": 47, "figur": 112, "file": [76, 121], "filenam": 76, "filter": 86, "final": [6, 10, 28, 29, 47, 49, 53, 54, 66, 76, 79, 81, 97, 98, 99, 102, 105, 108, 109], "finalizeupd": [10, 76], "find": [0, 3, 4, 6, 8, 12, 13, 28, 30, 31, 34, 35, 40, 41, 47, 48, 53, 54, 55, 57, 58, 60, 61, 62, 64, 67, 69, 73, 74, 75, 76, 78, 82, 83, 84, 86, 89, 91, 100, 101, 102, 103, 104, 105, 108, 110, 112, 115], "fine": [83, 109], "finer": 118, "finest": 65, "finit": [0, 3, 6, 24, 26, 27, 28, 29, 31, 33, 34, 36, 37, 38, 48, 52, 55, 56, 60, 63, 64, 65, 68, 69, 72, 73, 74, 75, 83, 84, 87, 99, 100, 101, 103, 105, 106, 108, 109, 113, 118], "finitecyl": 70, "first": [6, 8, 10, 13, 25, 27, 29, 32, 33, 36, 38, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 61, 62, 64, 65, 70, 76, 79, 80, 81, 82, 83, 84, 88, 89, 91, 97, 98, 100, 101, 102, 103, 104, 106, 108, 109, 112, 114, 115, 118, 121], "fit": [14, 62], "fix": [0, 10, 26, 28, 32, 39, 40, 48, 58, 62, 74, 76, 79, 82, 97, 105, 106, 109], "flag": [6, 10, 46, 55, 76], "flat": 71, "flatten": 121, "flip": 100, "float": [39, 76], "float32": 121, "float64": 76, "flow": [6, 28, 84, 112], "fluid": [12, 58], "flux": [34, 35, 55, 57, 73, 78, 86, 88, 89], "fnum": 76, "focu": 98, "follow": [1, 6, 8, 10, 12, 31, 33, 34, 35, 36, 37, 40, 41, 46, 47, 51, 53, 54, 63, 64, 65, 66, 73, 79, 81, 83, 87, 88, 91, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 111, 121], "foral": [0, 8, 9, 12, 13, 27, 28, 29, 30, 31, 32, 34, 35, 36, 40, 41, 49, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 64, 65, 67, 69, 73, 75, 76, 79, 82, 83, 84, 86, 89, 90, 91, 92, 95, 98, 100, 101, 102, 103, 104, 105, 109, 112, 113, 114, 115], "forc": [12, 20, 27, 58, 74, 82], "forg": 20, "form": [0, 2, 4, 6, 8, 12, 13, 24, 27, 28, 29, 31, 34, 35, 37, 38, 39, 41, 42, 43, 46, 48, 51, 53, 54, 55, 56, 57, 58, 62, 68, 73, 75, 78, 79, 81, 82, 83, 84, 88, 89, 91, 97, 100, 101, 104, 105, 106, 109, 113, 121], "formal": [76, 87, 100, 112], "format": 52, "formul": [3, 8, 12, 28, 31, 34, 40, 45, 46, 56, 57, 61, 64, 67, 69, 84, 87, 91, 100, 101, 114], "formula": [47, 79, 90, 98, 112], "forth": 2, "fortin": [46, 61, 72], "forward": [51, 52, 63, 91, 112], "found": [6, 46, 50, 64, 72, 84, 101, 105, 110], "four": [33, 49], "fourier": [41, 65, 79, 100, 102, 104], "fourth": [45, 46], "frac": [0, 1, 2, 4, 5, 6, 8, 12, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 41, 42, 43, 47, 48, 49, 50, 51, 53, 54, 55, 56, 61, 64, 65, 66, 69, 73, 75, 76, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 90, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 118, 121], "fraction": [30, 84, 100], "frame": 59, "framework": [12, 24, 121], "free": [4, 7, 10, 15, 61, 62, 67, 69, 73, 76, 82, 83, 88, 119, 120, 121], "freedof": [0, 1, 2, 3, 5, 7, 10, 11, 12, 14, 18, 33, 34, 35, 37, 38, 39, 43, 44, 55, 56, 58, 62, 63, 67, 69, 73, 76, 78, 80, 82, 84, 89, 91, 92, 103, 106, 108, 109, 110, 118], "freedom": [9, 10, 34, 39, 54, 61, 62, 71, 73, 76, 77, 80, 88, 91], "frequenc": [43, 64, 65, 73], "friedrich": [45, 53, 101, 106, 109, 112], "from": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 96, 97, 98, 100, 101, 103, 105, 106, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121], "front": 48, "frontend": [46, 72], "fruit": 20, "ftau": 70, "fubini": [97, 100], "fulfil": [8, 28, 29, 31, 33, 36, 53, 100, 104], "full": [33, 48, 64, 65, 84, 100, 105], "fullb": [119, 120], "fulli": 3, "func": [75, 80, 88, 121], "func2": 121, "funcf": 76, "function": [0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 13, 14, 15, 20, 22, 27, 29, 32, 33, 34, 35, 36, 37, 38, 39, 41, 45, 46, 47, 49, 50, 53, 54, 55, 57, 58, 63, 64, 65, 66, 69, 73, 75, 78, 79, 83, 84, 85, 88, 89, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 110, 111, 113, 114, 115, 118, 121], "fundament": [28, 100, 102], "further": [30, 33, 72, 83, 100], "furthermor": [29, 31, 65, 79, 88, 100], "furthoer": 34, "fv": 55, "fvec": 82, "g": [4, 11, 13, 22, 24, 28, 31, 33, 38, 39, 41, 49, 51, 53, 54, 57, 62, 66, 73, 74, 79, 81, 82, 84, 85, 89, 91, 98, 100, 101, 104, 105, 110, 111, 112, 113, 118, 121], "g_e": 62, "g_i": 100, "g_k": 100, "g_n": 100, "g_t": 62, "g_v": 62, "galerkin": [3, 8, 12, 28, 31, 34, 35, 36, 54, 63, 65, 72, 76, 79, 105, 114, 115, 119, 120], "game": 87, "gamg": [14, 18, 19, 22, 23], "gamma": [0, 29, 41, 55, 61, 79, 81, 101, 106, 107, 111], "gamma1": 47, "gamma2": 47, "gamma_": [4, 6, 38, 40, 42, 90, 100], "gamma_1": [28, 30, 47, 51, 81, 105], "gamma_2": [28, 30, 47, 51, 81, 105], "gamma_b": 100, "gamma_d": [0, 2, 8, 12, 55, 56, 57, 73, 74, 83, 84, 89, 92, 95, 101, 103, 104, 107], "gamma_i": 100, "gamma_n": [57, 73, 74, 89, 91, 100, 104], "gamma_r": [43, 51, 73, 82, 104], "gammai": [0, 55], "gamman": 57, "gammao": [0, 55], "gaug": 83, "gauss": [0, 12, 63, 64, 76, 105, 110], "gaussian": 81, "gb": 40, "geever": 118, "gener": [8, 10, 12, 15, 23, 27, 28, 30, 31, 33, 46, 48, 51, 52, 53, 61, 64, 65, 73, 75, 76, 77, 79, 83, 84, 86, 97, 99, 101, 103, 111, 112, 115, 121], "generatemesh": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "genuin": 67, "geo": [37, 38, 39, 44, 62, 70, 74, 78, 82, 83, 108, 118, 121], "geom": 10, "geom2d": [5, 37, 38, 39, 42, 43, 44, 47, 62, 63, 67, 69, 82], "geom_fre": [116, 121], "geometr": [0, 9, 32, 40, 64, 65, 70, 76, 78, 108, 110], "geometri": [40, 75, 76, 77, 121], "geoparamcf": 76, "geq": [6, 8, 12, 27, 28, 30, 31, 32, 33, 34, 47, 51, 53, 54, 64, 66, 67, 79, 84, 87, 102, 104], "get": [0, 3, 10, 15, 24, 26, 33, 34, 36, 38, 40, 46, 47, 49, 51, 53, 54, 55, 56, 61, 64, 65, 66, 76, 79, 80, 84, 86, 88, 89, 90, 92, 100, 103, 105, 106, 109, 110, 121], "getarrai": 23, "getbbboundari": 76, "getbboundari": [39, 76], "getboundari": [0, 38, 39, 55, 76, 78, 108], "getcurveord": 76, "getdata": 121, "getdof": [10, 38, 39, 66, 76, 108, 118], "getdofnr": [10, 76, 88, 110], "getf": [10, 11, 76], "gethpelementlevel": 76, "getintegrationrul": 118, "getmateri": [0, 38, 39, 44, 55, 76, 78, 108], "getn": [15, 22, 33, 35, 76], "getoperatorinfo": [120, 121], "getord": [10, 76], "getparentel": 76, "getparentfac": 76, "getparentvertic": 76, "getpc": [14, 23], "getperiodicnodepair": 76, "getpmltrafo": 76, "getsubvector": 23, "gettrac": [10, 76], "gettracetran": [10, 76], "gettrafo": [11, 76], "gf": [0, 1, 17, 18, 19, 34, 76, 121], "gfaux": 17, "gfcoars": 108, "gfconst": 106, "gfdom": [108, 109], "gfe": 116, "gfequ": 34, "gfetr": 116, "gfflux": [33, 35], "gfh": 116, "gfhtr": 116, "gfi": 108, "gfl2": 15, "gflam": 84, "gfp": [22, 81, 85, 121], "gfpf": 121, "gfpt": 121, "gfsigma": [18, 19], "gfstab": [119, 120], "gfstress": 74, "gftot": [37, 38, 39], "gftr": 121, "gfu": [0, 1, 2, 4, 5, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 63, 64, 66, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 108, 110, 113, 114, 115, 118, 119, 120, 121], "gfu0": [82, 108], "gfu1": 82, "gfu2": 86, "gfucorr": 18, "gfudual": 35, "gfuhat": [0, 17, 18, 19], "gfuref": 115, "gfv": 114, "gfvi": 17, "gg": 56, "gil": 12, "girault": 101, "github": [10, 11, 25, 46, 80], "give": [2, 6, 12, 20, 24, 26, 30, 34, 46, 62, 73, 75, 79, 82, 84, 87, 88, 90, 91, 100, 103, 104, 110], "given": [4, 6, 10, 28, 30, 32, 34, 41, 46, 47, 49, 50, 51, 52, 53, 54, 58, 64, 65, 66, 73, 76, 83, 87, 100, 105, 106, 109, 112, 113], "global": [0, 8, 9, 10, 11, 13, 15, 23, 24, 34, 37, 38, 42, 76, 79, 108, 109, 121], "globalinterfacespac": 0, "globalord": [10, 76], "globcoupl": 42, "globcouplingmat": 42, "globinv": 42, "globnum": 23, "globschur": 42, "glu": [0, 7, 39], "glue": [33, 34, 35, 78, 89, 108, 118], "gmre": 84, "go": [6, 41, 53, 61, 83], "goal": [40, 46, 47, 51, 66, 83, 105], "goe": 21, "good": [3, 8, 28, 30, 33, 35, 47, 50, 52, 56, 60, 66, 81, 86, 106, 109], "gopalakrishnan": 1, "got": [15, 20, 25, 56], "gov": 23, "grad": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 21, 22, 23, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 58, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 118, 121], "grade": 118, "gradient": [0, 13, 21, 33, 36, 37, 38, 45, 46, 52, 62, 69, 73, 77, 78, 82, 83, 84, 88, 100, 103, 121], "gradu": 70, "gradv": 70, "gram": [49, 115], "grape": 20, "graph": [15, 40, 52, 99], "grate": 45, "graviti": 75, "green": 35, "grew": 37, "grid": [10, 15, 40, 42, 47, 63, 64, 76, 112], "grid_siz": [4, 6, 12, 78, 83, 88], "gridfunct": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "grote": 118, "group": [20, 76], "grow": [8, 48, 103, 109, 112], "gtild": 84, "guarante": [51, 53], "gudi": 3, "guermond": 46, "gui": [44, 62, 67, 69, 70, 74], "guid": [33, 121], "h": [0, 1, 2, 3, 5, 8, 12, 17, 22, 30, 34, 35, 36, 38, 40, 42, 45, 46, 48, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 73, 74, 75, 77, 78, 79, 83, 84, 85, 86, 89, 91, 92, 95, 97, 98, 99, 102, 103, 104, 107, 108, 109, 112, 113, 116, 120, 121], "h0": 8, "h1": [0, 3, 10, 11, 12, 14, 15, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 58, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 84, 85, 89, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 118], "h1amg": 85, "h1lumpingfespac": 118, "h1norm": [96, 97, 103], "h1seminorm": [92, 96, 103], "h_": [83, 103], "h_0": [30, 34, 35, 41, 53, 58, 76, 100, 101, 104, 106, 109, 113], "h_1": 108, "h_e": [12, 34, 36], "h_i": 109, "h_k": [65, 66], "h_l": [63, 64, 65, 66], "h_t": [8, 12, 34, 36], "h_x": 71, "h_y": 71, "ha": [0, 2, 9, 12, 13, 25, 28, 29, 31, 33, 34, 40, 41, 51, 53, 63, 79, 82, 84, 86, 88, 91, 97, 98, 99, 100, 101, 104, 105, 106, 108, 111, 112, 114], "haas": 24, "hackbusch": [63, 64], "had": 73, "halv": 48, "hand": [0, 8, 10, 11, 21, 23, 24, 31, 34, 35, 36, 37, 41, 53, 64, 65, 73, 76, 82, 83, 89, 91, 98, 101, 102, 104, 105, 112, 121], "happen": [1, 86, 105], "hard": [40, 43, 105], "harmonic_extens": 91, "harmonic_extension_tran": 91, "hat": [0, 9, 13, 30, 34, 36, 51, 63, 66, 76, 81, 82, 83, 100, 106, 112, 121], "have": [1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 20, 29, 31, 36, 37, 38, 39, 40, 41, 46, 47, 48, 49, 51, 53, 54, 55, 58, 59, 61, 62, 63, 64, 65, 66, 69, 73, 76, 77, 79, 81, 82, 83, 84, 89, 90, 97, 99, 100, 101, 103, 104, 105, 106, 108, 109, 112, 118], "hcurl": [17, 62, 69, 70, 83], "hcurldiv": [18, 19], "hd": 8, "hdg": [2, 5, 7], "hdiv": [3, 7, 17, 18, 19, 33, 34, 35, 57, 86, 88, 89, 91], "hdivdiv": [62, 67, 69], "hdivdivsurfac": 70, "header_vis": 47, "heat": [45, 46, 73, 76, 78, 112, 114], "height": [23, 47, 109], "hellan": 45, "helling": [45, 62], "hello": 20, "help": [10, 19, 21, 26, 76, 85, 103, 121], "here": [1, 2, 4, 6, 10, 12, 24, 34, 40, 49, 56, 72, 73, 76, 88, 91, 100, 102, 113], "hermann": 105, "hermit": 9, "herrmann": 45, "hess": [3, 34], "hessenn": 3, "hessian": [3, 50, 84], "hesthaven": 121, "heun": 111, "heurist": 35, "hexhedra": 9, "hh": 108, "hhj": [45, 67], "hht": 45, "hidden_dof": [10, 18, 19, 76], "hidealldof": [10, 76], "hierarch": [63, 76, 88], "hierarchi": 63, "high": [4, 10, 24, 42, 53, 64, 76, 78, 88, 115], "higher": [4, 5, 6, 8, 12, 73, 76, 91, 100, 118, 121], "highest_order_dc": 1, "highli": [0, 33, 64, 78, 121], "hilbert": [8, 12, 27, 28, 29, 30, 31, 32, 36, 53, 54, 56, 65, 76, 79, 83, 99, 100, 101, 104, 113], "hint": [30, 102, 108], "hinv": 42, "hist": [33, 35, 49], "histori": 112, "hline": 111, "hmat": 42, "hochbruck": 115, "hochsteg": 25, "hold": [1, 8, 9, 12, 27, 28, 29, 31, 32, 33, 34, 41, 49, 50, 51, 53, 63, 64, 65, 66, 74, 76, 79, 81, 83, 90, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 109, 112], "hole": [35, 118], "holidai": 26, "holmholtz": 43, "home": 25, "homogen": [0, 33, 36, 41, 55, 73, 76, 82, 115], "honeydew": 20, "hood": 58, "hook": 74, "hope": [33, 103], "how": [0, 10, 20, 23, 30, 40, 47, 48, 56, 73, 77, 102], "how_to": 75, "howev": [1, 6, 41, 48, 51, 56, 57, 63, 64, 65, 82, 83, 84, 100, 103, 109, 118], "howto_linalg": 75, "howto_numpi": 75, "hp": [8, 46, 76, 103], "hpp": [10, 80], "hpref": 8, "html": [14, 25, 52, 75], "http": [14, 20, 23, 25, 46, 52, 65, 75, 105, 111, 118, 119, 120], "hu": 67, "huang": 67, "hub": [14, 15, 20, 21, 22, 23, 24], "huge": 121, "hv": [42, 48, 51, 81, 116, 121], "hv1": 42, "hv1glob": 42, "hv2": [42, 48, 51], "hv2glob": 42, "hv3": 51, "hvstab": 119, "hx": 63, "hy": 109, "hybrid": [45, 46, 54, 67, 72], "hyperbol": [12, 114], "hypr": 20, "i": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112, 113, 114, 115, 118, 119, 120, 121], "i_": [8, 9, 88, 105], "i_1": [8, 41], "i_2": 41, "i_h": [8, 12, 36, 54, 67, 68, 75, 83, 86, 109], "i_m": 8, "i_p": 79, "i_t": [8, 9, 12, 88], "id": [0, 8, 10, 14, 15, 20, 21, 23, 24, 42, 55, 70, 74, 76, 79, 100, 101, 105], "idea": [34, 36, 37, 40, 49, 65, 81, 91, 100, 101, 115, 118], "ideal": [51, 105], "ident": [48, 79, 81, 115], "identif": [10, 76], "identifi": [21, 24, 34, 38, 98, 105], "identitymatrix": [38, 39, 47, 82], "idiag": 108, "idnr": 76, "ifem": [25, 46, 72], "iff": [90, 109, 112], "ifpo": [4, 5, 6, 108], "ignor": 100, "ii": [34, 51, 105], "iii": [34, 46], "ij": [9, 38, 40, 42, 51, 79, 90, 100, 111], "il": 111, "ill": 115, "imagin": 55, "imaginari": 112, "imex": 5, "immedi": [31, 37, 49, 84, 101, 105, 106, 108, 109, 111], "impi_rt": 20, "implement": [0, 4, 12, 20, 26, 39, 46, 63, 66, 88, 105, 109, 110, 112, 115], "impli": [29, 30, 31, 32, 53, 55, 79, 83, 84, 99, 100, 101, 105, 106], "implicit": [5, 115], "import": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 28, 33, 34, 35, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 99, 103, 106, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121], "improv": [28, 31, 33, 46, 49, 51, 64, 65, 83, 92, 100, 108, 110, 111, 115], "includ": [10, 20, 21, 61, 73, 76, 78, 100, 110, 113], "inclus": [12, 67], "incompress": [12, 28, 58, 84], "inconsist": [38, 39], "increas": [8, 49, 64, 70, 79, 103, 114], "increment": [47, 113], "ind": 23, "inde": [33, 41, 79, 100, 101, 104, 105], "indefinit": 81, "independ": [0, 9, 12, 27, 40, 64, 79, 83, 100, 102, 106, 109, 121], "index": [8, 47, 75, 98, 121], "indexset": 23, "indi": 42, "indic": [23, 76, 84, 106], "individu": [8, 10, 40, 65, 74, 76, 100, 105, 108, 111], "indj": 42, "induc": [0, 28, 84, 106], "induct": [47, 64], "inequ": [12, 27, 30, 33, 53, 54, 64, 66, 71, 79, 81, 83, 86, 100, 101, 105, 106, 109], "inexact": [12, 82], "inf": [6, 45, 46, 53, 54, 86], "inf_": [8, 12, 28, 30, 31, 32, 37, 41, 48, 53, 54, 62, 65, 67, 79, 83, 100, 101, 105, 106, 108, 109], "infimum": [28, 83, 106], "infinit": [76, 87, 100, 101], "inflow": [6, 16, 18, 19], "inform": [8, 40], "infti": [41, 65, 79, 90, 98, 99, 100, 101, 102, 104, 109, 112], "inherit": [10, 28, 54, 76, 112], "initi": [4, 33, 47, 48, 112, 113, 114, 115], "inject": [6, 30, 31, 100], "inlet": [7, 12, 16, 18, 19, 22, 81, 85], "inlin": [64, 115], "inner": [0, 3, 12, 21, 27, 28, 29, 33, 35, 36, 48, 49, 51, 52, 55, 64, 65, 76, 79, 80, 83, 84, 98, 99, 101, 102, 103], "inner_solv": 91, "innerdof": [38, 39], "innerproduct": [3, 7, 12, 18, 19, 21, 22, 23, 24, 35, 42, 47, 49, 50, 51, 56, 58, 62, 66, 67, 69, 70, 74, 75, 81, 82, 85, 103, 106, 108, 115], "input": [10, 21, 33, 42, 64, 76], "insert": [8, 12, 36, 47, 56, 79, 84, 105, 108, 113, 114], "insid": [76, 100, 108, 109], "inspect": [76, 121], "instabl": 8, "instal": [25, 72, 103], "instanc": [10, 20, 76, 80], "instationari": [45, 46], "instead": [28, 30, 33, 34, 46, 47, 54, 55, 56, 76, 81, 82, 83, 100], "institut": 46, "insul": 73, "insur": 31, "int": [3, 6, 10, 12, 27, 28, 34, 35, 39, 41, 42, 53, 55, 56, 57, 58, 60, 61, 62, 67, 69, 73, 76, 79, 83, 84, 86, 87, 89, 91, 99, 102, 107, 113, 114, 118], "int32": 23, "int_": [0, 1, 2, 3, 4, 6, 8, 11, 12, 13, 28, 29, 30, 34, 35, 36, 38, 40, 41, 42, 43, 51, 53, 54, 55, 56, 57, 62, 65, 67, 70, 73, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 87, 88, 89, 90, 91, 96, 97, 98, 99, 100, 101, 102, 104, 106, 107, 111, 112, 113, 121], "int_0": [30, 41, 65, 79, 92, 97, 98, 100, 111], "int_a": 112, "int_i": [41, 79], "int_t": [1, 2, 6, 11, 12, 13, 34, 36, 54, 62, 67, 79, 86, 87, 88, 91, 112, 121], "int_x": 102, "integ": [55, 99], "integr": [0, 1, 2, 6, 8, 10, 12, 13, 15, 24, 33, 34, 35, 36, 45, 54, 55, 57, 65, 75, 76, 79, 80, 84, 88, 89, 90, 91, 92, 96, 97, 98, 103, 106, 111, 113, 118, 121], "integrand": [13, 79], "integrationrul": [76, 118], "intelmpi": 20, "intent": 101, "inter": [38, 39], "interact": [37, 47], "interconnect": 38, "interest": [8, 35, 51, 56, 69, 100], "interfac": [15, 20, 23, 25, 33, 38, 40, 42, 44, 45, 46, 61, 76, 78, 90, 100, 107, 108], "interface_dof": [18, 19], "interior": [1, 2, 66, 104, 118], "interleav": 72, "intern": [1, 2, 38, 69, 73, 74, 76], "interplai": 64, "interpol": [8, 9, 12, 17, 33, 36, 46, 54, 56, 65, 66, 73, 74, 75, 76, 83, 86, 88, 109, 118, 121], "interpret": [24, 35, 51, 62, 84], "intersect": [46, 99, 101], "intertest": [38, 39], "intertri": [38, 39], "interv": [9, 10, 41, 47, 76, 98, 100, 102, 111, 112, 113], "intord": [0, 55], "intpoint": 121, "intrang": [10, 18, 19, 76, 109], "intro": 25, "introduc": [0, 1, 2, 39, 47, 51, 53, 55, 56, 57, 61, 64, 79, 83, 88, 89, 100, 104, 111, 112, 114], "introduct": 72, "intrul": 118, "intuit": 65, "inv": [1, 2, 5, 14, 21, 38, 39, 42, 44, 63, 66, 73, 74, 77, 83, 108, 110, 121], "inva": [18, 19], "inva0": 108, "invent": 37, "invers": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 17, 18, 22, 33, 34, 35, 37, 38, 39, 42, 43, 44, 51, 52, 55, 56, 57, 58, 62, 63, 65, 66, 67, 69, 73, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 91, 92, 96, 97, 105, 106, 108, 109, 113, 114, 115, 116, 118, 121], "invert": [4, 8, 9, 37, 39, 56, 63, 73, 83, 91, 100, 108, 109, 121], "investig": 97, "invglob": 42, "invi": [108, 109], "invloc": 42, "invm": [4, 10, 76], "invmass": 116, "invmassbnd": [38, 39], "invmassp": 121, "invmassu": 121, "involv": [1, 2, 28, 63, 83], "invp": [119, 120, 121], "invstok": 7, "invu": [119, 120, 121], "io": [14, 25, 46], "ip": 24, "ipynb": [25, 46], "ipyparallel": [14, 15, 16, 18, 19, 21, 22, 23, 24, 42], "ipywidget": 47, "irregular": 33, "is_complex": [10, 76], "iset": 23, "isinst": 121, "isomorph": [28, 53, 105], "isomorphismu": 105, "isotrop": [10, 76], "issu": [26, 46, 60, 78, 89, 115], "iter": [10, 14, 22, 28, 37, 38, 39, 49, 50, 63, 65, 70, 76, 81, 82, 83, 85, 91, 105, 108, 110], "iteraton": 21, "its": [3, 8, 9, 10, 15, 20, 23, 27, 28, 32, 33, 34, 36, 40, 41, 46, 48, 50, 51, 55, 64, 65, 76, 83, 88, 90, 99, 100, 103, 105, 109, 115], "itself": [8, 15, 20, 105, 114], "itutori": 11, "j": [1, 5, 9, 13, 20, 30, 31, 34, 37, 38, 39, 42, 46, 48, 49, 51, 64, 67, 76, 79, 80, 83, 88, 99, 100, 105, 108, 109, 111, 112, 113, 114, 115, 118, 119, 120, 121], "j_1": 8, "j_m": 8, "j_t": [9, 106], "j_v": 28, "jaboci": 76, "jac": [105, 106], "jacobi": [13, 21, 47, 63, 64, 65, 79, 82, 83, 106, 110], "jacobian": [88, 112, 121], "ji": [13, 105, 111], "jinchao": [65, 105], "joachim": [25, 45, 46, 72], "johnson": [45, 46], "joint": 70, "joli": 118, "jschoeberl": [25, 46], "jump": [2, 34, 36, 87, 91, 121], "jumpdn": 3, "jupyt": [20, 25, 46, 72], "jupyterhub": 46, "jupyterlite_ngsolv": 25, "just": [2, 24, 25, 39, 60, 66, 73, 76, 84, 87, 121], "k": [1, 8, 13, 27, 28, 34, 41, 47, 48, 49, 50, 51, 58, 65, 66, 67, 79, 81, 84, 86, 87, 88, 91, 98, 99, 100, 101, 102, 104, 112], "kapidani": [119, 120], "kappa": [47, 48, 49, 63, 65, 83, 84, 106, 108, 109], "kappa_a": 105, "karush": [53, 84], "keep": [0, 15, 39, 55, 112], "kei": 12, "kept": 15, "ker": [27, 29, 82, 83, 101], "kernel": [27, 36, 40, 53, 54, 67, 82, 100, 101, 103, 106], "keyboardinterrupt": [119, 121], "keyword": [10, 76], "kind": [20, 41, 47, 73, 75, 79, 83, 89, 113], "kink": 76, "kirchhoff": [3, 69], "kkt": [53, 84], "kl": [79, 104], "know": [15, 20, 23, 33, 40, 41, 66, 76, 101, 105, 118], "knowledg": 8, "known": [28, 33, 46, 49, 61, 67, 69, 76, 91, 112, 113], "korn": 71, "krendl": 67, "krylov": [14, 84], "krylovsolv": 18, "krylovspac": [14, 18, 19, 21, 22, 38, 39, 42, 63, 74, 77, 82, 83, 84, 85, 108, 110], "ksp": [14, 23], "ksp_monitor": 18, "ksp_rtol": 18, "ksp_type": 18, "kuhn": [53, 84], "kutta": [4, 5, 45], "kutta_method": 111, "kwarg": [10, 55, 76, 121], "kx": [41, 104], "l": [8, 11, 12, 14, 15, 18, 19, 23, 27, 28, 29, 31, 46, 47, 51, 53, 63, 64, 65, 66, 68, 77, 79, 83, 84, 86, 89, 99, 100, 101, 103, 104, 105, 109, 111, 112, 118], "l2": [1, 2, 4, 5, 6, 7, 10, 12, 15, 18, 19, 34, 56, 57, 58, 76, 81, 86, 89, 91, 108, 109, 121], "l2gammanorm": 103, "l2norm": [92, 96, 103], "l2space": [10, 76], "l_": [80, 98, 99, 100, 101, 104, 109], "l_1": [98, 99], "l_2": [1, 4, 6, 12, 27, 28, 30, 31, 33, 34, 36, 40, 41, 45, 46, 53, 55, 56, 57, 58, 60, 61, 62, 64, 65, 66, 67, 69, 71, 75, 76, 83, 86, 89, 90, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 108, 109, 113], "l_p": 99, "l_u": 29, "la": [10, 42, 47, 63, 76, 81, 83, 106, 108, 109, 110], "lab": [25, 46], "label": [34, 36, 47, 79, 108], "lagrang": [8, 9, 38, 55, 61, 82, 83, 84, 91, 105], "lagrangian": [9, 45, 53, 68], "lam": [33, 34, 35, 38, 39, 42, 47, 55, 56, 63, 74, 78, 81, 83, 84, 89, 103, 106, 109, 110, 115, 121], "lam1": 92, "lam2": 106, "lambda": [0, 1, 13, 27, 34, 35, 38, 41, 47, 48, 49, 51, 55, 56, 64, 74, 78, 84, 86, 89, 92, 96, 97, 100, 101, 103, 105, 112, 121], "lambda_": [28, 34, 40, 80, 108], "lambda_0": [80, 100], "lambda_1": [27, 28, 48, 80, 103], "lambda_2": [28, 48, 55, 80, 103], "lambda_i": [47, 48, 51, 64, 80, 112], "lambda_j": 48, "lambda_k": [41, 100], "lambda_l": 78, "lambda_n": [27, 48], "lambda_r": 78, "lambda_t": 34, "lami": 108, "lammax": [22, 63, 81, 83, 85, 106, 109, 110], "lammin": [22, 63, 81, 83, 85, 103, 106, 109, 110], "lanczo": 47, "langer": 24, "languag": [10, 20], "laplac": [41, 64, 108], "larg": [0, 1, 2, 8, 30, 31, 33, 51, 52, 56, 64, 76, 77, 78, 83, 108, 112, 118], "larger": [15, 37, 40, 55, 92, 100], "largest": [13, 33, 47, 48, 51, 97, 103, 106], "last": [8, 10, 34, 35, 40, 49, 51, 58, 76, 79, 84, 97, 109, 119, 120, 121], "latenc": 40, "later": [6, 12, 29, 48, 54, 76, 84, 100, 112], "latest": [14, 72, 75], "launch": 25, "launcher": [14, 15, 20, 21, 22, 23, 24], "law": 74, "lax": [28, 30, 53, 101, 104], "layer": [12, 108], "lazarov": 1, "lbb": [53, 61, 84, 87], "ldot": [8, 9, 13, 30, 34, 47, 48, 49, 50, 51, 63, 66, 76, 79, 90, 98, 100, 104, 105, 111, 112, 113, 115], "lead": [1, 2, 5, 6, 8, 9, 12, 28, 29, 39, 48, 52, 53, 56, 63, 67, 79, 81, 83, 84, 88, 89, 91, 97, 99, 105, 106, 108, 111, 112], "learn": [23, 25, 52], "least": [40, 50, 54, 110], "leav": 78, "lebesgu": 27, "lectur": [6, 10, 25, 46, 65, 72, 81, 105], "left": [1, 2, 3, 5, 6, 8, 10, 12, 14, 24, 28, 30, 31, 34, 36, 38, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 61, 62, 64, 65, 66, 68, 69, 70, 73, 76, 78, 79, 80, 81, 82, 83, 84, 89, 90, 91, 92, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 110, 112, 113, 114, 121], "leftdomain": 44, "leftrightarrow": [13, 27], "legend": 47, "legendr": [76, 80], "lehrenfeld": [1, 7, 72], "lemma": [8, 9, 27, 28, 31, 32, 36, 37, 45, 63, 64, 65, 79, 83, 100, 101, 106, 108, 109, 112], "lemma_l2est": 79, "lemma_ortho": 79, "lemma_polext": 79, "lemma_sturmliouvil": 79, "len": [15, 35, 38, 39, 42, 50, 66, 108, 118, 121], "length": [33, 100], "leq": [1, 8, 9, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 47, 48, 49, 50, 51, 53, 54, 61, 64, 65, 66, 67, 71, 79, 80, 81, 83, 84, 86, 90, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 112, 115], "less": [34, 73, 98], "let": [8, 12, 20, 25, 27, 28, 29, 30, 31, 32, 33, 36, 37, 40, 41, 47, 48, 49, 50, 55, 62, 63, 64, 65, 66, 76, 79, 81, 82, 83, 84, 86, 92, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 109, 110, 113], "level": [8, 40, 46, 54, 64, 76, 103], "leveldof": 63, "lewi": 112, "lf": [1, 34], "lfequ": 34, "lgmap": 23, "lh": 79, "li": 100, "lib": 121, "libngpi": 76, "librari": [10, 11, 15, 23, 25, 72, 80, 121], "lift": 37, "like": [1, 2, 10, 15, 20, 24, 38, 40, 46, 50, 51, 61, 64, 73, 74, 76, 91, 109], "lim_": [2, 99, 112], "limit": [27, 41, 56, 69, 97, 100, 106], "linalg": 115, "line": [8, 9, 31, 32, 33, 35, 47, 50, 58, 87, 119, 120, 121], "linear": [1, 2, 5, 8, 9, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 35, 45, 47, 49, 50, 51, 52, 53, 56, 57, 63, 69, 70, 73, 77, 82, 84, 88, 89, 98, 100, 101, 102, 104, 105, 109, 111, 113, 114, 115, 121], "linearform": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 17, 18, 21, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 67, 69, 73, 74, 76, 77, 78, 80, 82, 83, 84, 86, 89, 91, 96, 108, 110, 113, 114], "linearli": 9, "linerar": 82, "linestyl": 47, "link": 64, "linspac": [47, 64, 112], "linux": 20, "lion": 105, "liouvil": 79, "lipschitz": [28, 92, 95, 97, 99, 100, 101], "list": [10, 15, 20, 23, 37, 38, 39, 42, 47, 76, 92, 96, 97, 103, 106, 108, 109, 110], "list_of_rung": 111, "lite": [25, 46], "literatur": [34, 36, 64, 65, 105, 121], "live": [55, 76], "ll": [33, 112], "lm": 111, "load": [10, 11, 40, 70, 74, 76, 80, 82], "loadstep": 70, "lobpcg": [92, 103], "loc": [98, 99], "local": [8, 9, 10, 12, 13, 15, 21, 23, 24, 33, 34, 35, 36, 37, 40, 42, 45, 46, 59, 63, 64, 66, 76, 77, 79, 81, 83, 98, 99, 100, 109, 121], "local2glob": 23, "local_mat": [23, 42], "local_vec": [24, 42], "localdof": 118, "localhcf": 76, "localip": 24, "localpr": 63, "locat": 9, "lock": [69, 83], "locmat": 23, "loembed": [10, 76], "log": [33, 47, 48, 49, 50, 51, 65, 80, 115], "logic": 108, "login": 46, "logo": 25, "long": [25, 40, 108], "longrightarrow": [68, 86], "look": [10, 11, 24, 50, 56, 73, 74, 76, 83, 100, 103], "loop": [1, 2, 6, 33, 51], "lospac": [10, 76], "lot": 8, "low": [64, 83, 115], "low_order_spac": [10, 76], "lower": [33, 34, 40, 51, 81, 86, 89, 103, 108, 109, 111], "lowest": [10, 67, 76, 87, 88, 91, 103], "lshape": 33, "lsrc": [119, 120], "lu": 101, "lucki": 86, "lump": [12, 45, 46], "lv": 101, "ly": 104, "m": [4, 5, 8, 9, 12, 13, 15, 18, 19, 20, 25, 27, 32, 46, 48, 51, 64, 70, 72, 79, 82, 83, 84, 96, 100, 103, 104, 105, 106, 108, 111, 112, 113, 114, 115, 118, 121], "m_": [51, 64, 105], "m_0": 64, "m_l": [63, 64, 66], "m_p": 121, "m_u": 121, "m_x": 108, "m_y": 108, "machin": 55, "maco": 20, "magic": [20, 67], "magnet": 83, "magnitud": [40, 109], "mai": [1, 2, 8, 9, 12, 20, 25, 33, 34, 40, 46, 58, 79, 83, 103], "mail": 46, "main": [8, 34], "maintain": 73, "major": 79, "make": [0, 8, 20, 62, 76, 78, 100, 112], "makebfi": 11, "makechamf": 74, "makestructured2dmesh": [92, 96, 97], "maketwomaterialmesh": 33, "manag": [20, 46], "mani": [20, 23, 46, 56, 64, 76, 82, 84, 105], "manifold": 104, "manual": 39, "map": [0, 8, 13, 14, 23, 27, 28, 38, 47, 60, 67, 76, 83, 88, 92, 96, 97, 100, 101, 102, 104, 105, 106, 121], "mappedintegrationpoint": 76, "mapsto": [27, 30, 53, 63, 75, 82, 88, 97, 102, 105, 109], "maptoallel": 76, "marini": [34, 88], "mark": [73, 76, 108], "mark_surface_el": 76, "markers": [1, 2], "mask": [0, 39, 78, 109], "mass": [4, 10, 12, 46, 66, 76, 112, 113, 116, 121], "massbnd": [38, 39], "massint": [38, 39], "master": [1, 7, 15, 72], "master_nr": 76, "masterinvers": 18, "mat": [0, 1, 2, 3, 5, 6, 7, 10, 11, 12, 14, 17, 18, 19, 21, 22, 23, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "mat00": [38, 39, 108], "mat01": [38, 39, 108], "mat02": [38, 39, 108], "mat10": [38, 39, 108], "mat11": [38, 39, 108], "mat12": [38, 39, 108], "mat20": [38, 39, 108], "mat21": [38, 39, 108], "mat22": [38, 39, 108], "mata": [22, 85], "match": [76, 100, 111], "materi": [0, 10, 25, 33, 37, 38, 39, 45, 46, 55, 72, 74, 75, 76, 78, 108, 118], "materialcf": [0, 33, 34, 35, 76, 78, 83, 89], "math": [43, 44, 64, 83, 119], "mathbb": [6, 8, 9, 12, 13, 27, 28, 30, 31, 32, 35, 36, 37, 48, 49, 50, 51, 52, 58, 63, 66, 74, 75, 76, 78, 79, 81, 82, 84, 88, 92, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 111, 112, 115, 121], "mathbf": [41, 84, 106], "mathcal": [34, 49, 65, 68, 81, 105], "mathemat": [5, 34, 46, 76, 103], "mathematik": 83, "matlab": 25, "matplotlib": [1, 2, 33, 46, 47, 48, 49, 50, 51, 64, 80, 103, 112, 115], "matric": [10, 11, 12, 14, 38, 51, 52, 53, 60, 64, 66, 76, 77, 103, 105, 112, 113, 115, 121], "matrix": [0, 1, 3, 4, 8, 9, 10, 12, 13, 14, 21, 23, 27, 30, 37, 38, 39, 42, 48, 49, 50, 51, 52, 53, 56, 61, 62, 63, 64, 66, 67, 73, 76, 77, 81, 82, 83, 84, 91, 96, 105, 106, 108, 109, 111, 112, 113, 115, 118, 119, 120, 121], "matrixvalu": 74, "max": [4, 5, 6, 7, 12, 15, 16, 18, 19, 22, 28, 33, 34, 35, 42, 47, 48, 74, 78, 81, 84, 85, 89, 106, 108, 109, 119, 120], "max_": [8, 47, 48, 64, 99], "max_i": 48, "max_it": [14, 23], "maxh": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "maxim": [27, 64, 74, 103, 109], "maximum": [47, 48], "maxit": [18, 19, 22, 38, 39, 47, 49, 77, 81, 82, 83, 84, 85, 92, 96, 97, 103, 110, 121], "maxstep": [84, 91], "maxwel": [34, 119, 120], "mayb": 55, "mbox": [8, 9, 12, 13, 27, 28, 31, 33, 35, 53, 98, 100, 104], "mc": 23, "mdcomp": 108, "mean": [2, 12, 29, 31, 36, 40, 42, 46, 48, 51, 54, 56, 58, 60, 76, 79, 82, 84, 91, 100, 103, 105, 106, 108, 112, 113, 121], "meanvalmat": 96, "meanvalu": 96, "measur": [8, 12, 33, 50, 51, 92, 101, 104], "mechan": [34, 46, 56, 103], "mechanid": 45, "medium": 44, "melon": 20, "member": 15, "membership": 108, "membran": 70, "memori": [40, 77, 121], "mesh": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 16, 17, 18, 19, 21, 22, 23, 24, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 100, 103, 106, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "mesh_siz": [0, 1, 2, 3, 5, 7, 22, 85], "meshgrid": 112, "meshnod": 76, "meshnoderang": 76, "meshpoint": 76, "meshrot": [0, 55], "messag": 20, "methdo": 68, "method": [1, 4, 10, 28, 33, 34, 35, 37, 39, 40, 49, 56, 62, 64, 65, 66, 69, 70, 72, 74, 77, 79, 82, 84, 86, 89, 91, 101, 110, 115, 119, 120, 121], "meti": [15, 108], "mform": [113, 114, 115, 118], "mg": [45, 63], "mgprecondition": 63, "michael": 59, "mid": 12, "middl": 33, "midpoint": 9, "might": 46, "milgram": [28, 30, 53, 101, 104], "milk": 6, "mimic": [6, 87], "min": [4, 5, 6, 7, 8, 12, 15, 16, 18, 19, 22, 28, 33, 34, 42, 47, 74, 78, 81, 84, 85, 89, 103, 106, 119, 120], "min_": [30, 34, 47, 49, 50, 53, 65, 84, 86], "min_u": 84, "mindlin": 45, "mini": 20, "miniconda3": 20, "minim": [29, 32, 34, 37, 40, 41, 45, 46, 47, 48, 50, 60, 65, 79, 83, 100, 105, 112], "minimum": [32, 50, 103, 105], "minion_nr": 76, "minr": 84, "minut": [25, 46], "minv": 118, "minva": 118, "miracl": 49, "misfit": 48, "miss": [3, 40, 87], "mix": [12, 34, 56, 61, 67, 70, 72, 81, 83, 86, 87, 91], "mixedflux": 89, "mixedtemp": 89, "mixtur": 79, "ml": [45, 63], "mlextens": 66, "mlprecondition": 63, "mlpreconditioner2": 63, "mm": 105, "mmat": 118, "mmax": 115, "modal": 76, "mode": [62, 76], "model": [6, 58, 69, 70, 73, 76, 77, 78, 118], "moder": 99, "modern": 40, "modif": [12, 79], "modifi": [0, 41, 47, 48], "modul": [10, 14, 20, 76, 85], "mollifi": 90, "moment": 67, "momentum": 58, "mone": 66, "monoton": [48, 51], "more": [0, 1, 2, 5, 10, 31, 40, 49, 52, 63, 65, 66, 73, 76, 79, 98, 99, 100, 101, 105, 106, 109, 110, 112, 118, 121], "morlei": [9, 12, 67], "most": [47, 58, 99, 108, 111, 119, 120, 121], "motiv": [62, 100, 112], "motor": 0, "move": [8, 14, 50, 73, 89], "moveto": [0, 8, 33, 34, 35, 55, 78, 89, 108, 118, 121], "mp": 121, "mpi": [10, 14, 15, 16, 18, 19, 21, 22, 23, 24, 42, 46, 76], "mpi4pi": [15, 16, 18, 19, 21, 22, 24, 42, 46], "mpi_comm": 76, "mpiaij": 23, "mpich": 20, "mpienginesetlaunch": [14, 15, 20, 21, 22, 23, 24], "msg_level": [16, 18, 19], "msm": 105, "msmall": 115, "mstabinv": [119, 120], "mstar": [5, 113, 114, 115], "mstarinv": [113, 114, 115], "mtcurl": 17, "mu": [27, 34, 38, 39, 53, 55, 56, 74, 83, 84, 121], "mu0": 115, "mu_": [38, 40], "much": [33, 37, 40, 50, 67], "mulder": 118, "mult": [42, 63, 109], "multi": [10, 13, 40, 46, 63, 76, 98, 108], "multidim": [76, 108, 109, 121], "multigrid": [40, 76, 83, 84], "multilevel": [46, 83], "multipl": [34, 49, 51, 63, 79, 105], "multiplel": 15, "multipli": [1, 2, 6, 13, 52, 55, 57, 63, 66, 76, 80, 81, 82, 100, 105, 113], "multiscal": 83, "multistep": 45, "multivector": [42, 82, 108, 115], "multtran": 109, "must": [12, 33, 34, 40, 41, 53, 54, 58, 60, 77, 83, 84, 103, 111, 115], "mv": 108, "mx": [37, 38, 39, 66, 108], "my": [10, 37, 38, 39, 108], "myassembl": 11, "myassemblematrix": 11, "myassemblevector": 11, "mydiffop": 10, "myelement": 10, "myfe": 10, "myfespac": 10, "myhighorderfespac": 80, "myhoel": 80, "myhof": 80, "myhofespac": 80, "mylap": 11, "mylaplac": 11, "mymatrix": 11, "mymodul": 10, "myneumannintegr": 11, "mysourc": 11, "n": [0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 31, 32, 34, 36, 40, 41, 43, 44, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 62, 63, 65, 67, 68, 69, 70, 73, 74, 76, 78, 79, 80, 82, 84, 87, 89, 90, 91, 96, 97, 98, 99, 100, 103, 104, 106, 108, 112, 113, 115, 116, 118, 121], "n2p": [14, 22], "n4ngla18embeddingtranspose": 120, "n4ngla9embedding": 120, "n6ngcomp9applymass": 120, "n_": [8, 63, 108], "n_1": [2, 36, 55, 91], "n_2": [36, 55, 91], "n_cut": 108, "n_e": 3, "n_i": [47, 90, 100, 105], "n_j": [90, 100], "n_l": [63, 66, 121], "n_r": 121, "n_t": [9, 13, 106], "nabla": [0, 1, 2, 3, 5, 6, 11, 12, 13, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 49, 50, 51, 53, 55, 56, 57, 58, 61, 62, 65, 67, 68, 69, 70, 73, 74, 75, 76, 78, 79, 83, 84, 86, 87, 88, 89, 90, 92, 95, 96, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 112, 113, 121], "nabla_": 100, "nabla_h": 6, "nabla_t": [62, 67], "nabla_x": 100, "naiver": 101, "name": [0, 7, 8, 10, 12, 13, 16, 18, 19, 22, 31, 33, 34, 35, 38, 55, 62, 67, 69, 74, 75, 76, 77, 78, 81, 85, 89, 101, 103, 105, 106, 108, 118, 120], "nameerror": 120, "nano": 46, "natur": [8, 24, 36, 46, 55, 57, 58, 62, 89, 110], "navier": [5, 12, 58], "navierstok": [16, 18, 19], "navierstokessimpl": [16, 18, 19], "navstok": 16, "nb": 108, "nbel": 108, "nbextens": 46, "nc": [12, 109], "nd": 101, "ndarrai": 76, "ndglob": 23, "ndof": [1, 2, 10, 11, 15, 18, 19, 22, 23, 33, 35, 38, 39, 42, 55, 58, 62, 63, 67, 76, 77, 80, 83, 85, 88, 108, 109, 121], "ndof_p": 121, "ndof_u": 121, "ndofglob": [10, 15, 22, 23, 76], "ndofloc": 42, "ndom": 108, "ne": 76, "nearest": 8, "nearli": [62, 112], "necessari": [31, 37, 101, 112], "necessarili": [27, 28, 30, 61, 98, 105], "neck": 40, "nedelec": [62, 68, 69, 83], "nedg": [76, 88], "need": [1, 3, 8, 10, 20, 24, 34, 36, 38, 47, 48, 49, 50, 51, 53, 54, 57, 58, 70, 76, 83, 88, 90, 91, 97, 98, 101, 103, 109, 111, 113, 114, 121], "neg": [6, 29, 49, 50, 56, 84, 101, 104, 108, 112], "neglect": 8, "neighbor": [12, 33, 36, 40, 108], "neighbour": [38, 39, 79], "neighbourhood": [98, 100], "nel": 70, "nepomnyashchikh": 105, "neq": [9, 27, 28, 29, 31, 32, 36, 49, 51, 52, 79, 84, 98, 100, 101, 103, 109], "ner": 118, "nest": [63, 64, 65, 66, 83, 108, 109], "netgen": [0, 5, 7, 8, 12, 15, 16, 18, 19, 22, 33, 34, 35, 37, 38, 39, 42, 43, 44, 46, 47, 55, 62, 63, 67, 69, 70, 72, 74, 76, 77, 78, 81, 82, 83, 85, 89, 103, 108, 113, 114, 115, 118, 121], "network": [40, 52], "neumann": [1, 28, 40, 41, 55, 57, 73, 78, 89, 91, 104, 108], "neunteufel": [59, 70], "never": [24, 46], "new": [3, 4, 10, 25, 33, 36, 46, 47, 48, 49, 55, 56, 57, 61, 76, 91, 112, 113, 121], "new_scen": 121, "newblock": [], "newer": 67, "newlin": 33, "newton": 70, "newtonminim": 70, "newtonsolv": 70, "next": [2, 6, 8, 10, 14, 23, 28, 29, 38, 48, 50, 53, 65, 66, 79, 81, 82, 87, 88, 98, 100, 105, 113], "nface": 76, "nfacet": 76, "ng": [10, 11, 72, 121], "ngcomp": [10, 76], "ngexcept": 58, "nglob": 23, "ngmesh": [14, 15, 22, 23, 76, 77, 83], "ngmg": [10, 76], "ngradu": 70, "ngradv": 70, "ngs2petsc": [14, 22], "ngs_element": 76, "ngs_object": [10, 76], "ngsglobal": [16, 18, 19], "ngshub_xx": 46, "ngsolv": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 117, 118, 119, 120, 121], "ngspetsc": [14, 18, 19, 22], "ngstd": [10, 76], "ni": [10, 13, 76], "nice": 108, "nitsch": [8, 45, 46, 64, 65, 79], "nn": [3, 67], "nnode": 76, "nodal": [8, 9, 10, 12, 13, 36, 66, 68, 76, 100, 109, 118, 121], "nodal_p1": 121, "node": [9, 10, 15, 33, 40, 73, 76, 110, 118], "node_typ": 76, "nodeid": [10, 76, 88], "non": [0, 1, 2, 5, 8, 9, 10, 28, 29, 31, 36, 37, 38, 53, 54, 56, 60, 62, 63, 64, 70, 73, 76, 84, 90, 91, 99, 100, 101, 104, 105, 106, 108, 109, 111, 112, 121], "nonassembl": [4, 5], "nonconform": 12, "none": [10, 15, 18, 20, 37, 38, 39, 47, 49, 63, 70, 76, 84, 108, 109, 121], "nonlinear": 45, "norm": [0, 1, 6, 12, 16, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 45, 46, 48, 50, 51, 52, 53, 62, 64, 65, 66, 74, 76, 79, 83, 84, 85, 87, 90, 96, 97, 99, 100, 104, 108, 109, 115, 116], "norm_natur": [14, 23], "normal": [0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 18, 19, 33, 34, 36, 40, 54, 55, 57, 60, 67, 69, 70, 84, 88, 89, 91, 92, 96, 97, 100, 103, 111, 116, 121], "normalfacetfespac": 3, "normtyp": [14, 23], "notat": [2, 28, 55, 63, 65, 73, 105, 109], "note": [3, 6, 20, 25, 31, 34, 35, 46, 48, 65, 72, 76, 79, 89, 103], "notebook": [20, 25, 26, 46, 72], "noth": [25, 73], "now": [0, 1, 5, 6, 8, 10, 12, 23, 24, 27, 28, 29, 30, 31, 34, 36, 41, 47, 51, 53, 54, 55, 56, 57, 62, 64, 65, 66, 69, 73, 76, 80, 81, 82, 83, 84, 86, 89, 90, 91, 100, 101, 103, 104, 106, 109, 110, 111, 112, 114], "np": [23, 47, 76, 112, 121], "nr": [15, 24, 37, 38, 39, 42, 108, 109], "nsurf": 70, "nsurfmat": 70, "nt": [62, 67], "nu": [7, 16, 18, 19, 74, 83], "null": [53, 82, 83, 99, 106], "num": [0, 15, 88, 92, 96, 97, 103, 108, 121], "number": [8, 10, 13, 15, 20, 23, 24, 36, 38, 40, 45, 46, 47, 48, 49, 63, 64, 65, 76, 77, 79, 83, 84, 91, 99, 103, 105, 106, 108, 109, 110], "numberspac": 34, "numer": [5, 12, 13, 36, 46, 49, 52, 58, 76, 83, 84, 86, 103, 106, 111, 112, 115, 121], "numerisch": 83, "numpd": [46, 65], "numpi": [23, 46, 47, 64, 75, 76, 112], "numset": [108, 118], "nv": [10, 76, 109], "nx": [92, 96, 97], "ny": [92, 96, 97], "nze": [1, 2, 118], "o": [1, 52, 63, 79, 81, 83, 84, 101, 105, 106, 108], "o_i": 99, "oberl": [], "obj": 121, "object": [10, 14, 15, 20, 23, 76, 121], "observ": [0, 27, 31, 33, 34, 36, 47, 48, 50, 51, 55, 62, 69, 75, 79, 83, 84, 89, 90, 92, 98, 102, 103, 108, 109], "obtain": [8, 12, 13, 15, 27, 28, 31, 32, 33, 35, 38, 41, 47, 48, 49, 50, 51, 53, 55, 56, 58, 61, 63, 65, 66, 67, 76, 79, 81, 83, 84, 88, 89, 91, 97, 100, 101, 103, 106, 108, 109, 112, 113, 114, 115, 121], "obvious": 38, "occ": [0, 7, 8, 12, 16, 18, 19, 22, 33, 34, 35, 55, 74, 77, 78, 81, 85, 89, 103, 108, 113, 114, 115, 118, 121], "occgeometri": [0, 7, 8, 12, 16, 18, 19, 22, 33, 34, 35, 55, 74, 77, 78, 81, 85, 89, 103, 108, 118, 121], "occur": 33, "od": [112, 113, 114, 115, 121], "odd": 79, "off": [56, 100], "offer": 23, "offic": 25, "offici": 72, "often": [0, 12, 27, 28, 46, 47, 52, 63, 105, 108, 112, 113], "og": 121, "old": [33, 48, 113], "older": 67, "oldstyl": [10, 76], "omain": 37, "omega": [0, 1, 2, 4, 6, 8, 9, 11, 12, 13, 27, 28, 29, 30, 33, 34, 35, 36, 40, 41, 43, 44, 51, 53, 55, 57, 58, 61, 65, 66, 73, 74, 75, 76, 78, 84, 86, 87, 89, 90, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 112, 113, 114], "omega_": 36, "omega_1": [13, 41, 55, 90, 107], "omega_2": [41, 55, 90, 107], "omega_i": [0, 24, 38, 40, 42, 90, 100, 104, 108, 109], "omega_j": [40, 90, 100, 109], "omega_k": 13, "omega_m": 13, "omega_n": 90, "omega_t": [33, 36], "omega_v": 34, "omega_x": 36, "ompi": 20, "onc": [12, 73, 75], "one": [0, 1, 2, 6, 8, 9, 10, 12, 13, 15, 20, 24, 28, 30, 31, 32, 33, 34, 35, 36, 40, 42, 47, 48, 49, 50, 51, 52, 53, 54, 55, 58, 62, 63, 64, 65, 66, 76, 78, 79, 82, 83, 84, 88, 89, 99, 101, 104, 105, 106, 108, 109, 110, 111, 121], "ones": [9, 31, 64, 66, 73, 89, 108, 112], "onli": [0, 1, 6, 7, 8, 10, 12, 13, 15, 24, 27, 33, 36, 37, 38, 39, 40, 41, 48, 49, 52, 54, 55, 62, 64, 65, 70, 76, 79, 82, 83, 84, 98, 100, 106, 109, 112, 113, 121], "onlin": [25, 46], "onlyonc": 76, "onstraint": 37, "onto": [12, 27, 29, 31, 33, 38, 45, 46, 53, 66, 79, 83, 100, 101, 102, 105, 109], "oosterle": 63, "opcurl": [62, 69], "opdiv": [1, 4, 5, 6, 53, 62, 67, 68, 69, 78, 86, 87, 88, 89, 90, 91], "opdiv_t": 62, "open": [20, 46, 76, 98, 99, 102, 109], "openmpi": 20, "openmpi4": 20, "oper": [1, 2, 3, 4, 8, 9, 10, 12, 15, 20, 21, 27, 28, 30, 31, 32, 34, 40, 41, 42, 45, 48, 51, 52, 53, 56, 61, 63, 64, 65, 66, 76, 79, 83, 84, 86, 90, 96, 97, 98, 101, 103, 104, 105, 108, 109, 118], "operatornam": [7, 8, 12, 13, 24, 27, 28, 29, 31, 34, 38, 41, 46, 49, 53, 54, 56, 57, 58, 59, 61, 62, 63, 66, 74, 76, 82, 83, 84, 86, 89, 97, 98, 100, 101, 103, 104, 105, 106, 108, 109, 112, 121], "oplu": 31, "opposit": [21, 29, 33, 51, 55, 57], "opt": [47, 48, 50], "optic": 46, "optim": [8, 28, 31, 34, 40, 47, 50, 51, 63, 66, 77, 79, 81, 83, 86], "option": [3, 10, 52, 55, 76, 103], "order": [0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 33, 34, 35, 40, 43, 44, 47, 48, 49, 50, 51, 55, 56, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 96, 97, 99, 100, 101, 103, 105, 106, 108, 109, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "order_equ": 34, "order_polici": [10, 76], "orderfac": 55, "orderinn": [17, 18, 19], "ordinari": [111, 112, 113], "org": [20, 52, 65, 72, 75, 105, 111, 118, 119, 120], "orient": 3, "origin": [28, 30, 31, 32, 34, 37, 49], "orthobrick": 83, "orthogon": [4, 8, 12, 27, 28, 29, 31, 32, 34, 36, 41, 49, 53, 54, 64, 65, 83, 86, 100, 101, 102, 103, 105, 115, 121], "orthonorm": [41, 47, 100, 101], "oscil": [0, 12, 64, 112], "ostermann": 115, "ot": 60, "other": [0, 2, 4, 5, 6, 10, 15, 21, 23, 24, 31, 37, 46, 51, 52, 54, 55, 63, 65, 73, 75, 76, 80, 83, 103, 105, 106, 108, 121], "otherp": 15, "otherwis": [29, 34, 58, 98, 106], "our": [10, 23, 46, 53, 54, 80, 100], "out": [1, 3, 6, 25, 37, 47, 64, 77, 80, 83, 90, 98, 121], "outer": [0, 33, 35, 55, 74, 77, 89, 100], "outflow": [6, 16, 18, 19, 57], "outlet": [7, 12, 16, 18, 19, 22, 81, 85], "outlin": [65, 109], "outperform": 1, "output": [21, 42, 64], "outsid": [100, 108, 109, 112], "outward": 6, "over": [0, 2, 6, 8, 12, 13, 21, 27, 33, 47, 49, 54, 55, 65, 76, 83, 87, 88, 108, 109, 112, 113, 121], "overal": 63, "overcom": [33, 91], "overestim": [33, 34], "overlap": [36, 37, 63, 73, 83, 90, 100], "overlin": [9, 12, 27, 29, 36, 40, 84, 90, 92, 97, 98, 99, 100, 102, 107, 108, 109], "overload": [10, 76], "overrid": 76, "overwritten": [10, 76], "own": [10, 20, 46, 80], "p": [0, 1, 2, 7, 8, 9, 10, 22, 27, 32, 33, 34, 42, 47, 49, 50, 51, 53, 54, 56, 58, 75, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 91, 99, 101, 102, 103, 106, 109, 110, 118, 121], "p2": 86, "p209": 115, "p2n": 14, "p_": [32, 33, 49, 63, 65, 66, 79, 100, 101, 121], "p_0": [12, 34, 49, 79, 88], "p_1": [49, 76, 79, 105, 113], "p_2": [79, 105], "p_h": [33, 54, 56, 86], "p_i": [49, 76, 79, 80, 105, 113], "p_j": [49, 76, 79, 80, 113], "p_k": [49, 79, 86], "p_l": [63, 64, 65, 66, 79], "p_m": [79, 105], "p_n": [49, 76, 79, 113], "p_su": 32, "p_sv": 32, "packag": [10, 25, 46, 108, 121], "pad": [73, 108], "page": [7, 47, 49, 80, 88], "pai": 109, "paid": 109, "pair": [54, 55, 58, 100, 112], "pajetrac": 118, "pap": [49, 81], "paper": 1, "papu": 81, "parabol": [12, 111, 113], "parallel": [11, 14, 15, 20, 23, 24, 40, 76, 77, 121], "parallel_statu": 42, "paralleldof": [10, 23, 24, 42, 76], "parallelmatrix": 42, "parallelogram": [48, 60], "parallelprecondition": 42, "parallelpreconditioner2": 42, "paramet": [0, 1, 2, 8, 10, 38, 45, 46, 47, 50, 51, 61, 69, 70, 76, 82, 84, 91, 97, 99], "parameter": 100, "paramt": [56, 91], "pardof": [15, 23, 42], "parent": [76, 105], "parsev": 100, "part": [0, 1, 2, 6, 12, 15, 34, 36, 55, 57, 61, 65, 73, 79, 83, 84, 86, 89, 90, 91, 98, 99, 101, 104, 105, 106, 112, 113, 121], "part_graph": 108, "partial": [0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 28, 33, 34, 36, 38, 40, 41, 42, 43, 46, 52, 53, 54, 55, 56, 57, 58, 61, 62, 65, 66, 67, 73, 75, 76, 78, 79, 84, 87, 89, 90, 92, 95, 97, 98, 99, 100, 101, 103, 104, 108, 113, 114, 118, 121], "partial_": 79, "partial_i": [71, 104], "partial_n": [0, 3, 34, 65, 73, 75], "partial_q": 53, "partial_t": 67, "partial_v": 53, "partial_x": [71, 97, 104], "particip": 20, "particular": [15, 40, 51, 58, 65, 88, 103], "partit": [15, 40, 100, 109], "partner": 15, "pasciak": [45, 46, 83, 85], "pass": [20, 113], "patch": [33, 34, 36, 83, 109], "patchwisesolv": 34, "path": [20, 25, 46, 108], "pattern": [10, 76], "pc": [18, 19], "pc_type": [18, 19], "pde": [5, 24, 28, 31, 40, 100, 105], "pdf": 118, "peak": [116, 121], "pechstein": [62, 67, 71], "pen": 0, "penalti": [0, 1, 2, 34, 39, 69, 76], "per": [8, 29, 39, 40, 42, 46, 49, 64, 78, 83, 88, 110, 121], "perform": [0, 1, 12, 15, 24, 40, 47, 64, 77, 100, 103, 106, 108, 113, 115, 118], "period": [0, 76], "permeabl": 83, "person": 20, "perturb": 64, "petsc": [22, 45, 46], "petsc4pi": [14, 20, 22, 23], "petsc_configure_opt": 20, "petscpc": [18, 19, 22], "pf": 121, "phase": 82, "phat": 121, "phd": 72, "phenomena": 76, "phi": [8, 9, 13, 83, 88, 97, 102, 104, 106, 112], "phi_t": [8, 9, 12, 13, 106], "phi_v": 34, "physic": [10, 40, 76, 82, 84, 88, 91], "pi": [8, 12, 41, 43, 44, 66, 73, 80, 100, 101, 102, 104, 119], "pi_": [1, 38, 47, 56, 64, 65, 66, 79, 83], "pi_0": [65, 66], "pi_h": [36, 109], "pi_i": 38, "pi_l": [65, 66], "pi_n": 79, "pi_p": 79, "pick": [28, 29, 31], "pickl": 20, "piec": [15, 58, 100], "piecewis": [6, 33, 36, 100, 108, 109], "pillwein": 34, "pinvit": [0, 92, 96, 97, 121], "piola": 121, "pip": [46, 103], "pip3": [20, 46], "pl": 118, "place": 109, "placement": 76, "plai": [86, 87, 103], "plan": 97, "plane": 70, "plate": 3, "platform": 20, "pleas": [26, 46], "plot": [33, 47, 48, 49, 50, 51, 64, 75, 80, 103, 110, 115], "plt": [1, 2, 33, 47, 48, 49, 50, 51, 64, 80, 103, 112, 115], "plu": [39, 49, 62, 64, 82], "plug": [89, 91], "pm": 84, "pmax": [37, 38, 39, 82], "pmin": [37, 38, 39, 82], "pml": 76, "pmltrafo": 76, "pnt": [70, 83], "po": [0, 55], "poincar\u00e9": [45, 101, 106], "point": [0, 6, 8, 9, 10, 12, 13, 20, 26, 28, 32, 35, 38, 48, 56, 58, 76, 80, 81, 82, 83, 91, 98, 101, 103, 105, 108, 111, 118], "poisson": [1, 2, 33, 36, 45, 46, 53, 73], "pol": 47, "polygon": 104, "polyhedr": 83, "polynomi": [1, 2, 6, 8, 9, 10, 13, 34, 36, 49, 75, 76, 80, 88, 91, 99, 100, 101, 103, 110], "pop": 49, "popular": 6, "portabl": 23, "pose": [28, 31, 34, 40, 41, 55, 100, 104], "posit": [0, 6, 27, 30, 34, 37, 50, 51, 52, 63, 64, 67, 69, 77, 78, 81, 84, 91, 92, 99, 101, 103, 104, 108, 112], "possibl": [6, 15, 25, 26, 28, 33, 34, 46, 100, 101, 108, 110], "possibli": 67, "post": 63, "posteriori": 67, "postprocess": [34, 86], "postsmooth": 63, "potenti": [76, 83], "power": [46, 48], "pp": 118, "pp43": 65, "practic": [40, 51, 108], "prager": 67, "pre": [0, 14, 18, 19, 20, 21, 22, 38, 39, 42, 46, 47, 49, 51, 63, 74, 76, 77, 81, 83, 84, 85, 91, 92, 96, 97, 103, 108, 109, 110], "pre2": 108, "pre3": 108, "prea": [22, 81, 82, 85], "prea1": [22, 85], "preambl": 45, "prec": [53, 54, 62, 90, 108, 109], "preceq": [1, 8, 12, 35, 36, 53, 64, 65, 66, 79, 83, 86, 100, 101, 104, 106, 108], "precis": [1, 2, 47, 55, 99, 100, 104], "precondit": [21, 40, 45, 46, 47, 49, 77, 81, 82, 105, 109, 110], "precondition": [10, 18, 19, 21, 22, 40, 45, 46, 47, 49, 52, 64, 74, 76, 77, 81, 83, 84, 85, 91, 108], "preconditiong": 105, "precontion": 108, "predefin": 76, "prefer": 13, "preform": [51, 64], "prelam": 84, "prepar": 25, "preschur": [22, 81, 85], "prescrib": [41, 82, 89], "present": [63, 64, 82, 101], "preserv": [12, 54, 79, 88, 100, 112], "presmooth": 63, "press": [25, 63], "pressur": [7, 12, 56, 58, 83, 84], "previou": [47, 49, 83, 99, 106, 109], "primal": [8, 35, 37, 38, 39, 57, 61, 62, 70, 91], "primari": 0, "prime": [13, 30, 36, 79, 88, 92, 97, 98, 100, 102, 106, 112], "principl": 27, "print": [0, 1, 2, 4, 10, 15, 18, 20, 21, 22, 23, 24, 38, 39, 42, 46, 47, 48, 50, 51, 55, 62, 63, 66, 67, 70, 76, 77, 78, 80, 81, 82, 83, 85, 86, 88, 92, 96, 97, 103, 106, 108, 109, 110, 115, 116, 118, 119, 120, 121], "printmast": 16, "printrat": [0, 14, 18, 19, 21, 22, 38, 39, 42, 63, 77, 82, 83, 84, 85, 91, 92, 96, 97, 103, 110], "prior": 103, "prism": 9, "prismat": 71, "problem": [8, 10, 12, 13, 21, 33, 34, 35, 37, 38, 39, 40, 41, 47, 50, 51, 60, 63, 64, 65, 67, 69, 70, 75, 76, 77, 79, 80, 81, 82, 86, 89, 91, 97, 100, 101, 103, 104, 105, 106, 107, 108, 109, 112], "proc": 20, "proc2dof": [15, 42], "proce": [34, 46, 51, 65, 106, 109, 114], "procedur": [27, 79], "process": [15, 20, 24], "processor": [15, 20, 24, 40, 121], "prod_": 104, "produc": [49, 79], "product": [12, 21, 27, 28, 29, 38, 48, 49, 50, 51, 52, 53, 64, 65, 76, 79, 83, 98, 99, 100, 101, 102, 103, 105, 109], "productmatrix": 120, "productspac": [10, 76], "profit": 77, "program": 110, "programm": 46, "progress": [60, 62], "proj": [82, 108], "project": [1, 8, 9, 12, 28, 29, 37, 45, 46, 56, 64, 65, 83, 86, 108, 109, 113, 118], "projectedmg": 63, "projector": [32, 45, 56, 64, 83, 84, 103, 105, 108, 109, 118], "projinv": 82, "projpr": 82, "prol": [63, 66], "prolong": [10, 63, 65, 66, 76, 83, 109], "prolongationop": 109, "promin": 111, "proof": [6, 8, 12, 27, 28, 29, 31, 32, 34, 36, 37, 53, 54, 64, 65, 66, 75, 79, 81, 83, 84, 87, 88, 90, 92, 98, 99, 100, 101, 104, 105, 106, 108, 109, 112], "propag": [48, 51, 105, 118], "proper": [8, 41, 76, 79, 88], "properli": 48, "properti": [10, 12, 28, 33, 45, 46, 50, 63, 65, 67, 76, 79, 83, 86, 89, 90, 109], "proport": 48, "prove": [8, 12, 29, 30, 32, 34, 48, 51, 53, 64, 65, 79, 83, 84, 90, 97, 99, 100, 101, 102, 103, 104, 105, 106, 109], "proven": [1, 12, 34, 47, 48, 51, 64, 65, 66, 79, 81, 83, 87, 90, 97, 100, 103, 106], "provid": [0, 3, 8, 10, 12, 13, 14, 15, 20, 29, 34, 35, 48, 51, 61, 63, 65, 72, 76, 79, 83, 84, 103, 108, 109, 121], "proxi": [3, 10, 76], "pru": 81, "psc": [14, 22, 23], "psc_f": 14, "psc_mat": 14, "psc_u": 14, "psi": [9, 64, 83], "psi_": [8, 9, 64], "psi_1": 9, "psi_h": 83, "psi_i": [13, 34, 83, 109], "psi_j": 9, "psi_l": 64, "psi_n": 9, "psi_t": [8, 9], "ptau": 70, "pu": [32, 109], "pull": [13, 100, 102, 112], "pure": [21, 24], "purpos": 23, "put": [11, 65, 115], "pv": 32, "pwd": 46, "px": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 42], "pxc": 66, "py": [46, 72, 98, 121], "pybind11_builtin": [10, 76], "pybind11_object": [10, 76], "pycapsul": [10, 76], "pymeti": 108, "pyngcor": [10, 76], "pypi": 20, "pyplot": [1, 2, 33, 47, 48, 49, 50, 51, 64, 80, 103, 112, 115], "pyramid": 9, "pythagora": 102, "pythagorean": 65, "python": [10, 14, 15, 20, 23, 25, 46, 72, 76, 108, 121], "python3": [20, 46, 121], "q": [7, 9, 10, 12, 22, 27, 30, 34, 53, 54, 55, 56, 58, 62, 78, 79, 81, 84, 85, 100, 101, 121], "q2": 86, "q_h": [7, 54, 86], "q_i": [34, 88], "qf": 121, "qquad": [0, 6, 8, 9, 12, 13, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 41, 43, 47, 49, 50, 51, 52, 53, 57, 62, 63, 64, 65, 66, 67, 73, 74, 76, 79, 80, 82, 83, 84, 88, 89, 92, 95, 97, 98, 100, 101, 102, 103, 104, 105, 106, 111, 112, 113, 114, 115], "qr": 115, "quad": [27, 30, 31, 32, 34, 40, 49, 53, 54, 55, 56, 64, 65, 69, 73, 76, 82, 88, 89, 90, 98, 100, 103, 105, 107, 109, 112], "quad_domin": [10, 80], "quadrat": [9, 50, 76, 81, 84, 105], "quadrilater": [9, 10, 62, 80], "qualiti": 51, "quantifi": 64, "quantiti": 51, "quasi": [8, 28, 31, 36, 54, 66, 83, 109], "queri": [10, 20, 76], "quickest": 20, "quickli": 66, "quotient": [51, 101, 103, 105, 112, 113], "r": [0, 1, 5, 6, 8, 9, 12, 13, 16, 18, 19, 21, 22, 24, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 44, 46, 47, 48, 49, 50, 51, 52, 58, 63, 68, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 88, 89, 92, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 109, 110, 112, 113, 115, 121], "r1": 34, "r2": 34, "r_": [34, 49], "r_0": 49, "r_e": 34, "r_h": 83, "r_i": [24, 51, 83, 105], "r_j": 88, "r_k": 49, "r_n": 49, "r_t": [8, 34], "radiu": [48, 51, 99, 102, 105], "rafetsed": 67, "rais": 121, "random": 64, "rang": [0, 10, 11, 14, 15, 18, 19, 20, 21, 22, 23, 31, 33, 35, 37, 38, 39, 40, 41, 42, 44, 47, 48, 49, 51, 55, 63, 64, 66, 70, 74, 76, 77, 79, 80, 81, 82, 83, 84, 88, 100, 103, 104, 105, 108, 109, 113, 114, 115, 118, 121], "rank": [14, 15, 16, 18, 19, 20, 21, 22, 24, 42, 84, 105, 106], "rare": 47, "rate": [8, 28, 48, 77, 79], "ratio": 8, "ration": 80, "raviart": [7, 9, 12, 61, 86, 88, 101], "rayleigh": [51, 103, 105], "raymond": 5, "rcll": 100, "re": [7, 34, 36, 48, 50, 51, 63, 82, 113], "reach": 40, "reaction": 28, "read": [25, 34, 72, 76, 84, 105, 113], "reader": [25, 81], "readi": [25, 83], "readonli": [10, 76], "readthedoc": 14, "real": [31, 112], "realiz": [4, 41], "rearrang": 61, "reason": [12, 56, 121], "recal": 24, "recast": 84, "receiv": [15, 42], "recent": [46, 55, 58, 62, 118, 119, 120, 121], "recommend": [72, 76], "reconstruct": [82, 86], "recov": [8, 33, 38, 53, 105], "rect": [12, 35, 37, 38, 39, 82, 89, 108, 118, 121], "rectangl": [0, 7, 12, 16, 18, 19, 33, 34, 35, 37, 38, 39, 55, 78, 81, 82, 85, 89, 97, 103, 108, 112, 118, 121], "rectangular": [24, 42, 53, 82, 105, 115], "recurr": [47, 79], "recurs": [33, 49, 63, 64, 65], "recv": 20, "redraw": [0, 4, 5, 16, 55, 70, 113, 114, 118, 119, 120, 121], "reduc": [0, 15, 30, 40, 45, 48, 49, 50, 64, 67, 73, 82, 101, 112, 114], "reduct": [15, 47, 48, 50, 79, 82], "ref": [12, 79, 100], "refer": [8, 9, 10, 12, 13, 36, 52, 62, 64, 88, 106, 112, 121], "refin": [10, 11, 14, 15, 18, 19, 22, 23, 35, 40, 63, 64, 65, 66, 75, 76, 77, 83, 84, 103, 108, 109, 118], "refinefromtre": 76, "refinehp": [8, 76], "refinementflag": 76, "reformul": [31, 49, 64, 105], "refract": 121, "reg": 78, "regex": 76, "regexpr": [10, 76], "region": [0, 10, 75, 76, 78, 108, 112, 121], "region_wis": 55, "regioncf": 76, "regular": [8, 9, 10, 33, 34, 39, 53, 62, 64, 76, 83, 84, 91, 98, 104, 105, 109], "reinforc": 91, "reinstal": 20, "reissner": [45, 62], "rel": [34, 112], "relat": [8, 13, 29, 47, 50, 64, 78, 101, 108], "relationship": 45, "relax": 50, "relev": 40, "reliabl": [33, 34, 35], "remain": 49, "remark": [12, 34, 82], "remov": [20, 33, 47], "renam": 56, "render": 46, "repeat": [27, 51], "rephras": [53, 54], "replac": [4, 12, 33, 36, 54, 56, 63, 65, 73, 75, 82, 101, 103, 106, 112, 113], "report": 26, "repositori": 80, "repres": [20, 24, 29, 34, 37, 49, 63, 65, 83, 98, 99, 105, 108, 109, 115], "represent": [24, 31, 45, 46, 47, 76, 103, 105, 106, 109, 115], "reproduc": 66, "reproven": 105, "requir": [3, 8, 12, 24, 35, 53, 61, 63, 65, 76, 82, 84, 86, 101, 104, 109, 121], "rescal": [12, 47, 109], "resf": [22, 81, 85], "resg": [22, 81, 85], "reshap": [0, 55], "residu": [14, 21, 33, 38, 39, 46, 47, 48, 50, 51, 63, 77, 83, 84, 91, 105, 110], "residuum": [34, 51, 73], "resolut": [10, 76], "reson": 46, "resourc": 121, "resp": 62, "respect": [27, 28, 31, 34, 35, 38, 41, 47, 48, 49, 53, 79, 84, 99, 100, 102, 105], "rest": [12, 36, 63, 66, 73, 83, 87, 104, 109], "restoresubvector": 23, "restrict": [1, 5, 13, 22, 30, 63, 66, 73, 76, 79, 85, 89, 98, 99, 100, 101, 104, 109], "result": [8, 15, 20, 24, 37, 51, 62, 65, 66, 69, 71, 76, 79, 81, 83, 97, 99, 100, 101, 106, 109, 110], "retriev": 15, "return": [0, 2, 3, 6, 7, 10, 15, 17, 18, 19, 24, 33, 42, 47, 49, 55, 58, 62, 63, 66, 67, 69, 70, 74, 76, 81, 103, 108, 109, 110, 112, 121], "rev": [65, 105], "revers": [7, 16, 18, 19, 81, 85, 105], "rewrit": [51, 53, 57, 61, 76, 100, 103], "rewritten": [49, 76], "rh": [79, 84, 91], "rho": [4, 10, 47, 48, 51, 76, 105], "rho_": [47, 48, 50], "rho_n": 47, "rhscurl": 17, "ri": [22, 85], "rich": [50, 53, 58], "richardson": [45, 46, 47, 49, 50, 52, 84], "riesz": [28, 31, 45, 46, 53], "right": [0, 1, 5, 6, 8, 10, 11, 12, 21, 23, 24, 28, 30, 31, 34, 35, 36, 37, 38, 41, 44, 47, 48, 49, 51, 52, 53, 55, 61, 62, 64, 65, 68, 69, 73, 76, 78, 79, 80, 81, 82, 83, 84, 89, 90, 91, 92, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 112, 113, 114, 121], "rightarrow": [2, 6, 8, 9, 12, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 50, 53, 54, 56, 58, 65, 66, 68, 69, 74, 75, 76, 78, 79, 83, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 109, 112, 113, 115], "rightdomain": 44, "rigid": [59, 62], "rigor": [33, 35], "ring": 46, "ring_resonator_import": [119, 120], "ritz": 49, "rk": 111, "rk2": [4, 111], "rk3": [111, 112], "rk4": [4, 111], "rll": 53, "robert": 118, "robin": [43, 51, 73, 104], "robust": [1, 2, 34, 46, 71, 110], "rodrigu": 79, "role": [57, 89, 103], "root": [15, 20, 28, 51], "rotat": [0, 8, 33, 55, 59, 61, 62, 67, 69, 70], "rotmat": [0, 55], "rotor": 0, "rough": 64, "roughli": 8, "roundoff": 49, "row": [12, 52, 56, 60, 61, 81, 82, 84, 111], "row_pardof": [14, 42], "rt": [7, 18, 19, 88, 91], "rt0": 88, "rt_0": 88, "rt_k": [86, 87, 88, 91], "rtol": [14, 23], "rule": [0, 8, 12, 13, 88, 100, 106, 109, 111, 114], "run": [20, 25, 26, 46, 48, 51], "rung": [4, 5, 45], "ruuth": 5, "s1": [47, 111], "s11": 62, "s2": 47, "s_0": 100, "s_1": 100, "s_i": 100, "s_j": [88, 100], "s_l": [64, 88], "s_m": 100, "saad": [47, 49], "sabin": [72, 80, 88], "saddl": [38, 56, 81, 82, 83, 91], "sai": [9, 28, 32, 38, 73, 98, 101], "same": [1, 2, 8, 9, 10, 12, 20, 21, 24, 28, 32, 34, 36, 38, 40, 41, 51, 55, 56, 63, 66, 68, 76, 79, 88, 89, 91, 97, 99, 101, 109, 110, 111, 112, 121], "sampl": 115, "satisfi": [27, 28, 30, 34, 36, 47, 51, 58, 61, 64, 65, 67, 69, 75, 76, 79, 81, 83, 84, 98, 109, 111, 112, 113], "scalar": [60, 82, 86, 112], "scale": [12, 27, 34, 36, 38, 42, 47, 64, 65, 77, 79, 81, 87, 97, 100, 108, 120, 121], "scaledchebi": 47, "scalematrix": 120, "scalingmat": 42, "scatter": [20, 44], "scene": [0, 4, 5, 16, 55, 76, 113, 114, 118, 119, 120, 121], "sch": [], "schmidt": [49, 115], "schoeberl": 25, "schuller": 63, "schur": [37, 38, 42, 82], "schurdir": [38, 39], "schwab": 79, "schwarz": [12, 27, 36, 37, 45, 46, 63, 64, 65, 66, 83, 100, 102, 109], "sch\u00f6berl": [7, 34, 45, 46, 62, 71, 72, 83, 119, 120], "scienc": 46, "scientif": [23, 46, 111], "scipi": [1, 2, 46, 52, 76, 115], "scipymat": [1, 2], "scope": 15, "scott": 46, "script": [76, 98], "scroll": 37, "sd": 70, "search": [30, 34, 50, 55, 60, 61, 69, 76, 83, 103, 113], "secod": 6, "secon": 57, "second": [4, 5, 6, 10, 31, 33, 40, 50, 51, 55, 56, 58, 64, 67, 70, 76, 81, 83, 84, 86, 89, 97, 98, 103, 106, 109, 110, 112, 114, 115, 118], "secondari": 0, "secondord": 10, "section": [46, 50, 52, 75, 109], "sector": [8, 104], "see": [1, 5, 10, 12, 31, 37, 46, 53, 54, 58, 59, 60, 65, 73, 76, 79, 80, 81, 84, 88, 89, 100, 103, 115], "seem": 103, "seen": [6, 49, 53, 63, 64, 83, 91, 99], "segment": 9, "seidel": [63, 64, 105, 110], "select": 76, "self": [1, 2, 10, 42, 48, 51, 63, 66, 76, 109, 121], "semi": [33, 36, 83, 99, 100, 101, 103, 106, 109, 112, 113], "send": [20, 40, 46], "sens": [9, 34, 55, 62, 64, 76, 78, 83, 100, 121], "sensit": 47, "sent": 15, "separ": [2, 41, 84], "septemb": 72, "sequenc": [27, 31, 32, 33, 63, 66, 75, 97, 99, 100, 101, 103], "sequenti": [24, 63], "seri": [41, 65, 100, 102, 104], "serial": 20, "server": 46, "set": [0, 3, 4, 5, 7, 9, 10, 12, 13, 14, 15, 17, 18, 19, 22, 24, 27, 28, 30, 33, 34, 35, 37, 38, 39, 41, 48, 51, 53, 54, 55, 56, 57, 58, 62, 63, 64, 66, 70, 73, 75, 76, 79, 80, 81, 83, 85, 87, 88, 91, 98, 99, 101, 104, 105, 106, 108, 109, 113, 114, 115, 116, 118, 121], "set_minmax": 121, "set_titl": 47, "set_ylim": 47, "set_yscal": 47, "setcouplingtyp": [10, 18, 19, 76], "setdefinedon": [10, 76], "setdeform": [0, 55, 76], "setelementord": 76, "setislocalmat": 23, "setlgmap": 23, "setmateri": 44, "setminu": [100, 107, 108, 109], "setnormtyp": [14, 23], "setnumthread": [4, 5], "setoper": [14, 23], "setord": [10, 58, 76, 81], "setparallelstatu": 42, "setpml": 76, "setrandom": [48, 51, 64], "setrefinementflag": [33, 35, 76], "setsiz": 23, "settoler": [14, 23], "settyp": [14, 23], "setup": [21, 77, 78, 97, 108, 109], "setupkorn": 103, "setuppoincareproblem": 103, "setupproblem": 103, "setuptraceproblem": 103, "sever": [5, 12, 40, 58, 64, 108], "shape": [1, 7, 8, 9, 10, 16, 18, 19, 22, 24, 33, 34, 35, 42, 46, 63, 76, 77, 78, 80, 81, 85, 88, 89, 103, 108, 109, 118, 121], "share": [15, 24, 110], "sharp": [100, 103, 106], "shat": 82, "shear": [69, 70], "shift": [0, 25, 47, 112, 114], "ship": [14, 20], "short": [33, 63, 76, 88, 100, 108, 112, 115], "should": [10, 33, 46, 76, 100, 108, 121], "show": [10, 12, 29, 30, 31, 33, 34, 36, 41, 46, 47, 53, 64, 65, 74, 79, 81, 84, 97, 102, 106, 108, 109, 112], "shown": [34, 40, 53, 60, 64, 65, 79, 83, 90, 105], "shutdown": [14, 15, 20, 21, 22, 23, 24], "siam": [24, 65, 105], "side": [0, 8, 10, 11, 12, 21, 23, 24, 30, 31, 33, 34, 35, 36, 37, 41, 53, 55, 64, 65, 73, 76, 78, 79, 82, 89, 91, 98, 100, 101, 102, 103, 104, 105, 106, 112, 121], "sigma": [18, 19, 37, 47, 48, 53, 57, 60, 61, 62, 64, 67, 68, 69, 70, 71, 74, 76, 81, 83, 84, 86, 87, 88, 89, 90, 91, 109, 112, 119, 120], "sigma1": 47, "sigma_": [34, 62, 67, 90, 91], "sigma_h": [62, 67, 68, 71, 86, 87, 91], "sigma_n": [34, 57, 62, 89, 90], "sigman": 70, "sign": [12, 55, 81], "signatur": [10, 76], "silenc": [38, 39], "sim": 73, "simeq": [1, 8, 12, 34, 36, 65, 100, 101, 103, 104], "similar": [12, 21, 27, 33, 36, 40, 50, 52, 61, 63, 64, 65, 74, 78, 79, 83, 88, 91, 100, 102, 108, 113, 114, 115], "similarli": [2, 24, 76, 81], "simpl": [0, 3, 6, 20, 24, 31, 48, 51, 73, 83, 100, 105, 109], "simpler": 12, "simplest": [33, 58], "simpli": [3, 27, 50, 79, 83], "simplic": 79, "simplici": [76, 79], "simplif": 12, "simplifi": [30, 47, 51, 109], "simul": [44, 46], "simultan": 8, "sin": [0, 6, 41, 55, 66, 73, 80, 100, 101, 102, 104, 119], "sinc": [1, 2, 3, 4, 7, 8, 10, 12, 15, 27, 28, 31, 32, 33, 34, 36, 37, 39, 40, 48, 49, 50, 51, 60, 61, 64, 65, 76, 79, 83, 84, 86, 88, 97, 98, 100, 101, 104, 105, 106, 108, 109, 111, 112, 114, 115, 121], "sine": [100, 108], "singl": [1, 24, 45, 51, 65, 111], "singular": [33, 35, 39, 40, 58, 111], "sinum": 118, "sinv": 84, "sinwel": 62, "sisc": 118, "sit": [8, 9, 110], "site": 121, "situat": [51, 52], "six": 62, "size": [1, 2, 8, 12, 15, 20, 23, 47, 48, 63, 65, 75, 76, 77, 79, 83, 84, 97, 103, 105, 106, 108, 109, 112, 114], "skel_dof2proc": 42, "skel_pardof": 42, "skeleton": [0, 2, 6, 38, 121], "sketch": [27, 101], "skew": 61, "skew2vec": [18, 19], "skip": [2, 12, 30, 31, 58, 65, 83, 98], "sleep": [0, 55, 113, 114, 115, 118], "slider": [74, 76], "slightli": [62, 112], "small": [8, 33, 35, 40, 45, 46, 49, 56, 64, 69, 78, 79, 83, 84, 86, 103, 108, 109, 115, 118, 121], "smaller": [20, 37, 64, 88, 108, 109, 118], "smallest": [0, 24, 27, 34, 47, 97, 103, 106, 108], "smooth": [6, 8, 9, 13, 33, 41, 60, 63, 64, 90, 98, 99, 100, 104, 106], "smoothback": 63, "smoother": 64, "smoothli": 26, "smuggl": 1, "so": [1, 2, 3, 6, 7, 27, 29, 33, 51, 52, 53, 55, 66, 73, 76, 77, 79, 82, 83, 91, 105, 106, 108, 109, 111], "sobolev": [8, 64, 79, 90, 101, 103, 104], "softwar": 20, "sol": [22, 66, 81, 85, 91], "sol_lam": 55, "sol_sigma": 86, "sol_u": [55, 86], "solid": [45, 46, 77, 103], "solsigma": 70, "solut": [0, 1, 2, 6, 8, 19, 21, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 46, 47, 48, 49, 50, 53, 54, 58, 64, 65, 73, 75, 76, 79, 82, 83, 84, 86, 91, 100, 104, 105, 108, 109, 110, 111, 112, 113, 115, 121], "solv": [4, 5, 8, 10, 14, 18, 23, 28, 30, 34, 35, 37, 38, 40, 45, 46, 52, 53, 56, 63, 64, 70, 73, 77, 80, 82, 84, 85, 86, 100, 104, 105, 109, 111, 112, 114, 115], "solvabl": [28, 30, 31, 53], "solveestimatemark": [33, 35], "solveiniti": 16, "solvem": [10, 76], "solver": [0, 1, 5, 14, 18, 23, 24, 40, 49, 83, 84, 91, 92, 96, 97, 103, 108, 109, 121], "solverparamet": 18, "solvestok": 58, "some": [0, 3, 6, 8, 10, 12, 28, 29, 30, 31, 32, 34, 35, 39, 41, 46, 48, 49, 54, 63, 64, 73, 75, 76, 81, 82, 86, 89, 99, 100, 101, 104, 105, 109, 110, 111, 112, 113, 121], "someth": [20, 102], "sometim": 8, "soon": 51, "sophist": [1, 2], "sort": 33, "sourc": [6, 20, 33, 34, 35, 36, 43, 44, 46, 55, 57, 76, 89, 113], "sp": [1, 2, 121], "space": [0, 2, 3, 6, 7, 8, 9, 10, 12, 14, 27, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 47, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 69, 73, 75, 78, 79, 80, 82, 83, 84, 86, 88, 89, 91, 98, 103, 104, 106, 109, 110, 114, 115, 121], "span": [9, 31, 49, 76, 82, 100, 104, 106, 108, 115], "spars": [1, 2, 23, 47, 52, 76, 77, 108], "sparsecholeski": [0, 1, 2, 5, 38, 39, 44, 62, 77, 85, 103, 109, 121], "sparsematrixd": 42, "sparseschur": 42, "sparsiti": [10, 76], "spatial": [63, 113, 114], "spd": [47, 48, 50, 51, 52, 64, 84], "special": [15, 83, 99], "specialcf": [0, 1, 2, 3, 4, 5, 6, 7, 17, 18, 19, 22, 34, 57, 62, 67, 69, 70, 85, 91, 116, 121], "specif": [10, 65, 76], "specifi": [6, 34, 73, 76, 104, 111], "spectral": [48, 49, 51, 64, 81, 105, 106, 108], "spectrum": [37, 47, 81, 84, 109], "spend": 103, "sphere": 121, "spiteri": 5, "splinegeometri": [44, 62], "split": [20, 31, 33, 37, 38, 39, 40, 45, 46, 51, 53, 63, 64, 73, 83, 98, 100, 105, 106, 108, 109], "splitelements_alfeld": 76, "springer": 63, "spy": [1, 2], "sqrt": [0, 1, 30, 31, 33, 34, 35, 41, 47, 49, 50, 51, 52, 64, 65, 79, 80, 81, 84, 88, 92, 97, 100, 103, 111, 112, 119], "sqrt2": 100, "squar": [0, 8, 28, 41, 51, 55, 65, 76, 101, 103, 104, 106, 108], "src": [20, 119], "ss": 111, "stab_e": 112, "stab_i": 112, "stab_improvede": 112, "stab_rk3": 112, "stab_trapez": 112, "stabil": [1, 2, 49, 60, 61, 86, 109, 111, 115], "stabl": [6, 30, 45, 46, 54, 56, 60, 66, 79, 83, 109, 111, 112], "stackrel": [68, 86], "stage": 111, "stand": [37, 39], "standard": [10, 20, 21, 23, 34, 57, 60, 77, 80, 88, 91, 108], "star": 46, "start": [0, 1, 2, 14, 15, 20, 21, 22, 23, 24, 26, 28, 30, 33, 34, 40, 46, 48, 55, 64, 65, 76, 80, 81, 86, 100, 109, 111, 121], "start_and_connect": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24], "state": [28, 31, 60, 98, 99, 100, 101, 103, 112], "statement": [27, 104], "static": [10, 76], "stationari": [4, 45, 46, 103, 105], "stdflux": 89, "stdout": [14, 15, 20, 21, 22, 23, 24], "stdtemp": 89, "steel_": 76, "steeper": 66, "stefano": 14, "stenberg": 61, "stencil": 118, "step": [4, 5, 8, 12, 13, 21, 34, 35, 36, 40, 45, 46, 47, 48, 50, 51, 63, 64, 70, 79, 81, 82, 84, 105, 106, 108, 109, 111, 115], "steven": 5, "stiff": [112, 113, 118], "still": [39, 46, 60, 63, 82, 100], "stoke": [5, 45, 46, 56, 84, 101], "stokesa": [18, 19], "stop": [33, 49, 121], "storag": 24, "store": [24, 52, 82, 108, 109, 121], "str": [10, 20, 37, 38, 39, 47, 76, 108], "straight": [12, 91], "strain": 74, "strang": 56, "strategi": [8, 48], "stream": 20, "strength": 6, "strengthen": 105, "stress": [60, 61, 70, 74], "stretch": 62, "strictli": [51, 111], "string": [10, 76], "strip": 121, "strong": [0, 33, 55, 64, 76, 88, 98, 100], "stronger": [64, 100], "structur": [10, 34, 45, 46, 56], "student": 46, "studi": [8, 40, 58, 75, 100], "sturm": 79, "sub": [2, 8, 12, 23, 24, 27, 30, 32, 34, 37, 38, 39, 40, 42, 49, 53, 54, 63, 65, 73, 76, 77, 79, 82, 83, 84, 88, 98, 103, 104, 106, 108, 109, 111, 112, 113], "subdivis": 9, "subdomain": 108, "submatrix": 91, "subplot": 47, "subset": [6, 8, 9, 12, 15, 27, 29, 31, 34, 36, 37, 40, 41, 47, 49, 54, 63, 64, 66, 67, 68, 76, 79, 81, 83, 84, 88, 92, 95, 97, 98, 100, 101, 102, 104, 105, 106, 109, 111, 112, 113], "subspac": [27, 28, 29, 31, 45, 46, 65, 105], "substep": 118, "substitut": [13, 47, 51, 52, 58, 65, 79], "substructur": 37, "subtrac": 81, "subtract": 12, "succ": 53, "succeq": [6, 8, 36, 53], "successfulli": 26, "sudo": 20, "suffici": [0, 1, 2, 8, 31], "suggest": [8, 46, 100], "suit": 64, "sum": [13, 15, 22, 24, 33, 34, 35, 36, 41, 49, 52, 55, 64, 65, 66, 74, 83, 85, 98, 100, 102, 105, 106, 108, 109, 111, 112, 121], "sum_": [2, 6, 8, 9, 12, 13, 24, 33, 34, 35, 36, 38, 40, 41, 42, 47, 49, 51, 62, 65, 66, 76, 79, 87, 90, 91, 99, 100, 101, 102, 104, 105, 106, 111, 113, 121], "sum_i": [76, 100, 105, 106, 109, 111], "sum_j": [48, 105], "sum_k": [65, 100], "sum_l": [65, 100], "sum_t": [1, 2, 3, 12, 24, 34, 36, 54, 62, 67, 86, 87, 91, 106, 112], "sum_v": [34, 62], "sumlocdof": 15, "summand": 106, "summat": [12, 15], "summatrix": [120, 121], "summer": 46, "sumup": 15, "sundai": 26, "sung": 3, "sup": [6, 27, 45, 46, 53, 54, 86], "sup_": [12, 27, 28, 29, 30, 31, 34, 36, 37, 41, 47, 48, 53, 54, 75, 83, 84, 86, 87, 90, 97, 103, 105, 112], "sup_u": [6, 37, 84], "sup_v": [29, 34, 53], "super": [33, 42, 63, 76, 109, 121], "supp": 98, "supplement": 72, "support": [3, 10, 76, 90, 98, 99, 100, 108, 109], "suppos": 33, "supremum": [30, 75], "supset": 37, "surfac": [6, 35, 70, 76], "surfacel2": 55, "surject": [30, 31, 53, 100], "swap": [30, 89, 103], "switch": 3, "sym": [56, 60, 62, 67, 74, 82, 103], "symbol": [10, 76], "symbolicbfi": [62, 67, 69, 70], "symbolicenergi": 70, "symboliclfi": [62, 67, 69], "symmetr": [0, 1, 2, 5, 8, 27, 28, 30, 31, 37, 45, 46, 48, 51, 52, 53, 60, 61, 62, 63, 64, 67, 69, 70, 74, 77, 81, 84, 91, 101, 103, 112, 121], "symmetri": 45, "symplect": 121, "syng": 67, "system": [1, 5, 27, 31, 34, 37, 38, 41, 42, 46, 47, 49, 50, 51, 52, 54, 56, 57, 58, 61, 67, 76, 77, 81, 82, 84, 89, 91, 111, 112, 113, 114], "t": [0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 16, 22, 24, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 46, 47, 49, 50, 51, 52, 54, 56, 57, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 76, 79, 81, 82, 83, 84, 85, 87, 88, 89, 91, 96, 97, 100, 103, 105, 106, 108, 109, 111, 112, 113, 114, 115, 118, 119, 120, 121], "t_": [4, 13, 34, 36, 47, 111, 112], "t_0": [34, 47, 112], "t_1": [9, 34, 36, 47, 112], "t_2": [34, 36], "t_envelop": 119, "t_i": [9, 34], "t_j": [9, 111, 112, 113], "t_l": 65, "t_m": 9, "t_n": [4, 34, 47, 112], "tabl": [10, 76], "table_i": [10, 76], "tag": 20, "take": [6, 8, 14, 25, 29, 31, 32, 33, 35, 36, 40, 41, 46, 53, 54, 62, 65, 76, 78, 84, 87, 89, 99, 100, 104, 106, 108, 112], "taken": [40, 47, 103], "talk": [40, 100], "tall": 105, "tang": [7, 17, 18, 19, 62, 67, 69, 70], "tangenti": [7, 33, 61, 68, 70], "tangentialfacetfespac": [7, 17, 18, 19, 116], "target": 35, "tartar": [45, 101, 103, 104], "taskmanag": [4, 16, 34, 44, 70, 74, 77, 83, 85, 118, 119, 120, 121], "tau": [4, 5, 18, 19, 28, 34, 47, 53, 57, 60, 61, 62, 67, 69, 70, 79, 83, 86, 88, 89, 91, 105, 111, 112, 113, 114, 115, 118, 119, 120], "tau_": [47, 62, 67, 91], "tau_1": [47, 91], "tau_2": [47, 91], "tau_h": [62, 67, 86, 91], "tau_i": 47, "tau_j": 112, "tau_k": 47, "tau_n": [47, 57, 88, 89, 91], "taun": 70, "tauopt": 47, "taylor": [58, 111, 112], "td": 67, "tdnn": [45, 69, 70], "teach": [25, 46], "team": [15, 20, 42], "tear": 38, "technic": [83, 88, 109], "techniqu": [6, 35, 40, 45, 46, 64, 65, 79, 81, 100, 101, 105, 112], "technologi": 20, "telescop": 65, "tell": [29, 76], "temperatur": [73, 76, 78, 113], "tend": [4, 5, 16, 113, 114, 115, 118, 119, 120, 121], "tensor": [60, 62, 74, 76, 79], "term": [0, 1, 2, 4, 5, 6, 7, 8, 12, 13, 30, 33, 34, 36, 40, 46, 47, 48, 49, 55, 56, 58, 62, 64, 65, 69, 73, 76, 79, 81, 83, 84, 86, 87, 89, 90, 91, 97, 100, 101, 104, 108, 121], "termin": 33, "test": [1, 2, 6, 8, 10, 13, 42, 55, 57, 58, 64, 73, 76, 89, 100, 113, 115], "testfunct": [0, 1, 3, 5, 7, 10, 12, 18, 19, 34, 44, 55, 57, 58, 62, 67, 69, 70, 76, 86, 89, 90, 91, 121], "testspac": [38, 39, 116, 121], "tet": 62, "tetrahedr": [10, 110, 118], "tetrahedra": 9, "text": [6, 30, 34, 43, 47, 48, 49, 50, 51, 53, 55, 57, 58, 60, 61, 62, 63, 65, 67, 68, 70, 73, 74, 76, 79, 80, 82, 83, 84, 88, 89, 90, 100, 102, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115], "textbook": [72, 101], "tfrac": [8, 30, 32, 40, 47, 49, 50, 53, 56, 61, 74, 79, 84, 92, 96, 97, 100, 103, 105, 108, 109, 112, 114, 121], "th": [20, 34, 51, 67, 79, 105, 109], "than": [1, 10, 15, 33, 34, 37, 55, 65, 66, 76, 88, 98, 99, 100, 108], "thank": [64, 81, 86, 108, 109, 121], "thei": [1, 7, 12, 24, 30, 63, 64, 73, 79, 83, 108, 109, 111, 121], "them": [3, 10, 21, 24, 46], "theo_subdomainh1": 100, "theorem": [8, 12, 28, 30, 31, 32, 36, 37, 45, 46, 53, 54, 64, 65, 66, 76, 79, 81, 83, 84, 88, 90, 99, 101, 102, 103, 105, 108, 109], "theoret": 46, "theori": [25, 31, 51, 63, 64, 65, 72, 79, 101, 108, 109], "therefor": 32, "thermal": 73, "thesi": [1, 7, 59, 62, 72], "theta": 47, "thi": [0, 1, 2, 5, 6, 7, 8, 10, 11, 12, 13, 20, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 60, 61, 62, 63, 64, 65, 66, 67, 69, 72, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 97, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115, 121], "thick": [69, 70], "thin": [71, 121], "think": [40, 52, 53, 63], "third": [2, 6, 53, 86, 91, 98, 99, 111], "thm": 101, "thoma": [7, 9, 61, 86, 88], "thread": 11, "three": [47, 62, 73, 79, 82, 88], "threorem": 45, "through": [25, 32, 35, 88], "thu": [1, 2, 3, 4, 6, 12, 13, 27, 28, 29, 31, 32, 34, 40, 46, 47, 48, 50, 51, 53, 54, 55, 56, 57, 58, 62, 64, 65, 66, 69, 73, 76, 79, 80, 83, 84, 86, 87, 88, 90, 91, 92, 98, 99, 100, 101, 103, 104, 105, 106, 110, 112, 115, 118, 121], "tightli": 67, "tild": [9, 30, 33, 37, 41, 42, 50, 51, 66, 73, 79, 84, 97, 100, 104, 108], "till": 8, "time": [0, 1, 4, 6, 8, 9, 12, 13, 15, 24, 25, 27, 28, 30, 31, 35, 37, 40, 49, 52, 53, 54, 55, 58, 61, 63, 64, 68, 75, 76, 79, 81, 82, 84, 97, 98, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 115, 119, 120, 121], "timer2list": 121, "timestep": 16, "tk": 62, "tn": 2, "tnt": [0, 2, 4, 6, 10, 11, 12, 14, 17, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 47, 48, 49, 50, 51, 55, 56, 63, 64, 66, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "todo": 20, "togeth": [0, 7, 8, 10, 11, 24, 27, 34, 49, 63, 76, 81, 83, 103, 106], "tol": [47, 49, 74, 81, 85], "told": 47, "toler": 33, "too": [53, 58, 121], "tool": [46, 63, 65, 83, 102, 105, 108, 109], "toolbar_vis": 47, "toolkit": 23, "top": [10, 34, 44, 56, 62, 73, 76, 80, 82], "topologi": 110, "tordman": 118, "total": [9, 55, 57, 119, 120], "touch": 46, "toward": [8, 76], "tpeak": 119, "tr": [8, 12, 38, 41, 74, 90, 96, 97, 100, 101, 103, 104], "trace": [3, 6, 12, 34, 45, 46, 53, 55, 62, 70, 74, 76, 79, 96, 108, 121], "traceback": [58, 119, 120, 121], "tracenorm": 97, "traceop": [116, 121], "traceoper": [10, 76, 121], "tracespac": [10, 76], "traction": 74, "tran": 70, "transfer": [14, 83, 100, 121], "transform": [8, 9, 10, 12, 13, 45, 46, 51, 76, 82, 85, 97, 100, 106], "transfrom": 81, "transistor": 40, "transmiss": 73, "transport": [5, 45, 46, 111], "transpos": [13, 63, 84, 109], "trapezoid": [111, 114], "travel": 121, "treat": [2, 3, 5, 55], "treatment": [5, 74], "trial": [6, 10, 76, 78], "trialfunct": [0, 1, 3, 5, 7, 10, 12, 18, 19, 34, 44, 55, 57, 58, 62, 67, 69, 70, 76, 86, 89, 91, 121], "trialspac": [38, 39, 116, 121], "triangl": [8, 9, 10, 33, 34, 36, 54, 66, 83, 86, 88, 106, 118], "triangul": [8, 9, 76, 110], "triangular": [10, 51, 52, 53, 67, 80, 88, 110, 111, 118], "trick": [1, 8], "tricki": [1, 2, 55, 63], "trig": [58, 81, 118], "trigonometr": [0, 47], "tripl": 9, "trivial": [27, 53, 66, 83, 87, 99, 100, 101, 106], "trottenberg": [63, 64], "true": [0, 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 28, 29, 33, 34, 35, 36, 42, 43, 44, 47, 48, 50, 51, 53, 54, 55, 56, 62, 63, 66, 67, 69, 70, 73, 74, 76, 81, 82, 83, 84, 85, 88, 91, 92, 100, 101, 103, 104, 108, 109, 110, 113, 114, 115, 116, 118, 121], "try": [25, 30, 33, 46, 58, 75, 83], "tt": 70, "tu": [25, 45, 46, 56], "tu_1": 27, "tucker": [53, 84], "tupl": [10, 20, 76], "tutori": [11, 20, 24, 25, 75], "tuwel": 46, "tuwien": [10, 11], "tv": 27, "twice": [64, 82, 104], "two": [0, 2, 8, 9, 21, 24, 30, 32, 34, 36, 38, 47, 48, 50, 52, 63, 73, 75, 76, 80, 82, 88, 91, 97, 100, 101, 105, 106, 108, 109, 114], "typ": 121, "type": [10, 14, 21, 23, 24, 34, 65, 67, 76, 78, 84, 109, 121], "type1": 62, "typic": [1, 2, 9, 40, 52, 53, 76, 88, 105, 113, 114], "typo": 55, "u": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 115, 118, 121], "u0": [73, 113, 114, 115, 118], "u1": [22, 70, 85], "u2": [70, 86], "u3": 70, "u_": [6, 24, 31, 34, 36, 42, 53, 64, 65, 76, 92, 95, 97, 104, 106, 115], "u_0": [4, 32, 53, 65, 66, 73, 82, 100, 106, 113, 114, 115], "u_1": [2, 27, 29, 30, 31, 32, 53, 55, 61, 76, 82, 91, 107, 113, 115], "u_2": [2, 27, 29, 30, 31, 55, 61, 65, 91, 107], "u_d": [0, 2, 8, 41, 53, 55, 57, 73, 74, 83, 84, 89, 104, 113], "u_f": [73, 83, 106, 108, 109], "u_g": 100, "u_h": [6, 8, 12, 13, 28, 29, 31, 33, 34, 35, 36, 54, 56, 62, 65, 76, 83, 86, 87, 91, 103, 105, 108, 109, 112, 113], "u_i": [13, 24, 31, 38, 40, 42, 64, 76, 82, 83, 100, 105, 106, 108, 109, 113, 115], "u_j": [38, 40, 100, 105], "u_k": [41, 65, 98, 100, 101], "u_l": [29, 64, 65, 66, 78, 100, 101], "u_m": [27, 100, 115], "u_n": [12, 27, 31, 76, 97, 100, 113], "u_p": 79, "u_r": [31, 78], "u_t": [6, 62], "u_x": 82, "ud": [2, 73], "uext": 66, "uh": 70, "uhat": [0, 1, 5, 7, 18, 19, 91], "ui": [37, 38, 39], "uin": [7, 16, 18, 19, 22, 81, 85], "ulrich": 63, "umberto": 14, "umfpackinvers": 58, "un": 115, "unbound": [98, 99, 119, 120], "unconstrain": 82, "under": [8, 12, 47, 79, 84, 99], "underbrac": [1, 28, 29, 32, 47, 48, 51, 56, 79, 86, 101], "underli": 65, "underlin": [8, 13, 63, 65, 84, 105, 109], "understand": [84, 90], "understood": [24, 34, 69, 76, 78, 87, 102], "undocu": 55, "unew": 118, "unfortun": [35, 39], "uniform": [8, 12, 15, 33, 48, 65, 77], "uniformli": [12, 100, 103], "union": 76, "uniqu": [24, 27, 28, 29, 30, 31, 32, 50, 53, 76, 79, 83, 88, 90, 104, 105], "unisolv": 34, "unit": [8, 11, 75, 76, 79, 102, 104, 105, 108], "unit_cub": [46, 55, 116], "unit_squar": [0, 1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 15, 17, 21, 23, 24, 42, 43, 47, 48, 49, 50, 51, 55, 56, 57, 58, 63, 64, 66, 67, 69, 73, 76, 80, 84, 86, 88, 91, 103, 106, 108, 109, 110, 113, 114, 115, 118, 121], "uniti": 109, "unknown": [33, 40, 58, 59, 77, 79, 112, 113, 114], "unless": [83, 98], "unlik": 37, "unnecessari": 33, "unset": 76, "unsetdeform": [0, 55, 76], "unsetpml": 76, "uold": 118, "up": [1, 6, 8, 12, 13, 14, 15, 24, 28, 30, 31, 49, 51, 58, 63, 66, 79, 83, 97, 98, 99, 101, 109, 111, 112, 113, 121], "updat": [0, 10, 11, 21, 47, 49, 50, 51, 55, 58, 63, 76, 81, 83, 114], "updatedoft": [10, 76], "upgrad": [20, 103], "upost": 86, "upper": [33, 34, 35, 51, 81, 89, 103, 109], "upwind": [4, 6], "uri": 5, "us": [0, 1, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 21, 24, 25, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 67, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 90, 91, 97, 98, 100, 101, 102, 103, 105, 106, 108, 109, 111, 112, 113, 118, 121], "usag": 77, "user": 46, "usethread": 15, "usual": [1, 6, 9, 13, 33, 35, 40, 55, 68, 81, 84, 86, 108, 111], "ut": 115, "uup": [4, 5, 6], "uvec": 70, "uvert": [37, 39], "uxi": 82, "uzawa": 84, "v": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 115, 118, 121], "v0": [114, 118], "v1": [3, 22, 23, 70, 85], "v1loc": 23, "v2": [3, 23, 70, 86], "v2loc": 23, "v3": 70, "v4": 20, "v_": [8, 9, 12, 13, 31, 34, 42, 53, 63, 64, 67, 79, 83, 100, 103], "v_0": [8, 53, 63, 65, 66, 67, 82, 83, 100, 101, 104, 114], "v_1": [2, 28, 30, 31, 55, 63, 66, 84, 100, 101, 107], "v_2": [2, 28, 30, 31, 55, 84, 107], "v_d": [8, 101, 104], "v_g": 31, "v_h": [6, 7, 8, 12, 13, 24, 28, 29, 31, 34, 35, 36, 54, 62, 65, 68, 76, 83, 84, 86, 87, 91, 103, 105, 106, 109, 112, 113], "v_i": [31, 38, 40, 42, 83, 104, 105, 108, 109], "v_j": [40, 99, 105, 108], "v_k": 65, "v_l": [63, 64, 65, 66, 121], "v_m": [32, 115], "v_n": [3, 27, 32, 104, 121], "v_r": [31, 121], "v_t": [8, 9, 24, 34, 62, 88], "val": [23, 42, 76], "valid": 98, "valu": [0, 1, 2, 3, 12, 15, 24, 27, 31, 33, 34, 35, 36, 41, 42, 47, 48, 50, 55, 58, 61, 62, 64, 66, 73, 76, 78, 79, 88, 89, 90, 97, 100, 101, 103, 105, 106, 108, 109, 112, 113, 121], "van": 118, "vanish": [12, 34, 36, 49, 79, 80, 86, 98, 100], "varepsilon": [5, 27, 32, 47, 48, 49, 56, 60, 62, 69, 71, 74, 83, 90, 99, 103, 106, 107], "varepsilon_": 70, "vari": [78, 112], "variabl": [1, 3, 10, 15, 20, 37, 38, 39, 40, 46, 51, 53, 54, 55, 56, 57, 61, 64, 76, 89, 91, 112, 113, 121], "variant": [84, 105, 121], "variat": [8, 12, 30, 34, 45, 46, 53, 60, 67, 73, 75, 76, 78, 82, 83, 84, 86, 87, 91, 101, 104, 105, 114], "varphi": [9, 13, 32, 34, 62, 64, 80, 90, 98, 99, 100], "varphi_": [8, 9, 13, 64, 66, 79, 90, 106], "varphi_i": [9, 12, 13, 36, 80, 99, 105, 106, 108], "varphi_j": [12, 13, 105], "varphi_l": 64, "varphi_t": [9, 13, 62], "vb": [76, 78], "vdot": [105, 111], "vec": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "vecmap": 14, "vector": [3, 4, 6, 10, 11, 12, 14, 21, 23, 27, 29, 33, 34, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 60, 62, 63, 64, 66, 69, 70, 73, 75, 76, 78, 82, 83, 84, 88, 96, 100, 101, 105, 106, 108, 109, 115, 121], "vectorh1": [0, 17, 19, 22, 33, 55, 56, 58, 74, 81, 82, 85, 103], "vectori": 72, "vectorl2": [18, 19, 116, 121], "vectormap": 14, "vectorvalu": 12, "vegt": 118, "vel": 7, "veloc": [7, 12, 16, 58, 59, 84, 114], "venv": 46, "veri": [0, 6, 21, 24, 33, 46, 49, 51, 55, 64, 65, 66, 74, 76, 82, 83, 89, 98, 103, 108, 112, 115], "verif": 75, "verifi": [12, 28, 30, 41, 48, 50, 54, 64, 83, 89, 99, 100, 103, 106, 111, 112], "versa": [104, 121], "version": [1, 2, 3, 8, 20, 46, 50, 72, 79, 81, 103, 105, 110, 121], "vert": 10, "vertex": [9, 12, 33, 34, 36, 37, 39, 62, 76, 80, 83, 104, 109, 110], "vertic": [8, 12, 33, 36, 37, 39, 54, 59, 67, 69, 76, 79, 104, 109, 110], "vf": [7, 34], "vhat": [0, 1, 3, 5, 7, 18, 19, 91], "vi": [37, 38, 39], "via": [1, 3, 20, 23, 28, 30, 40, 46, 47, 51, 64, 65, 79, 99, 105, 106, 108, 109, 115, 121], "vice": [104, 121], "view": 14, "violat": [12, 36], "virtual": [46, 55], "viscos": 7, "visibl": [10, 76], "visual": [15, 17, 25, 109], "vnum": 76, "vol": [10, 11, 15, 22, 33, 34, 35, 38, 39, 76, 78, 96, 108, 109], "vol_or_bnd": [10, 76], "voltag": 82, "volum": [0, 6, 8, 55, 75, 76, 78, 108], "vorb": [10, 76, 78], "vt": 7, "vvert": [37, 39], "w": [0, 3, 4, 5, 6, 8, 9, 12, 18, 19, 21, 27, 28, 30, 31, 32, 33, 34, 35, 41, 42, 47, 49, 51, 53, 63, 64, 67, 68, 69, 70, 73, 75, 79, 81, 82, 88, 98, 100, 101, 104, 105, 109, 113, 118, 119, 120, 121], "w_": [49, 83, 99], "w_0": 53, "w_2": 99, "w_e": 100, "w_h": [12, 28, 31, 35, 67, 68, 69, 83], "w_i": [51, 83, 105, 109], "w_k": [49, 66], "w_l": [65, 66], "w_n": [3, 49], "w_p": 99, "wa": [1, 28, 36, 37, 65, 100, 103, 105], "wai": [1, 3, 12, 13, 25, 31, 33, 35, 75, 82, 83, 106, 110], "wall": [7, 12, 16, 18, 19, 22, 81, 85], "want": [5, 20, 25, 34, 35, 40, 49, 54, 55, 66, 76, 82, 83, 84, 101, 103, 113, 121], "warbuton": 121, "warn": [38, 39, 55], "watch": 77, "wave": [43, 46, 112, 118], "wavi": 6, "wb_fulledg": 76, "wb_withedg": [76, 77], "we": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 69, 73, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 97, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115, 118, 121], "weak": [8, 12, 29, 31, 43, 46, 56, 57, 58, 73, 78, 83, 88, 90, 98, 99, 100, 106, 113], "weaker": [8, 31, 55], "weakli": 55, "webglscen": 121, "webgui": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 33, 34, 35, 37, 38, 39, 42, 43, 46, 47, 48, 49, 55, 56, 57, 58, 63, 64, 66, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "webgui_jupyter_widget": [46, 121], "weight": [8, 13, 34, 35, 79, 104, 111], "well": [12, 27, 36, 40, 41, 51, 61, 62, 64, 65, 67, 76, 79, 81, 83, 84, 90, 97, 100, 104, 108, 109, 112, 115, 118, 121], "were": [32, 62, 91, 99], "wf": 34, "what": [1, 2, 3, 8, 23, 28, 30, 34, 36, 50, 51, 75, 76, 77, 86, 87, 88, 89, 91, 105], "wheel": 20, "when": [0, 25, 40, 47, 48, 63, 88, 98], "whenev": 48, "where": [0, 1, 2, 3, 6, 8, 9, 12, 20, 27, 33, 34, 35, 36, 37, 39, 42, 51, 54, 56, 61, 62, 63, 64, 65, 66, 68, 69, 73, 76, 78, 79, 80, 82, 83, 84, 100, 102, 103, 105, 106, 108, 109, 112, 113, 115, 118, 121], "whether": 6, "which": [0, 1, 5, 8, 10, 12, 15, 20, 23, 27, 28, 29, 30, 32, 34, 35, 37, 38, 40, 46, 47, 49, 50, 54, 55, 58, 60, 63, 64, 65, 66, 76, 79, 81, 82, 83, 84, 88, 97, 98, 99, 100, 102, 103, 105, 106, 108, 109, 110, 112, 113], "while": [4, 5, 16, 33, 35, 48, 50, 51, 58, 63, 64, 119, 120, 121], "who": 105, "whole": [53, 55, 73, 90, 100, 104, 105], "whose": 100, "why": [30, 55, 100], "widehat": [1, 3, 5, 8, 9, 12, 13, 36, 81, 88, 91, 99, 106], "wider": 9, "widetild": [9, 33, 37, 42, 47, 73, 86, 99, 100, 104, 109], "widget": [47, 121], "width": [23, 40, 66, 109], "wien": [25, 45, 46], "wiki": [52, 111], "wikipedia": [52, 111], "wind": [4, 6], "window": 20, "winther": [60, 83], "wip": 76, "wirebasket": 76, "wirebasket_dof": [18, 19], "wise": [6, 15, 24, 33, 34, 37, 56, 58, 60, 70, 76, 86, 88, 106, 108, 121], "witch": 37, "within": [1, 10, 15, 20, 25, 29, 46, 65, 80, 101, 108, 111, 121], "without": [0, 21, 25, 30, 34, 39, 46, 50, 54, 65, 84], "wolfgang": 63, "work": [6, 15, 24, 25, 26, 40, 46, 47, 62, 70, 100, 108, 118], "worker": 15, "world": [15, 20], "worri": 73, "wors": 33, "worth": 103, "would": [5, 6, 35, 54], "wr": [49, 81], "wrap": 14, "write": [11, 13, 28, 47, 51, 54, 56, 64, 76, 79, 82, 105, 106, 109, 112], "written": [28, 86, 111, 115], "wrn": [49, 81], "wrt": 54, "www": [20, 23], "x": [0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 28, 30, 34, 36, 37, 38, 39, 41, 42, 43, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 62, 63, 64, 66, 67, 69, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 85, 86, 88, 89, 91, 92, 96, 97, 98, 99, 100, 101, 102, 103, 104, 108, 109, 110, 112, 113, 114, 115, 116, 118, 121], "x2": 86, "x_": [8, 12, 49], "x_0": 49, "x_1": [13, 61, 98], "x_2": 61, "x_i": [8, 12, 33, 36, 51], "x_j": [12, 51], "x_k": [13, 49], "x_m": 13, "x_n": [49, 98], "xaux": [17, 19], "xc": 66, "xi": [64, 79, 100], "xlabel": [103, 115], "xlam": 34, "xscale": 33, "xsigma": 34, "xu": [31, 65, 105], "xw": 34, "xwf": 34, "xx": 46, "xy": 75, "y": [0, 2, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 22, 30, 34, 37, 38, 39, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 55, 63, 66, 70, 74, 75, 76, 78, 79, 80, 81, 84, 85, 88, 89, 92, 96, 97, 100, 101, 102, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118, 121], "y_": [111, 112], "y_0": [112, 115], "y_j": [111, 112], "y_k": 112, "yet": [0, 49, 66], "ylabel": 115, "you": [10, 20, 25, 26, 46, 51, 73, 75, 77, 103, 108], "young": [30, 81], "your": [20, 25, 26, 46, 75, 77, 102], "yousef": 47, "yscale": [33, 48, 49, 50, 51, 80, 115], "z": [22, 29, 30, 41, 47, 48, 49, 53, 64, 74, 76, 83, 85, 100, 101, 104, 105, 111, 112, 115, 116, 118, 121], "z_": 101, "z_0": [41, 100], "z_i": 105, "z_k": [41, 100], "z_l": 100, "za": 111, "zaglmayr": [72, 80, 88], "zampini": 14, "zerbinati": 14, "zero": [1, 2, 12, 53, 58, 66, 73, 81, 84, 100, 106, 108, 112, 121], "zhao": 83, "zikatanov": 31, "zip": [33, 38, 39, 42, 80, 103, 115], "znew": 118, "zold": 118, "zulehn": 67, "zz": 33}, "titles": ["40. Nitsche\u2019s Method for boundary and interface conditions", "42. Hybrid DG for elliptic equations", "41. DG - Methods for elliptic problems", "44. Fourth Order Equation", "39. Instationary Transport Equation", "43. Splitting Methods for the time-dependent convection diffusion equation", "38. Stationary Transport Equation", "45. H(div)-conforming Stokes", "24. Finite element error analysis", "20. Finite Element Method", "21. Implementation of Finite Elements", "23. Implement our own system assembling", "25. Non-conforming Finite Element Methods", "22. Finite element system assembling", "88. NGSolve - PETSc interface", "84. Distributed Meshes and Spaces", "<no title>", "<no title>", "<no title>", "<no title>", "83. Introduction to MPI with mpi4py", "86. Iteration methods in parallel", "89. Solving Stokes in parallel", "87. Using PETSc", "85. Consistent and Distributed Vectors", "This repository contains an interactive introduction to the Finite Element Method", "<no title>", "7. Basic properties", "10. Coercive variational problems and their approximation", "9. Riesz representation theorem and symmetric variational problems", "12. Exercises", "11. Inf-sup stable variational problems", "8. Projection onto subspaces", "26. A posteriori error estimates", "29. Equilibrated Residual Error Estimates", "28. Goal driven error estimates", "27. The residual error estimator", "82. BDDC - Preconditioner", "80. FETI methods", "81. FETI-DP", "77. Introduction to Non-overlapping Domain Decomposition", "78. Traces spaces", "Implement a parallel BDDC preconditioner", "Helmholtz Equation", "Grating", "An Interactive Introduction to the Finite Element Method", "An Interactive Introduction to the Finite Element Method", "61. The Chebyshev Method", "58. The Richardson Iteration", "62. Conjugate Gradients", "59. The Gradient Method", "60. Preconditioning", "57. Basic Iterative Methods", "35. Abstract Theory", "36. Abstract theory for mixed finite element methods", "33. Boundary Conditions", "37. Parameter Dependent Problems", "34. Mixed Methods for second order equations", "32. Stokes Equation", "Non-linear dynamics", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\nHellinger Reissner mixed formulation", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\n\\(\\DeclareMathOperator{\\opcurl}{curl}\\)\n\\(\\DeclareMathOperator{\\eps}{\\varepsilon}\\)\nReduced symmetry methods", "Tangential displacement normal normal stress continuous finite elements", "69. Multigrid and Multilevel Methods", "71. Analysis of the Multigrid Iteration", "70. Analysis of the multi-level preconditioner", "72. Multi-level Extension", "The Hellan Herrmann Johnson Method for Kirchhoff plates", "Relationship between HHJ and TDNNS", "Reissner Mindlin Plates", "Nonlinear Shells", "3D-TDNNS Elasticity and Reissner Mindlin Plate elements", "Preamble", "2. Boundary Conditions", "5. 3D Solid Mechanics", "6. Exercises", "1. Solving the Poisson Equation", "4. Iterative Solvers", "3. Variable Coefficients", "30. hp - Finite Elements", "31. Implementation of High Order Finite Elements", "74. The Bramble-Pasciak Transformation", "75. A Small Number of Constraints", "76. Parameter Dependent Problems", "73. Structure of Saddle-point Problems", "<no title>", "49. Finite Element Error Analysis", "50. Error Analysis in \\(L_2 \\times H^1\\)", "48. Finite Elements in \\(H(\\operatorname{div})\\)", "46. Application of the abstract theory", "47. The function space \\(H(\\operatorname{div})\\)", "51. Hybridization Techniques", "Friedrichs\u2019 Inequality", "<no title>", "<no title>", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "13. Generalized derivatives", "14. Sobolev spaces", "15. Trace theorems and their applications", "16. Equivalent norms on \\(H^1\\) and on sub-spaces", "19. Exercises", "18. Experiments with norms", "17. The weak formulation of the Poisson equation", "63. Additive Schwarz Methods", "64. Some Examples of ASM preconditioners", "68. Exercise: Robust preconditioners", "66. Domain Decomposition with minimal overlap", "67. Overlapping Domain Decomposition Methods", "65. Schwarz preconditioners for high order finite elements", "Runge Kutta Methods", "Single-step methods", "52. Heat Equation", "53. Wave Equation", "Exponential Integrators for Parabolic Equations", "<no title>", "<no title>", "54. Mass-lumping and Local time-stepping", "56. Nano-optics: A ring-resonator", "Nano-optics: A ring-resonator", "55. Discontinuous Galerkin for the Wave Equation"], "titleterms": {"": [0, 38, 41, 103], "0": 3, "1": [12, 68, 76, 87, 100, 101, 106], "1d": [79, 102], "2": [100, 109], "2d": 30, "3d": [71, 74], "A": [33, 45, 46, 82, 107, 119, 120], "One": 30, "The": [12, 15, 20, 33, 36, 45, 46, 47, 48, 50, 51, 62, 64, 67, 76, 79, 81, 90, 96, 97, 100, 104, 105, 118], "With": 108, "abstract": [45, 46, 53, 54, 89], "ad": 108, "addit": [45, 105], "algebra": 75, "algorithm": [33, 64, 66], "alpha": 48, "an": [25, 45, 46], "analysi": [8, 45, 64, 65, 83, 86, 87, 108, 109], "appendix": 45, "applic": [89, 100], "approxim": [8, 28, 31, 64], "around": 8, "asm": 106, "assembl": [11, 13], "b": 107, "base": [79, 108, 112], "basi": [76, 80], "basic": [27, 52], "bddc": [37, 42], "between": 68, "bilienarform": 75, "bilinear": [1, 30, 76], "bisect": 33, "block": [30, 84, 105, 110], "bottom": 41, "bound": 105, "boundari": [0, 2, 8, 41, 53, 55, 56, 57, 66, 73, 83], "brambl": [81, 102], "build": 30, "butcher": 111, "c": 3, "calcul": 11, "can": 11, "case": 79, "chebyshev": 47, "class": 15, "classif": 112, "cl\u00e9ment": 36, "coars": [83, 108, 109], "coeffici": [33, 78], "coefficientfunct": 75, "coerciv": [1, 28, 30], "commut": 68, "comparison": 109, "complement": 84, "complex": 30, "comput": [13, 34, 66, 75], "conda": 20, "condit": [0, 2, 8, 30, 53, 55, 56, 57, 73, 83, 112], "conform": [7, 12], "conjug": 49, "consist": 24, "constrain": [53, 84], "constraint": 82, "contain": 25, "continu": [1, 3, 62], "convect": 5, "converg": [64, 111], "correct": [45, 46, 83], "curl": 61, "cycl": 64, "data": 66, "dd": 109, "declaremathoper": [60, 61], "decomposit": [40, 45, 46, 66, 108, 109], "definit": 100, "delta": 34, "depend": [5, 45, 46, 56, 83], "deriv": [30, 98], "detail": 118, "dg": [1, 2], "diagon": [106, 111], "diagram": 68, "differenti": [45, 121], "diffus": 5, "dirichlet": [2, 8, 53, 56, 83], "discontinu": [6, 45, 46, 121], "discret": [1, 54, 121], "displac": 62, "distribut": [15, 24], "div": [7, 33, 60, 61, 88, 90], "diverg": 62, "domain": [40, 41, 45, 46, 90, 100, 102, 108, 109], "dp": 39, "driven": 35, "dual": [60, 75, 89], "dynam": 59, "edg": [33, 41, 100], "ee": 112, "effici": [36, 66, 121], "eigenvalu": 121, "elast": [45, 71, 74], "element": [8, 9, 10, 11, 12, 13, 15, 25, 45, 46, 54, 58, 62, 71, 76, 79, 80, 86, 88, 110], "ellipt": [1, 2], "enough": 30, "ep": 61, "eqival": 102, "equat": [1, 3, 4, 5, 6, 12, 43, 45, 46, 53, 57, 58, 67, 76, 83, 104, 112, 113, 114, 115, 121], "equilibr": 34, "equival": 101, "error": [8, 33, 34, 35, 36, 46, 54, 62, 79, 86, 87], "essenti": 73, "estim": [8, 33, 34, 35, 36, 46, 54, 62, 65, 79, 103], "euler": [112, 113], "evalu": 102, "exampl": [30, 35, 54, 56, 84, 106, 111], "exercis": [11, 30, 75, 102, 107, 111, 112, 121], "expand": 49, "experi": [48, 75, 103, 109], "explicit": [111, 112], "exponenti": 115, "extend": 66, "extens": [66, 100], "feti": [38, 39], "finest": 63, "finit": [8, 9, 10, 12, 13, 15, 25, 45, 46, 54, 58, 62, 76, 79, 80, 86, 88, 110], "first": [12, 30], "flux": [33, 83], "form": [1, 30, 76], "formul": [58, 60, 62, 74, 76, 83, 89, 104, 113], "fortin": 54, "fourth": 3, "fraction": 102, "framework": 34, "free": 65, "friedrich": [92, 95, 102, 103], "from": [30, 63], "function": [11, 62, 76, 80, 90, 102, 112], "galerkin": [6, 45, 46, 113, 121], "gauss": 51, "gener": [34, 98, 100, 105], "geometr": [8, 75], "geometri": 118, "get": 75, "goal": 35, "grade": 8, "gradient": [49, 50, 51], "graph": 108, "grate": 44, "green": 33, "grid": [83, 108, 109], "gridfunct": 75, "h": [7, 33, 41, 68, 76, 87, 88, 90, 100, 101, 106], "happen": 20, "hdg": 1, "heat": 113, "hellan": 67, "helling": 60, "helmholtz": [43, 45], "herrmann": 67, "hhj": 68, "high": [8, 46, 80, 110], "hilbert": [102, 105], "hp": 79, "hybrid": [0, 1, 3, 91], "hyperbol": 112, "i": 30, "ie": 112, "implement": [10, 11, 42, 80, 121], "implicit": [111, 112, 113], "improv": 112, "incompress": 56, "inequ": [45, 92, 95, 96, 97, 102, 103], "inf": [30, 31], "inner": 24, "inspect": 11, "instal": [20, 46], "instationari": 4, "integr": [11, 100, 112, 115], "interact": [25, 45, 46], "interfac": [0, 14, 55], "interior": 3, "interpol": [41, 79, 100, 102], "introduct": [20, 25, 40, 45, 46], "invers": 103, "ipyparallel": 20, "iter": [21, 45, 46, 47, 48, 51, 52, 64, 77, 84], "jacobi": [51, 105], "johnson": 67, "kirchhoff": 67, "korn": 103, "krylov": 49, "kutta": 111, "l_2": [8, 79, 87, 106], "lagrang": 40, "laplac": 121, "lbb": 54, "legendr": 79, "lemma": [12, 102, 105], "level": [63, 65, 66, 83, 109], "librari": 20, "lift": 34, "linear": [59, 74, 75, 76, 112], "linearform": 75, "literatur": 46, "local": [86, 118], "lump": 118, "mark": 33, "mass": 118, "materi": 56, "mathbb": 105, "matric": [13, 24, 63], "matrix": [11, 24], "maxwel": 83, "mean": [30, 102], "mechan": 74, "mesh": [8, 15, 33, 108], "method": [0, 2, 3, 5, 6, 9, 12, 21, 25, 38, 45, 46, 47, 50, 51, 52, 53, 54, 57, 60, 61, 63, 67, 76, 83, 105, 108, 109, 111, 112, 113, 114, 118], "mid": 112, "mindlin": [69, 71], "minim": [30, 49, 53, 84, 108, 109], "mix": [45, 46, 53, 54, 57, 60, 89], "ml": 65, "model": 75, "mpi": 20, "mpi4pi": 20, "multi": [65, 66], "multigrid": [45, 46, 63, 64], "multilevel": 63, "multipl": 24, "n": 105, "nano": [119, 120], "natur": [41, 73], "nearli": [56, 65], "netgen": 75, "newmark": 114, "ngsolv": [14, 33, 34, 46, 76], "nitsch": 0, "nn": 62, "non": [12, 40, 45, 46, 59], "nonlinear": 70, "norm": [8, 41, 75, 101, 103, 106], "normal": [62, 90], "number": 82, "numer": [45, 97], "one": 100, "onto": [30, 32], "opcurl": [61, 68], "opdiv": [60, 61], "opencascad": 75, "oper": [24, 36, 54, 100, 121], "operatornam": [33, 88, 90], "optic": [119, 120], "optim": [48, 64, 65], "order": [3, 8, 30, 45, 46, 53, 57, 80, 110], "ordinari": 45, "orthogon": 79, "our": 11, "over": 100, "overlap": [40, 45, 46, 105, 108, 109, 110], "own": 11, "p": 12, "packag": 20, "parabol": [45, 112, 115], "parallel": [21, 22, 42, 45, 46], "paralleldof": 15, "paramet": [40, 48, 56, 83], "part": 100, "partit": 108, "pasciak": 81, "penalti": [3, 56, 83], "petsc": [14, 20, 23], "piec": 62, "piola": 88, "pip": 20, "plate": [45, 67, 69, 71], "poincar\u00e9": [96, 102, 103], "point": [45, 46, 84, 102, 112], "poisson": [76, 104], "polynomi": [47, 79], "post": 86, "posteriori": [33, 46], "practic": 45, "preambl": 72, "precondit": [51, 63, 84], "precondition": [14, 37, 38, 42, 51, 63, 65, 82, 105, 106, 107, 109, 110], "primal": [60, 89], "problem": [2, 28, 29, 30, 31, 45, 46, 49, 53, 56, 83, 84], "process": 86, "product": 24, "project": [32, 63, 79, 82], "proof": 97, "properti": [27, 64], "prove": 54, "r": 105, "real": 30, "recoveri": 33, "red": 33, "reduc": 61, "refin": [8, 33], "regular": 65, "reissner": [60, 69, 71], "relationship": 68, "relax": 48, "reliabl": 36, "repeat": 30, "repositori": 25, "represent": 29, "residu": [34, 36], "reson": [119, 120], "richardson": [21, 48, 51], "riesz": 29, "ring": [119, 120], "rk2": 112, "robust": [83, 107], "rule": 112, "rung": 111, "saddl": [45, 46, 84], "scale": 102, "schur": 84, "schwarz": [105, 110], "second": [12, 30, 45, 46, 53, 57], "segment": 80, "seidel": 51, "shell": [45, 70], "shift": 104, "sigma": 34, "simpl": [111, 112], "singl": 112, "singular": 8, "small": [82, 106], "smooth": 62, "smoother": 83, "smoth": 64, "sobolev": [45, 46, 76, 99, 100, 102], "solid": 74, "solv": [12, 22, 49, 76, 121], "solver": [45, 46, 77], "some": [45, 106], "space": [15, 41, 45, 46, 49, 58, 76, 90, 99, 100, 101, 102, 105, 108, 113], "split": 5, "squar": [100, 102], "stabil": [6, 112], "stabl": 31, "start": 75, "stationari": 6, "step": [20, 102, 112, 113, 114, 118], "stoke": [7, 12, 22, 53, 54, 58, 83], "strang": 12, "stress": 62, "structur": 84, "sub": [41, 45, 46, 90, 100, 101, 105], "subspac": 32, "sup": [30, 31], "symmetr": 29, "symmetri": 61, "system": [11, 13, 30, 53], "tableau": 111, "tangenti": 62, "tdnn": [62, 68, 71], "techniqu": 91, "term": 106, "test": 121, "theorem": [29, 100, 104], "theori": [45, 46, 53, 54, 89], "thi": 25, "three": 20, "time": [5, 45, 46, 87, 113, 114, 118], "trace": [41, 90, 97, 100, 103], "transform": [81, 88], "transport": [4, 6], "trapezoid": 112, "triangl": [12, 79, 80], "two": 83, "upper": 105, "us": [11, 20, 23], "v": 64, "valu": [30, 102], "varepsilon": 61, "variabl": [33, 78], "variat": [28, 29, 31, 58, 62, 74, 113], "vector": [13, 24], "verif": 97, "verlet": 118, "version": 102, "vertex": 8, "visual": 76, "wave": [45, 114, 121], "we": 11, "weak": [76, 104], "wise": 62, "within": 53, "without": 20, "work": 75, "zhu": 33, "zienkiewicz": 33}}) \ No newline at end of file +Search.setIndex({"alltitles": {"3D Solid Mechanics": [[74, "d-solid-mechanics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[71, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]], "A Small Number of Constraints": [[82, "a-small-number-of-constraints"]], "A posteriori error estimates": [[33, "a-posteriori-error-estimates"], [46, null]], "A practical introduction": [[45, "a-practical-introduction"]], "Abstract Theory": [[45, "abstract-theory"], [46, null], [53, "abstract-theory"]], "Abstract theory for mixed finite element methods": [[54, "abstract-theory-for-mixed-finite-element-methods"]], "Adding a coarse grid space": [[108, "adding-a-coarse-grid-space"]], "Additional methods": [[45, "additional-methods"]], "Additive Schwarz Methods": [[105, "additive-schwarz-methods"]], "Algorithm": [[66, "algorithm"]], "An Interactive Introduction to the Finite Element Method": [[45, "an-interactive-introduction-to-the-finite-element-method"], [46, "an-interactive-introduction-to-the-finite-element-method"]], "Analysis of the 2-level method:": [[109, "analysis-of-the-2-level-method"]], "Analysis of the DD preconditioner": [[109, "analysis-of-the-dd-preconditioner"]], "Analysis of the Multigrid Iteration": [[64, "analysis-of-the-multigrid-iteration"]], "Analysis of the method": [[108, "analysis-of-the-method"]], "Analysis of the multi-level preconditioner": [[65, "analysis-of-the-multi-level-preconditioner"]], "Appendix": [[45, "appendix"]], "Application of the abstract theory": [[89, "application-of-the-abstract-theory"]], "Approximation of Dirichlet boundary conditions": [[8, "approximation-of-dirichlet-boundary-conditions"]], "Approximation of coercive variational problems": [[28, "approximation-of-coercive-variational-problems"]], "Approximation of inf-sup stable variational problems": [[31, "approximation-of-inf-sup-stable-variational-problems"]], "BDDC - Preconditioner": [[37, "bddc-preconditioner"]], "Basic Iterative Methods": [[52, "basic-iterative-methods"]], "Basic properties": [[27, "basic-properties"]], "Basis functions for the segment:": [[80, "basis-functions-for-the-segment"]], "Basis functions for the triangle:": [[80, "basis-functions-for-the-triangle"]], "Block-Jacobi and general additive Schwarz preconditioners": [[105, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Block-preconditioning": [[84, "block-preconditioning"]], "Boundary Conditions": [[55, "boundary-conditions"], [73, "boundary-conditions"]], "Boundary conditions:": [[57, "boundary-conditions"]], "Bramble-Hilbert Lemma": [[102, "bramble-hilbert-lemma"]], "Building systems from building-blocks": [[30, "building-systems-from-building-blocks"]], "Butcher tableaus of simple methods": [[111, "butcher-tableaus-of-simple-methods"]], "Chebyshev polynomials": [[47, "chebyshev-polynomials"]], "Coercive examples": [[30, "coercive-examples"]], "Coercive variational problems and their approximation": [[28, "coercive-variational-problems-and-their-approximation"]], "Commuting diagram for H^1 - H(\\opcurl)": [[68, "commuting-diagram-for-h-1-h-opcurl"]], "Comparison to DD with minimal overlap": [[109, "comparison-to-dd-with-minimal-overlap"]], "Computation of element-vectors and element-matrices": [[13, "computation-of-element-vectors-and-element-matrices"]], "Computation of the lifting \\| \\sigma^\\Delta \\|": [[34, "computation-of-the-lifting-sigma-delta"]], "Computing dual norms": [[75, "computing-dual-norms"]], "Conjugate Gradients": [[49, "conjugate-gradients"]], "Consistent and Distributed Vectors": [[24, "consistent-and-distributed-vectors"]], "Consistent vectors:": [[24, "consistent-vectors"]], "Constrained minimization problem": [[53, "constrained-minimization-problem"]], "Constrained minimization problems": [[84, "constrained-minimization-problems"]], "Continuity and discrete coercivity of the HDG bilinear-form": [[1, "continuity-and-discrete-coercivity-of-the-hdg-bilinear-form"]], "Convergence of Runge Kutta methods": [[111, "convergence-of-runge-kutta-methods"]], "DG - Methods for elliptic problems": [[2, "dg-methods-for-elliptic-problems"]], "Diagonal preconditioner for L_2-norm": [[106, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[106, "diagonal-preconditioner-for-the-h-1-norm"]], "Diagonally implicit Runge-Kutta methods:": [[111, "diagonally-implicit-runge-kutta-methods"]], "Dirichlet boundary conditions": [[2, "dirichlet-boundary-conditions"]], "Dirichlet boundary conditions as mixed system": [[53, "dirichlet-boundary-conditions-as-mixed-system"]], "Dirichlet boundary conditions by penalty:": [[83, "dirichlet-boundary-conditions-by-penalty"]], "Discontinuous Galerkin Methods": [[45, "discontinuous-galerkin-methods"], [46, null]], "Discontinuous Galerkin for the Wave Equation": [[121, "discontinuous-galerkin-for-the-wave-equation"]], "Discontinuous Galerkin method": [[6, "discontinuous-galerkin-method"]], "Distributed Meshes and Spaces": [[15, "distributed-meshes-and-spaces"]], "Distributed finite element spaces": [[15, "distributed-finite-element-spaces"]], "Distributed vectors and matrices": [[24, "distributed-vectors-and-matrices"]], "Domain Decomposition with Lagrange parameters": [[40, "domain-decomposition-with-lagrange-parameters"]], "Domain Decomposition with minimal overlap": [[108, "domain-decomposition-with-minimal-overlap"]], "Dual mixed formulation": [[89, "dual-mixed-formulation"]], "Dual mixed method": [[60, "dual-mixed-method"]], "Efficiency of the residual error estimator": [[36, "efficiency-of-the-residual-error-estimator"]], "Efficient implementation:": [[121, "efficient-implementation"]], "Efficiently computable multi-level decomposition": [[66, "efficiently-computable-multi-level-decomposition"]], "Eigenvalues of the discretized Laplace-operator": [[121, "eigenvalues-of-the-discretized-laplace-operator"]], "Eqivalent versions of the Poincar\u00e9 inequality": [[102, "eqivalent-versions-of-the-poincare-inequality"]], "Equilibrated Residual Error Estimates": [[34, "equilibrated-residual-error-estimates"]], "Equilibration in NGSolve": [[34, "equilibration-in-ngsolve"]], "Equivalent norms on H^1 and on sub-spaces": [[101, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Error Analysis in L_2 \\times H^1": [[87, "error-analysis-in-l-2-times-h-1"]], "Error estimate of the L_2 projection": [[79, "error-estimate-of-the-l-2-projection"]], "Error estimates in L_2-norm": [[8, "error-estimates-in-l-2-norm"]], "Error estimates:": [[62, "error-estimates"]], "Essential boundary conditions": [[73, "essential-boundary-conditions"]], "Example": [[35, "example"]], "Example: Dirichlet boundary condition by penalty": [[56, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Finite elements for Stokes": [[54, "example-finite-elements-for-stokes"]], "Example: Nearly incompressible materials": [[56, "example-nearly-incompressible-materials"]], "Examples": [[84, "examples"], [111, "examples"]], "Exercise A:": [[107, "exercise-a"]], "Exercise B:": [[107, "exercise-b"]], "Exercise:": [[11, "exercise"], [111, "exercise"], [121, "exercise"]], "Exercise: Robust preconditioners": [[107, "exercise-robust-preconditioners"]], "Exercises": [[30, "exercises"], [75, "exercises"], [102, "exercises"]], "Exercises:": [[112, "exercises"]], "Expanding the Krylov-space": [[49, "expanding-the-krylov-space"]], "Experiment with CoefficientFunctions": [[75, "experiment-with-coefficientfunctions"]], "Experiments with BilienarForms and LinearForms": [[75, "experiments-with-bilienarforms-and-linearforms"]], "Experiments with norms": [[103, "experiments-with-norms"]], "Experiments with overlapping DD": [[109, "experiments-with-overlapping-dd"]], "Experiments with the Richardson iteration": [[48, "experiments-with-the-richardson-iteration"]], "Explicit Euler method (EE)": [[112, "explicit-euler-method-ee"]], "Explicit methods:": [[111, "explicit-methods"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[112, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Exponential Integrators for Parabolic Equations": [[115, "exponential-integrators-for-parabolic-equations"]], "Extending boundary data": [[66, "extending-boundary-data"]], "Extension operators": [[100, "extension-operators"]], "FETI methods": [[38, "feti-methods"]], "FETI-DP": [[39, "feti-dp"]], "Finite Element Error Analysis": [[86, "finite-element-error-analysis"]], "Finite Element Method": [[9, "finite-element-method"], [46, null]], "Finite Element Spaces": [[58, "finite-element-spaces"]], "Finite Elements in H(\\operatorname{div})": [[88, "finite-elements-in-h-operatorname-div"]], "Finite element error analysis": [[8, "finite-element-error-analysis"]], "Finite element error estimates": [[54, "finite-element-error-estimates"]], "Finite element system assembling": [[13, "finite-element-system-assembling"]], "Flux recovery in H(\\operatorname{div})": [[33, "flux-recovery-in-h-operatorname-div"]], "Flux-recovery error estimates with NGSolve": [[33, "flux-recovery-error-estimates-with-ngsolve"]], "Fourth Order Equation": [[3, "fourth-order-equation"]], "Fractional Sobolev spaces": [[102, "fractional-sobolev-spaces"]], "Friedrichs\u2019 Inequality": [[92, "friedrichs-inequality"], [95, "friedrichs-inequality"]], "Friedrichs\u2019 inequality": [[102, "friedrichs-inequality"], [103, "friedrichs-inequality"]], "Friedrichs\u2019 inequality in 1D": [[102, "friedrichs-inequality-in-1d"]], "Friedrichs\u2019 inequality on the square": [[102, "friedrichs-inequality-on-the-square"]], "Galerkin method in space": [[113, "galerkin-method-in-space"]], "General definition:": [[100, "general-definition"]], "General framework": [[34, "general-framework"]], "Generalized derivatives": [[98, "generalized-derivatives"]], "Geometric mesh refinement": [[8, "geometric-mesh-refinement"]], "Geometry with local details:": [[118, "geometry-with-local-details"]], "Get started with Netgen-Opencascade geometric modeling": [[75, "get-started-with-netgen-opencascade-geometric-modeling"]], "Goal driven error estimates": [[35, "goal-driven-error-estimates"]], "Graded meshes around vertex singularities": [[8, "graded-meshes-around-vertex-singularities"]], "Graph-based mesh partitioning": [[108, "graph-based-mesh-partitioning"]], "Grating": [[44, "grating"]], "H(\\operatorname{div}) on sub-domains": [[90, "h-operatorname-div-on-sub-domains"]], "H(div)-conforming Stokes": [[7, "h-div-conforming-stokes"]], "H^1-norm with small L_2-term": [[106, "h-1-norm-with-small-l-2-term"]], "Heat Equation": [[113, "heat-equation"]], "Hellan-Herrmann-Johnson method": [[67, "hellan-herrmann-johnson-method"]], "Helmholtz Equation": [[43, "helmholtz-equation"]], "High Order Finite Elements": [[46, null]], "High order elements": [[8, "high-order-elements"]], "Hybrid DG for elliptic equations": [[1, "hybrid-dg-for-elliptic-equations"]], "Hybrid Interfaces": [[0, "hybrid-interfaces"]], "Hybridization Techniques": [[91, "hybridization-techniques"]], "Hybridized C^0-continuous interior penalty method:": [[3, "hybridized-c-0-continuous-interior-penalty-method"]], "Implement a parallel BDDC preconditioner": [[42, "implement-a-parallel-bddc-preconditioner"]], "Implement our own system assembling": [[11, "implement-our-own-system-assembling"]], "Implementation of Finite Elements": [[10, "implementation-of-finite-elements"]], "Implementation of High Order Finite Elements": [[80, "implementation-of-high-order-finite-elements"]], "Implicit Euler method (IE)": [[112, "implicit-euler-method-ie"]], "Implicit Euler time-stepping": [[113, "implicit-euler-time-stepping"]], "Inf-sup stable variational problems": [[31, "inf-sup-stable-variational-problems"]], "Inner products:": [[24, "inner-products"]], "Installing MPI and PETSc without conda": [[20, "installing-mpi-and-petsc-without-conda"]], "Installing NGSolve": [[46, "installing-ngsolve"]], "Installing conda-packages using pip": [[20, "installing-conda-packages-using-pip"]], "Installing with conda": [[20, "installing-with-conda"]], "Instationary Transport Equation": [[4, "instationary-transport-equation"]], "Integration by parts": [[100, "integration-by-parts"]], "Interface conditions": [[55, "interface-conditions"]], "Interfaces": [[0, "interfaces"]], "Interpolation space H^s": [[41, "interpolation-space-h-s"]], "Interpolation spaces": [[100, "interpolation-spaces"]], "Introduction to MPI with mpi4py": [[20, "introduction-to-mpi-with-mpi4py"]], "Introduction to Non-overlapping Domain Decomposition": [[40, "introduction-to-non-overlapping-domain-decomposition"]], "Inverse estimates": [[103, "inverse-estimates"]], "Iteration Methods": [[45, "iteration-methods"], [46, null]], "Iteration methods in parallel": [[21, "iteration-methods-in-parallel"]], "Iterative Solvers": [[45, "iterative-solvers"], [77, "iterative-solvers"]], "Jacobi and Gauss Seidel Preconditioners": [[51, "jacobi-and-gauss-seidel-preconditioners"]], "Kirchhoff Plate equation": [[67, "kirchhoff-plate-equation"]], "Korn\u2019s inequality": [[103, "korn-s-inequality"]], "Legendre Polynomials": [[79, "legendre-polynomials"]], "Linear Algebra": [[75, "linear-algebra"]], "Linear elasticity": [[74, "linear-elasticity"]], "Linear stability classification": [[112, "linear-stability-classification"]], "Literature": [[46, "literature"]], "Local post-processing": [[86, "local-post-processing"]], "Marked edge bisection": [[33, "marked-edge-bisection"]], "Mass-lumping and Local time-stepping": [[118, "mass-lumping-and-local-time-stepping"]], "Matrix vector multiplication:": [[24, "matrix-vector-multiplication"]], "Maxwell equations:": [[83, "maxwell-equations"]], "Mean-value interpolation": [[102, "mean-value-interpolation"]], "Mesh refinement algorithms": [[33, "mesh-refinement-algorithms"]], "Minimization problem": [[30, "minimization-problem"]], "Mixed Finite Element Methods": [[45, "mixed-finite-element-methods"], [46, null]], "Mixed Methods for Elasticity": [[45, "mixed-methods-for-elasticity"]], "Mixed Methods for Plates and Shells": [[45, "mixed-methods-for-plates-and-shells"]], "Mixed Methods for Second Order Equations": [[45, "mixed-methods-for-second-order-equations"], [46, null]], "Mixed Methods for second order equations": [[57, "mixed-methods-for-second-order-equations"]], "Mixed method for second order equation": [[53, "mixed-method-for-second-order-equation"]], "Multi-level Extension": [[66, "multi-level-extension"]], "Multigrid Methods": [[45, "multigrid-methods"], [46, null]], "Multigrid Preconditioning": [[63, "multigrid-preconditioning"]], "Multigrid and Multilevel Methods": [[63, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[63, "multilevel-preconditioner"]], "NGSolve - PETSc interface": [[14, "ngsolve-petsc-interface"]], "Nano-optics: A ring-resonator": [[119, "nano-optics-a-ring-resonator"], [120, "nano-optics-a-ring-resonator"]], "Natural boundary conditions": [[73, "natural-boundary-conditions"]], "Natural trace space": [[41, "natural-trace-space"]], "Nearly optimal analysis of the ML - preconditioner": [[65, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Newmark time-stepping method": [[114, "newmark-time-stepping-method"]], "Nitsche\u2019s Method for boundary and interface conditions": [[0, "nitsche-s-method-for-boundary-and-interface-conditions"]], "Nitsche\u2019s method:": [[0, "nitsche-s-method"]], "Non-conforming Finite Element Methods": [[12, "non-conforming-finite-element-methods"]], "Non-linear dynamics": [[59, "non-linear-dynamics"]], "Non-overlapping Domain Decomposition Methods": [[45, "non-overlapping-domain-decomposition-methods"], [46, null]], "Nonlinear Shells": [[70, "nonlinear-shells"]], "Normal-trace of functions in H(\\operatorname{div})": [[90, "normal-trace-of-functions-in-h-operatorname-div"]], "Numerical analysis of Parabolic Equations": [[45, "numerical-analysis-of-parabolic-equations"]], "Numerical analysis of Wave Equations": [[45, "numerical-analysis-of-wave-equations"]], "Numerical verification": [[97, "numerical-verification"]], "One sup is enough": [[30, "one-sup-is-enough"]], "Optimal analysis of the multi-level preconditioner": [[65, "optimal-analysis-of-the-multi-level-preconditioner"]], "Optimal convergence of the V-cycle": [[64, "optimal-convergence-of-the-v-cycle"]], "Optimizing the relaxation parameter \\alpha": [[48, "optimizing-the-relaxation-parameter-alpha"]], "Ordinary differential equations": [[45, "ordinary-differential-equations"]], "Orthogonal polynomials on triangles": [[79, "orthogonal-polynomials-on-triangles"]], "Overlapping DD Methods with coarse grid": [[109, "overlapping-dd-methods-with-coarse-grid"]], "Overlapping Domain Decomposition Methods": [[109, "overlapping-domain-decomposition-methods"]], "Overlapping blocks": [[110, "overlapping-blocks"]], "PETSc preconditioner for NGSolve": [[14, "petsc-preconditioner-for-ngsolve"]], "Parallel Solvers": [[45, "parallel-solvers"], [46, null]], "Parameter Dependent Problems": [[56, "parameter-dependent-problems"], [83, "parameter-dependent-problems"]], "Penalty formulation for the Stokes equation:": [[83, "penalty-formulation-for-the-stokes-equation"]], "Penalty formulation for the flux:": [[83, "penalty-formulation-for-the-flux"]], "Piola Transformation": [[88, "piola-transformation"]], "Poincar\u00e9 inequality": [[102, "poincare-inequality"], [103, "poincare-inequality"]], "Poincar\u00e9 inequality in 1D": [[102, "poincare-inequality-in-1d"]], "Point evaluation functional": [[102, "point-evaluation-functional"]], "Poisson equation in NGSolve:": [[76, "poisson-equation-in-ngsolve"]], "Preamble": [[72, "preamble"]], "Preconditioner for S": [[38, "preconditioner-for-s"]], "Preconditioning": [[51, "preconditioning"]], "Primal mixed formulation": [[89, "primal-mixed-formulation"]], "Primal mixed method:": [[60, "primal-mixed-method"]], "Projected preconditioner": [[82, "projected-preconditioner"]], "Projection based interpolation": [[79, "projection-based-interpolation"]], "Projection based interpolation on triangles": [[79, "projection-based-interpolation-on-triangles"]], "Projection matrices from the finest level": [[63, "projection-matrices-from-the-finest-level"]], "Projection onto subspaces": [[32, "projection-onto-subspaces"]], "Proof of the trace inequality:": [[97, "proof-of-the-trace-inequality"]], "Proving discrete LBB by the Fortin operator": [[54, "proving-discrete-lbb-by-the-fortin-operator"]], "Red-Green Refinement": [[33, "red-green-refinement"]], "Regularity-free estimate": [[65, "regularity-free-estimate"]], "Reissner Mindlin Plates": [[69, "reissner-mindlin-plates"]], "Relationship between HHJ and TDNNS": [[68, "relationship-between-hhj-and-tdnns"]], "Reliability of the residual error estimator": [[36, "reliability-of-the-residual-error-estimator"]], "Repeat the exercise in 2D.": [[30, "repeat-the-exercise-in-2d"]], "Richardson iteration": [[21, "richardson-iteration"]], "Riesz representation theorem and symmetric variational problems": [[29, "riesz-representation-theorem-and-symmetric-variational-problems"]], "Robust coarse-grid correction": [[83, "robust-coarse-grid-correction"]], "Robust smoothers": [[83, "robust-smoothers"]], "Robust two-level methods for parameter dependent problems": [[83, "robust-two-level-methods-for-parameter-dependent-problems"]], "Runge Kutta Methods": [[111, "runge-kutta-methods"]], "Saddle-point Problems": [[45, "saddle-point-problems"], [46, null]], "Scaled domain": [[102, "scaled-domain"]], "Schur complement iteration": [[84, "schur-complement-iteration"]], "Schwarz preconditioners for high order finite elements": [[110, "schwarz-preconditioners-for-high-order-finite-elements"]], "Second inf-sup condition means onto": [[30, "second-inf-sup-condition-means-onto"]], "Shift theorems": [[104, "shift-theorems"]], "Simple methods based on the integral equation": [[112, "simple-methods-based-on-the-integral-equation"]], "Single step methods and hyperbolic equations": [[112, "single-step-methods-and-hyperbolic-equations"]], "Single step methods and parabolic equations": [[112, "single-step-methods-and-parabolic-equations"]], "Single-step methods": [[112, "single-step-methods"]], "Sobolev Spaces": [[46, null]], "Sobolev spaces": [[99, "sobolev-spaces"]], "Sobolev spaces over sub-domains": [[100, "sobolev-spaces-over-sub-domains"]], "Solving Stokes in parallel": [[22, "solving-stokes-in-parallel"]], "Solving Stokes\u2019 equation with the non-conforming P^1-triangle": [[12, "solving-stokes-equation-with-the-non-conforming-p-1-triangle"]], "Solving the Poisson Equation": [[76, "solving-the-poisson-equation"]], "Solving the minimization problem": [[49, "solving-the-minimization-problem"]], "Solving the wave equation:": [[121, "solving-the-wave-equation"]], "Some Examples of ASM preconditioners": [[106, "some-examples-of-asm-preconditioners"]], "Some inequalities in Sobolev spaces": [[45, "some-inequalities-in-sobolev-spaces"]], "Splitting Methods for the time-dependent convection diffusion equation": [[5, "splitting-methods-for-the-time-dependent-convection-diffusion-equation"]], "Stability conditions": [[112, "stability-conditions"]], "Stability function": [[112, "stability-function"]], "Stability of the method": [[6, "stability-of-the-method"]], "Stationary Transport Equation": [[6, "stationary-transport-equation"]], "Step function": [[102, "step-function"]], "Stokes Equation": [[58, "stokes-equation"]], "Stokes equation within the abstract theory": [[53, "stokes-equation-within-the-abstract-theory"]], "Structure of Saddle-point Problems": [[84, "structure-of-saddle-point-problems"]], "Sub-space Correction Methods": [[45, "sub-space-correction-methods"], [46, null]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[105, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "Symmetric variational problems": [[29, "symmetric-variational-problems"]], "TDNNS Variational formulation:": [[62, "tdnns-variational-formulation"]], "Tangential displacement normal normal stress continuous finite elements": [[62, "tangential-displacement-normal-normal-stress-continuous-finite-elements"]], "Testing the differential operators": [[121, "testing-the-differential-operators"]], "The 1D case": [[79, "the-1d-case"]], "The Additive Schwarz Lemma": [[105, "the-additive-schwarz-lemma"]], "The Algorithm": [[64, "the-algorithm"]], "The Approximation Property": [[64, "the-approximation-property"]], "The Bramble-Pasciak Transformation": [[81, "the-bramble-pasciak-transformation"]], "The Chebyshev Method": [[47, "the-chebyshev-method"]], "The Chebyshev iteration": [[47, "the-chebyshev-iteration"]], "The Cl\u00e9ment- operator": [[36, "the-clement-operator"]], "The Finite Element Method": [[76, "the-finite-element-method"]], "The First Lemma of Strang": [[12, "the-first-lemma-of-strang"]], "The Galerkin Method": [[45, "the-galerkin-method"], [46, null]], "The Gradient Method": [[50, "the-gradient-method"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[67, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "The Helmholtz equation": [[45, "the-helmholtz-equation"]], "The MPI library": [[20, "the-mpi-library"]], "The ParallelDofs class": [[15, "the-paralleldofs-class"]], "The Poincar\u00e9 inequality": [[96, "the-poincare-inequality"]], "The Richardson Iteration": [[48, "the-richardson-iteration"]], "The Second Lemma of Strang": [[12, "the-second-lemma-of-strang"]], "The Smothing Property": [[64, "the-smothing-property"]], "The Sobolev space H^1, linear and bilinear forms": [[76, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Trace Inequality": [[97, "the-trace-inequality"]], "The Verlet method:": [[118, "the-verlet-method"]], "The Zienkiewicz Zhu error estimator": [[33, "the-zienkiewicz-zhu-error-estimator"]], "The divergence of nn-continuous piece-wise smooth functions:": [[62, "the-divergence-of-nn-continuous-piece-wise-smooth-functions"]], "The function space H(\\operatorname{div})": [[90, "the-function-space-h-operatorname-div"]], "The installation happens in three steps:": [[20, "the-installation-happens-in-three-steps"]], "The preconditioned Richardson iteration": [[51, "the-preconditioned-richardson-iteration"]], "The preconditioned gradient method": [[51, "the-preconditioned-gradient-method"]], "The residual error estimator": [[36, "the-residual-error-estimator"]], "The trace space H^{1/2}": [[100, "the-trace-space-h-1-2"]], "The trace space on one edge of the square": [[100, "the-trace-space-on-one-edge-of-the-square"]], "The upper-bound by the overlap": [[105, "the-upper-bound-by-the-overlap"]], "The weak formulation of the Poisson equation": [[104, "the-weak-formulation-of-the-poisson-equation"]], "This repository contains an interactive introduction to the Finite Element Method": [[25, "this-repository-contains-an-interactive-introduction-to-the-finite-element-method"]], "Time-dependent Problems": [[45, "time-dependent-problems"]], "Time-dependent problems": [[46, null]], "Trace inequality": [[103, "trace-inequality"]], "Trace norm on bottom edge": [[41, "trace-norm-on-bottom-edge"]], "Trace norm on boundary sub-domains": [[41, "trace-norm-on-boundary-sub-domains"]], "Trace theorems and their applications": [[100, "trace-theorems-and-their-applications"]], "Traces spaces": [[41, "traces-spaces"]], "Trapezoidal method": [[112, "trapezoidal-method"]], "Two-level analysis for Maxwell equations": [[83, "two-level-analysis-for-maxwell-equations"]], "Using PETSc": [[23, "using-petsc"]], "Using ipyparallel": [[20, "using-ipyparallel"]], "Variable Coefficients": [[78, "variable-coefficients"]], "Variable coefficients": [[33, "variable-coefficients"]], "Variational Formulation": [[58, "variational-formulation"]], "Variational formulation in space": [[113, "variational-formulation-in-space"]], "Variational formulation:": [[74, "variational-formulation"]], "Vector operations:": [[24, "vector-operations"]], "Visualizing the basis functions": [[76, "visualizing-the-basis-functions"]], "Wave Equation": [[114, "wave-equation"]], "Weak formulation": [[76, "weak-formulation"]], "With the coarse space": [[108, "with-the-coarse-space"]], "Work with GridFunctions": [[75, "work-with-gridfunctions"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[60, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "\\DeclareMathOperator{\\opdiv}{div}\n\\DeclareMathOperator{\\opcurl}{curl}\n\\DeclareMathOperator{\\eps}{\\varepsilon}\nReduced symmetry methods": [[61, "declaremathoperator-opdiv-div-declaremathoperator-opcurl-curl-declaremathoperator-eps-varepsilon-reduced-symmetry-methods"]], "complex-valued problem as real system": [[30, "complex-valued-problem-as-real-system"]], "hp - Finite Elements": [[79, "hp-finite-elements"]], "inf-sup condition": [[30, "inf-sup-condition"]], "inf-sup condition of the first-order derivative bilinear-form": [[30, "inf-sup-condition-of-the-first-order-derivative-bilinear-form"]], "use our own integrators for element matrix calculation:": [[11, "use-our-own-integrators-for-element-matrix-calculation"]], "use our own matrix assembling function:": [[11, "use-our-own-matrix-assembling-function"]], "we can inspect the element matrix:": [[11, "we-can-inspect-the-element-matrix"]]}, "docnames": ["DG/Nitsche", "DG/elliptic", "DG/elliptic_stdDG", "DG/fourthorder", "DG/instationary", "DG/splitting", "DG/stationary", "DG/stokes", "FEM/erroranalysis", "FEM/finiteelements", "FEM/myFiniteElement", "FEM/myassembling", "FEM/nonconforming", "FEM/systemassembling", "MPIparallel/PETSc_interface", "MPIparallel/distmesh", "MPIparallel/hdivnstsimple/NavierStokesMT", "MPIparallel/hdivnstsimple/TestConvert", "MPIparallel/hdivnstsimple/veclaplace", "MPIparallel/hdivnstsimple/veclaplace-auxiliary", "MPIparallel/intro", "MPIparallel/paralleliteration", "MPIparallel/parallelstokes", "MPIparallel/petsc", "MPIparallel/vectypes", "README", "Untitled", "abstracttheory/BasicProperties", "abstracttheory/Coercive", "abstracttheory/RieszRepresentation", "abstracttheory/exercises", "abstracttheory/infsup", "abstracttheory/subspaceprojection", "aposteriori/aposteriori", "aposteriori/equilibrated", "aposteriori/goaldriven", "aposteriori/residualEE", "domaindecomposition/BDDC", "domaindecomposition/FETI", "domaindecomposition/FETI-DP", "domaindecomposition/introduction", "domaindecomposition/traces", "ex20210609", "helmholtz/absorbing", "helmholtz/grating", "iFEM", "intro", "iterative/Chebyshev", "iterative/Richardson", "iterative/conjugategradients", "iterative/gradientmethod", "iterative/preconditioning", "iterative/simple", "mixed/abstract", "mixed/abstractfem", "mixed/boundary", "mixed/pardep", "mixed/secondorder", "mixed/stokes", "mixedelasticity/dynamics", "mixedelasticity/hellingerreissner", "mixedelasticity/reducedsymmetry", "mixedelasticity/tdnns", "multigrid/algorithms", "multigrid/analysisMG", "multigrid/analysisML", "multigrid/multilevel-extension", "plates/hhj", "plates/hhj-tdnns", "plates/reissnermindlin", "plates/shells", "plates/tdnnsplate", "preamble", "primal/boundary_conditions", "primal/elasticity3D", "primal/exercises", "primal/first_example", "primal/solvers", "primal/subdomains", "pversion/hpfem", "pversion/myhighorder", "saddlepoint/bramblepasciak", "saddlepoint/fewconstraints", "saddlepoint/parameterdependent", "saddlepoint/structure", "saddlepoint/useBP", "secondorder/erroranalysis", "secondorder/erroranalysisl2h1", "secondorder/finiteelements", "secondorder/framework", "secondorder/hdiv", "secondorder/hybridization", "sobolev/Friedrichs", "sobolev/Untitled", "sobolev/Untitled1", "sobolev/Untitled2", "sobolev/poincare", "sobolev/trace", "sobolevspaces/GeneralizedDerivatives", "sobolevspaces/SobolevSpaces", "sobolevspaces/Traces", "sobolevspaces/equivalentnorms", "sobolevspaces/exercises", "sobolevspaces/experiments", "sobolevspaces/preciseweak", "subspacecorrection/asm", "subspacecorrection/examples", "subspacecorrection/exercise", "subspacecorrection/minimaldd", "subspacecorrection/overlapping", "subspacecorrection/pversion", "timedependent/ODEs/runge_kutta", "timedependent/ODEs/singlestep", "timedependent/intro/heatequation", "timedependent/intro/waveequation", "timedependent/parabolic/exponential", "timedependent/waves/Maxwell-DG", "timedependent/waves/Untitled", "timedependent/waves/lts", "timedependent/waves/ringresonator", "timedependent/waves/testpml/ringresonator", "timedependent/waves/wave-leapfrogDG"], "envversion": {"sphinx": 61, "sphinx.domains.c": 3, "sphinx.domains.changeset": 1, "sphinx.domains.citation": 1, "sphinx.domains.cpp": 9, "sphinx.domains.index": 1, "sphinx.domains.javascript": 3, "sphinx.domains.math": 2, "sphinx.domains.python": 4, "sphinx.domains.rst": 2, "sphinx.domains.std": 2, "sphinx.ext.intersphinx": 1}, "filenames": ["DG/Nitsche.ipynb", "DG/elliptic.ipynb", "DG/elliptic_stdDG.ipynb", "DG/fourthorder.ipynb", "DG/instationary.ipynb", "DG/splitting.ipynb", "DG/stationary.ipynb", "DG/stokes.ipynb", "FEM/erroranalysis.ipynb", "FEM/finiteelements.ipynb", "FEM/myFiniteElement.ipynb", "FEM/myassembling.ipynb", "FEM/nonconforming.ipynb", "FEM/systemassembling.ipynb", "MPIparallel/PETSc_interface.ipynb", "MPIparallel/distmesh.ipynb", "MPIparallel/hdivnstsimple/NavierStokesMT.ipynb", "MPIparallel/hdivnstsimple/TestConvert.ipynb", "MPIparallel/hdivnstsimple/veclaplace.ipynb", "MPIparallel/hdivnstsimple/veclaplace-auxiliary.ipynb", "MPIparallel/intro.ipynb", "MPIparallel/paralleliteration.ipynb", "MPIparallel/parallelstokes.ipynb", "MPIparallel/petsc.ipynb", "MPIparallel/vectypes.ipynb", "README.md", "Untitled.ipynb", "abstracttheory/BasicProperties.ipynb", "abstracttheory/Coercive.ipynb", "abstracttheory/RieszRepresentation.ipynb", "abstracttheory/exercises.ipynb", "abstracttheory/infsup.ipynb", "abstracttheory/subspaceprojection.ipynb", "aposteriori/aposteriori.ipynb", "aposteriori/equilibrated.ipynb", "aposteriori/goaldriven.ipynb", "aposteriori/residualEE.ipynb", "domaindecomposition/BDDC.ipynb", "domaindecomposition/FETI.ipynb", "domaindecomposition/FETI-DP.ipynb", "domaindecomposition/introduction.ipynb", "domaindecomposition/traces.ipynb", "ex20210609.ipynb", "helmholtz/absorbing.ipynb", "helmholtz/grating.ipynb", "iFEM.ipynb", "intro.md", "iterative/Chebyshev.ipynb", "iterative/Richardson.ipynb", "iterative/conjugategradients.ipynb", "iterative/gradientmethod.ipynb", "iterative/preconditioning.ipynb", "iterative/simple.ipynb", "mixed/abstract.ipynb", "mixed/abstractfem.ipynb", "mixed/boundary.ipynb", "mixed/pardep.ipynb", "mixed/secondorder.ipynb", "mixed/stokes.ipynb", "mixedelasticity/dynamics.ipynb", "mixedelasticity/hellingerreissner.ipynb", "mixedelasticity/reducedsymmetry.ipynb", "mixedelasticity/tdnns.ipynb", "multigrid/algorithms.ipynb", "multigrid/analysisMG.ipynb", "multigrid/analysisML.ipynb", "multigrid/multilevel-extension.ipynb", "plates/hhj.ipynb", "plates/hhj-tdnns.ipynb", "plates/reissnermindlin.ipynb", "plates/shells.ipynb", "plates/tdnnsplate.ipynb", "preamble.ipynb", "primal/boundary_conditions.ipynb", "primal/elasticity3D.ipynb", "primal/exercises.ipynb", "primal/first_example.ipynb", "primal/solvers.ipynb", "primal/subdomains.ipynb", "pversion/hpfem.ipynb", "pversion/myhighorder.ipynb", "saddlepoint/bramblepasciak.ipynb", "saddlepoint/fewconstraints.ipynb", "saddlepoint/parameterdependent.ipynb", "saddlepoint/structure.ipynb", "saddlepoint/useBP.ipynb", "secondorder/erroranalysis.ipynb", "secondorder/erroranalysisl2h1.ipynb", "secondorder/finiteelements.ipynb", "secondorder/framework.ipynb", "secondorder/hdiv.ipynb", "secondorder/hybridization.ipynb", "sobolev/Friedrichs.ipynb", "sobolev/Untitled.ipynb", "sobolev/Untitled1.ipynb", "sobolev/Untitled2.ipynb", "sobolev/poincare.ipynb", "sobolev/trace.ipynb", "sobolevspaces/GeneralizedDerivatives.ipynb", "sobolevspaces/SobolevSpaces.ipynb", "sobolevspaces/Traces.ipynb", "sobolevspaces/equivalentnorms.ipynb", "sobolevspaces/exercises.ipynb", "sobolevspaces/experiments.ipynb", "sobolevspaces/preciseweak.ipynb", "subspacecorrection/asm.ipynb", "subspacecorrection/examples.ipynb", "subspacecorrection/exercise.ipynb", "subspacecorrection/minimaldd.ipynb", "subspacecorrection/overlapping.ipynb", "subspacecorrection/pversion.ipynb", "timedependent/ODEs/runge_kutta.ipynb", "timedependent/ODEs/singlestep.ipynb", "timedependent/intro/heatequation.ipynb", "timedependent/intro/waveequation.ipynb", "timedependent/parabolic/exponential.ipynb", "timedependent/waves/Maxwell-DG.ipynb", "timedependent/waves/Untitled.ipynb", "timedependent/waves/lts.ipynb", "timedependent/waves/ringresonator.ipynb", "timedependent/waves/testpml/ringresonator.ipynb", "timedependent/waves/wave-leapfrogDG.ipynb"], "indexentries": {}, "objects": {}, "objnames": {}, "objtypes": {}, "terms": {"": [8, 9, 12, 18, 19, 25, 27, 28, 29, 30, 32, 34, 39, 45, 46, 53, 54, 64, 65, 70, 71, 72, 74, 75, 76, 77, 80, 81, 82, 84, 92, 100, 101, 102, 104, 105, 111, 112, 115, 118, 121], "0": [0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "00": [53, 76], "0000000000000007": 96, "0000000001010011": 39, "00000000e": 76, "00000001011111010111110": 119, "0000001001000110": 39, "0000010011010000": 39, "0000010110100000": 39, "0000101000001100": 39, "0000110000": 0, "00010095216118821378": 83, "0001011001000000": 39, "00010127534375465337": 115, "00010129028658340466": 48, "00010270136193037442": 48, "00010317733972589769": 84, "00010321518710120788": 84, "00010396456155120892": 84, "00010413209609269955": 48, "00010508880471851492": 85, "00010543202872418047": 21, "00010558276299811141": 48, "000107053640393076": 48, "00010854500989614187": 48, "00010870660188901045": 21, "00010949506857410921": 76, "00010989556910275802": 38, "00011005715705197291": 48, "00011159037138615583": 48, "00011314494646081535": 48, "00011375493321065046": 110, "00011455858334166559": 38, "00011472117993089234": 48, "00011598796595447701": 77, "00011616801306622024": 80, "00011631937360145951": 48, "00011658034877248662": 21, "00011793983348546948": 48, "00011958286986273097": 48, "00012008137501908114": 85, "00012124879733930423": 48, "0001229056447701414": 14, "00012293793490808928": 48, "00012465060600999609": 48, "00012635246751698076": 83, "0001263871385961654": 48, "00012814786519089072": 48, "00012890926660311867": 21, "00012993312295558898": 48, "00013119349751905874": 39, "00013174325375354127": 48, "00013357860421564497": 48, "0001338648614332045": 77, "00013426732432066454": 39, "0001354395258069982": 48, "00013732637489447186": 48, "00013923951281523983": 48, "00014029678008718827": 84, "00014117930594623312": 48, "00014254435985880764": 21, "00014314612577460858": 48, "00014514034896921082": 48, "00014716235745298453": 48, "00014753949588550174": 85, "0001487969376679355": 81, "00014885168670214898": 39, "0001492125384764743": 48, "0001498362645389152": 77, "00015014932580892883": 83, "00015055318197175534": 39, "00015104356593841114": 38, "0001512912846923413": 48, "000153398994230856": 48, "0001555360707766003": 48, "0001576247469778465": 21, "00015770292364614483": 48, "00015989996786680568": 48, "00016212762425659415": 48, "00016400656081982313": 115, "00016421847717886735": 84, "00016434676866042806": 84, "00016438631950522982": 48, "00016491550344460967": 77, "00016602077647740788": 84, "00016667648625635052": 48, "00016757613019838458": 83, "000168998563190804": 48, "00017135299511121462": 48, "00017374023302765528": 48, "00017430461490466542": 21, "00017616073424457884": 48, "00017718883102318885": 110, "0001774124343507029": 63, "00017741243435071894": 63, "00017853792896837772": 77, "0001786149624489968": 48, "00018078405517481033": 91, "0001811033877998466": 48, "00018362648701864706": 48, "0001852656638868365": 83, "00018618474348148243": 48, "00018877864731223603": 48, "0001914086954771383": 48, "00019208909401800605": 77, "0001927549479218684": 21, "00019297576676091736": 84, "0001940753918806743": 48, "0001958658710676972": 85, "00019677924746292868": 48, "00019849103261233912": 38, "00019952078029812246": 48, "00020230051569474947": 48, "0002028582087546903": 21, "00020511898629702412": 48, "00020797673218777952": 48, "00020810720521774676": 77, "00021087430099283637": 48, "00021211111708816004": 38, "00021280037638335268": 81, "0002131654979901198": 21, "0002135880486764573": 84, "00021381224798692054": 48, "00021606301815433576": 83, "00021679113620092907": 48, "00021981153653095173": 48, "0002207732294693555": 115, "00022281477996417238": 84, "00022287402784859796": 48, "00022597919711308994": 48, "00022912763948485545": 48, "00023057965122062087": 77, "00023231995844078663": 48, "00023555676589107158": 48, "00023574705190481792": 21, "00023883868229783445": 48, "00024216633679531238": 48, "0002455403673118469": 48, "0002489614206936301": 48, "0002524301528301413": 48, "0002544824048447953": 84, "00025594722878148363": 48, "0002561028796712345": 85, "0002595133229074687": 48, "00026073407364023984": 21, "0002631291189986039": 48, "00026555485540291635": 77, "00026679531040890817": 48, "00027051260019073385": 48, "00027428170123139963": 48, "00027461820027847346": 83, "0002781033363919083": 48, "000281978238647573": 48, "0002859071512307771": 48, "00028670864286594716": 84, "0002867283203485827": 84, "00028838783661312803": 21, "0002898908277756472": 48, "00029393003246500445": 48, "0002980255401792292": 48, "00030217813664752045": 48, "0003026291735590164": 84, "0003063886186010778": 48, "00031065779392879563": 48, "0003112297948991817": 77, "000314986481834931": 48, "00031900021889388756": 21, "00031937551299935366": 48, "00032163144527917325": 38, "00032219893579614505": 85, "00032382572973983466": 48, "00032833798617694766": 48, "00033291314840126285": 48, "0003375520946429079": 48, "00034225571544383636": 48, "0003437652718687495": 110, "0003470249138323727": 48, "0003518606055004487": 48, "0003528984291593283": 21, "0003567637189834118": 48, "00036173519584250553": 48, "0003641682638948373": 84, "0003658814417513913": 83, "00036677599084996784": 48, "00036683687047396396": 77, "000371887072176918": 48, "00037706942158399916": 48, "00037798057948934126": 76, "0003823240346148249": 48, "00038765192079231786": 48, "0003894308112507474": 21, "0003902376619710067": 38, "00039045108702023503": 21, "00039291080381269393": 85, "0003930541038178998": 48, "0003985316217737422": 48, "00040408552732787813": 48, "0004097168879425399": 48, "0004154267860854548": 48, "0004212163194443691": 48, "00042485165355397067": 81, "0004253862507363e": 48, "00042708660114481857": 48, "00042780257753367363": 77, "0004320763401729045": 21, "0004330387599710549": 48, "00043907394059037843": 48, "0004407752179327705": 84, "000443926110546035": 84, "00044476274680689597": 84, "0004451933037808829": 48, "0004467008934476962": 76, "00045139802666244205": 48, "00045310590428783134": 76, "0004576893029314665": 48, "00046060801777727964": 83, "0004640683430989626": 48, "000470536374732366": 48, "0004761165147485691": 84, "0004770946427009313": 48, "0004778431788232897": 84, "00047825313447910644": 21, "00048374440942494824": 48, "00048575900897660407": 39, "0004904869551287794": 48, "0004933831969703863": 83, "0004942564003676569": 77, "0004973235780976221": 48, "0004994633778866232": 85, "0005": 121, "0005002942167523764": 84, "0005042555949383612": 48, "0005112843408443743": 48, "0005184111698643815": 48, "0005256374551755312": 48, "0005295374859014327": 21, "0005329645893605536": 48, "0005403939846894724": 48, "0005434954855783924": 110, "0005479270734053628": 48, "0005555653080149196": 48, "000563310161583251": 48, "0005645223644237875": 77, "0005668987307889644": 84, "0005669357974305203": 84, "0005711631280335377": 48, "0005791257224512458": 48, "0005865868529358018": 21, "0005871994813932063": 48, "0005953859632015506": 48, "000603686748322705": 48, "0006042081173435654": 83, "0006070238087351294": 76, "00061210343963129": 48, "0006150757230139204": 85, "000620637662759349": 48, "0006292910664307944": 48, "0006352763333349191": 77, "0006380653228011725": 48, "0006469621278029652": 48, "0006501978632935306": 21, "0006559832014964314": 48, "0006568584384695376": 84, "0006602691617452845": 91, "0006651302884261447": 48, "0006744051579833437": 48, "0006815596893261466": 81, "000683642014896617": 83, "0006838096047742041": 48, "0006933454489940485": 48, "000703014536807873": 48, "0007041293195152594": 39, "0007128187407370412": 48, "000721366424845875": 21, "0007227599600523393": 48, "000725752752707929": 39, "0007272861126943784": 77, "0007281408027218528": 115, "0007328401211737334": 48, "0007372917166384283": 39, "0007430611780766196": 48, "0007534251127049187": 48, "0007639339353912823": 48, "0007745896852841808": 48, "000778573195676947": 85, "0007815351507657772": 83, "000785394430782293": 48, "0007957605690849792": 38, "0007963502699764032": 48, "0007967222328500014": 84, "0007970718790778335": 84, "0007978295322313947": 21, "0008013860460216729": 21, "0008074593310986968": 48, "0008155136133605154": 14, "0008187237729799078": 48, "0008301457855142871": 48, "0008329679456139172": 77, "0008417275901325724": 48, "0008534714402833544": 48, "0008634299729498884": 84, "0008653796219227064": 48, "0008774544540124627": 48, "0008896982890273583": 48, "0008920130395837511": 21, "0008992713275613841": 38, "0009021135134710553": 48, "0009022414011419461": 84, "0009147025484014691": 48, "000927467849965456": 48, "0009404119099430902": 48, "000953537256301915": 48, "0009668464537611202": 48, "0009803421043661597": 48, "0009850043882244734": 76, "0009893154642094026": 77, "0009922984038274098": 83, "000994026848073895": 48, "0009957540133089203": 21, "001": [4, 16, 18, 19, 42, 108, 118], "0010069624297562696": 39, "0010079033633484853": 48, "0010219743677684654": 48, "0010257501072204825": 84, "0010362426186451216": 48, "0010507109136523886": 48, "0010534825624365811": 85, "001065382091468879": 48, "0010802590324318184": 48, "0010818082293433016": 63, "0010818082293433783": 63, "0010953446592037578": 48, "001095452880817425": 84, "00110000000001111111111111111111111111111111111111": 108, "0011010100000000": 39, "0011106419374519126": 48, "0011163969057570045": 21, "00112175": 82, "0011261538765406645": 48, "0011307102211392996": 84, "0011334165967012847": 84, "00114188353023771": 48, "0011428466557880732": 110, "0011578339974337177": 48, "0011740084228764713": 48, "001190409997919305": 48, "0011912144262735045": 77, "0012070419612846086": 48, "0012135653881944293": 81, "0012239075998425912": 48, "0012410102494057379": 48, "0012583532955394456": 48, "0012601136081994504": 21, "0012759401743892452": 48, "0012809618903024942": 84, "0012937743735248989": 48, "0013118594328021408": 48, "0013223771120878588": 76, "0013301989452423582": 48, "0013487965579305366": 48, "0013676559729323798": 48, "0013787077033072886": 85, "0013867809482308554": 48, "0013982543263219135": 83, "001406175298682715": 48, "0014258428969956122": 48, "0014383176229297973": 21, "0014397425850613988": 77, "0014457876747266": 48, "0014660136233021089": 48, "0014865247950606303": 48, "001486650169981125": 38, "0014958478667025682": 21, "0015073253043182886": 48, "0015284193284582851": 48, "0015498111090446118": 48, "0015715049529611042": 48, "0015837080803695188": 91, "0015921163492495247": 38, "001593505233576355": 48, "0016028677378609755": 80, "0016158163919351772": 48, "0016384429379776749": 48, "0016389793743943e": 48, "0016613894517866516": 48, "0016652310329207363": 85, "0016785598117133912": 21, "0016846605848640193": 48, "0017082610614374385": 48, "0017321956797978454": 48, "0017484379251981065": 77, "0017564693136689225": 48, "0017754830163756964": 76, "001781086913609521": 48, "0018060535084499963": 48, "001815036053916696": 84, "0018222848385735526": 110, "0018231395803895414": 84, "0018313742067635904": 48, "0018570541983739639": 48, "0018830987558998633": 48, "001909513236338424": 48, "0019135138262121013": 84, "0019363030826879543": 48, "00195038440947e": 48, "0019634738256118496": 48, "0019910310851446664": 48, "001996326380935e": 48, "002000033787596275": 83, "002018980572441813": 48, "0020473280915745116": 48, "0020542580850501455": 85, "0020760795413710255": 48, "00209015543474033": 38, "0021046499274098534": 77, "0021052409173061668": 48, "002131411608940556": 21, "002134818313440375": 48, "0021648179244101325": 48, "0021952460474713676": 48, "002217939823003e": 48, "002226109084597644": 48, "0022470117450723555": 81, "0022574135446350964": 48, "0022787217503585614": 110, "002289166045515667": 48, "0023213733165311098": 48, "002354042200669435": 48, "002385951928666893": 21, "0023871796570160172": 48, "0023928342549886e": 84, "0024127234859912e": 38, "0024207927632217833": 48, "0024548887180404605": 48, "0024894748439375424": 48, "0025083332705242": 77, "002524558589773273": 48, "002560147533562283": 48, "0025621895472479556": 85, "0025962493853125836": 48, "002632871989946613": 48, "0026700233303073207": 48, "002707711530251991": 48, "0027214089686471436": 81, "0027459448578374505": 48, "00274976526375405": 38, "0027644465369766743": 84, "0027847317285992117": 48, "0028240807089284594": 48, "0028410571431563554": 76, "002864000519549906": 48, "002904500039104571": 48, "002938211266571423": 83, "0029400048222059856": 77, "002945588307840723": 48, "002987274531417232": 48, "003029568084823295": 48, "003072478516418594": 48, "003116015552098299": 48, "003160189099587527": 48, "0031661513061312222": 85, "0032050092528697147": 48, "0032504862967541446": 48, "003295602986137172": 76, "0032966307115871057": 48, "0033429816790807e": 48, "003343453178112618": 48, "003390964582487553": 48, "0034196513449910823": 77, "0034274924070966e": 38, "0034391760214572452": 48, "003488098807697223": 48, "0035184047718742643": 84, "003533916534866e": 48, "0035377444753272353": 48, "0035742193069537506": 110, "0035849111823151934": 115, "003588124785603924": 48, "0036392517327986876": 48, "0036905492669662104": 21, "003691137550267643": 48, "0037437947167205878": 48, "003797235962696573": 48, "003828543006036e": 48, "0038514742772535683": 48, "003906522914880215": 48, "003962395402637919": 48, "004019105547541827": 48, "004022073286663706": 77, "004044215842409425": 76, "00406449190545115": 103, "004076667444189602": 48, "004135095482647261": 48, "004194404356601538": 48, "004200374856019856": 76, "004204370475404719": 84, "004254609071788941": 48, "004300349628057608": 91, "0043157249547117315": 48, "0043653384802368215": 85, "00436698": 24, "004377767661651467": 48, "004440753187991515": 48, "00444576": 24, "004463903340199571": 83, "004504697877859927": 48, "004569618434104862": 48, "004623704351628802": 84, "004635531928614874": 48, "004702455812997286": 48, "004766219873666417": 77, "0047704079296280195": 48, "004812013162339133": 84, "004823102886311118": 84, "004839406523086855": 48, "0048515408982188635": 14, "0048687652674056035": 21, "0049094702519929204": 48, "0049385274075988955": 38, "004980618201255226": 48, "005": [44, 113], "005050120921221e": 48, "00505286989475429": 48, "005055833317030188": 76, "005065786308053268": 38, "0051166744123425645": 76, "005126245308471082": 48, "005200764884080136": 48, "0052764495430249175": 48, "00532115": 24, "0053533207010933665": 48, "0054314002835130675": 48, "00544510021893": 38, "005481794529587": 84, "005486673124053477": 84, "005510710740585726": 48, "005512321246644729": 110, "005591275063881841": 48, "005596514201958335": 77, "005673116803016881": 48, "005756260083031563": 48, "00579803836835753": 81, "005840729622399467": 48, "00587075e": 76, "0058884106331422": 85, "005926550751686317": 48, "006013749432886288": 48, "006047521099574e": 48, "006052884938201": 84, "006102352279461678": 48, "006183878700617e": 48, "006192386577113453": 48, "006250616150810586": 76, "006268014060966591": 81, "0062838803053114995": 48, "0063391913992392e": 38, "0063768621596142555": 48, "00641114": 24, "0064397891421690305": 77, "006471361574809355": 48, "006567408748907632": 48, "006601850082293865": 83, "006665034668024815": 48, "00666667": 24, "00666667e": 76, "00668231e": 76, "00674705": 24, "0067642711321864776": 48, "0068651507820937825": 48, "006967707126888932": 48, "007071974572961421": 48, "007175095266953569": 39, "007177988453838204": 48, "00720533141359291": 84, "0072391428746526e": 48, "007285785061202624": 48, "007352872683189109": 77, "007363468914282e": 83, "007395401677089779": 48, "007506876607308029": 48, "007538948850570068": 85, "007620249216139159": 48, "007735559962372405": 48, "007767462483043e": 38, "00785285043673079": 48, "00794492344759875": 84, "007972163400751166": 48, "008093542827183131": 48, "008167506192283442": 76, "008217033941975723": 48, "008250496171208304": 77, "008342683267924778": 48, "00838466": 24, "008470538670058644": 48, "008600649402844286": 48, "008733066159301757": 48, "008770264329065e": 48, "008858810330184645": 110, "008867841122120142": 48, "008983784519869005": 77, "009005028016874708": 48, "0091029727105706e": 81, "00914468216745215": 48, "0091636403030867e": 84, "009243880281609152": 85, "00924986": 11, "009286860553797623": 48, "00929141": 24, "00941187124058379": 63, "009411871240584515": 63, "00943162187210645": 48, "0094491081610007e": 48, "009547208619791e": 83, "009567470112709246": 83, "00957902659759177": 48, "009639280718573347": 77, "009656245121461408": 76, "009729137049970023": 48, "00976904": 24, "009775262366193539": 76, "009882017461817013": 48, "00998139": 11, "01": [46, 47, 64, 69, 76, 106, 114, 118], "010000000000006035": 106, "010037734049959545": 48, "0101": 78, "0101101000000000": 39, "010139292127639758": 76, "010196355090081538": 48, "0101979369744123e": 81, "010205668737527e": 77, "010357950994738508": 48, "010359419640115403": 77, "0105225943949912": 48, "01059709e": 76, "010690360225888162": 48, "010835322288439926": 85, "01086132581604804": 48, "01103557098161611": 48, "011161378149839e": 48, "011166401565346e": 48, "011192460122905604": 84, "011213178124896332": 48, "011394232337990156": 48, "011407488793323194": 115, "01148717272069801": 77, "01157882151180759": 48, "0115905": 24, "011767036450854718": 48, "011768153572898": 48, "011843671131637522": 110, "011958970994245996": 48, "0121547221434405": 48, "012187751064729127": 21, "01229555425486588": 38, "01231228e": 76, "0123374": 24, "012354390197259415": 48, "012363": 24, "012479782578389e": 84, "012556324138378401": 81, "012558078894809354": 48, "012765895567014494": 48, "0128688782012293e": 84, "012900680539926e": 77, "012920478670266024": 85, "012977951297551641": 48, "013060236654630365": 77, "013194361094089788": 48, "0133493": 24, "013415244070862429": 48, "0134339": 24, "0134663": 24, "013640723643751601": 48, "013738193462247741": 91, "013747268116727e": 48, "013811543530733e": 84, "013867502737470573": 83, "013870927739243541": 48, "0139698": 24, "014105989018833337": 48, "01413108426448e": 48, "0143129": 24, "01434604512072099": 48, "01450892008875497": 77, "014591238920965869": 48, "014841718816667147": 48, "0148526": 24, "015097639034237173": 48, "0151684": 24, "0152942": 24, "015336450193677e": 48, "015359159966462567": 48, "015471598348598e": 83, "01554149e": 76, "0155923933842377e": 48, "015626448542843247": 48, "0156347": 24, "01570390414750895": 77, "0157114": 24, "01574233336548143": 81, "015899678638718388": 48, "0159095": 24, "016023404407948e": 48, "016142253725977114": 83, "01617903153000453": 48, "0162840947818825e": 48, "016328325811686722": 85, "016464696402088225": 48, "016756870923679827": 48, "017055761899451082": 48, "0173201345490836e": 48, "01736158601934315": 48, "0174519": 24, "01767457072797152": 48, "017677856690926363": 77, "017768": 11, "017929160922748995": 110, "017994955245204867": 48, "018322991779705176": 48, "0183463": 24, "018470559015394374": 80, "0185708": 24, "0185867": 24, "01865894699243924": 48, "0187142": 24, "0189391": 82, "019003103789169966": 48, "0190510611836723e": 48, "01918863e": 76, "01919012190952205": 39, "019355763553309795": 48, "0194531211449646e": 48, "0194674123506714e": 48, "019659": 24, "01971724897913322": 48, "019749": 24, "0197734469007467e": 39, "01983965339095311": 83, "02": [37, 44, 64, 76, 85, 115, 118], "02005492668175809": 38, "020087907739980797": 48, "020188714252706304": 77, "020264232604018606": 103, "020264234576601235": 103, "020347788031209e": 83, "02046811734348254": 48, "02065791386774517": 85, "020831301863592395": 106, "020858291715411852": 48, "021258890368765174": 48, "021302779591812e": 39, "02167043155414069": 48, "021680172914178e": 39, "02209351174170106": 48, "02252883553082078": 48, "0228366": 24, "022852060313368e": 83, "022977263391911276": 48, "023059227507159427": 77, "0231229": 24, "0231253": 24, "0232373": 24, "0232447712545553e": 85, "0232821": 24, "0233311": 24, "0233791": 24, "023439891145440477": 48, "0235113218041585e": 48, "023620592653494303": 21, "0238686459851815e": 48, "023918188335835474": 48, "023921286926102014": 47, "023978513225437e": 48, "0241589": 24, "024273223974312075": 77, "0243206": 24, "02435467077440151": 39, "024384741600227022": 106, "024414250705698613": 48, "024547976218031113": 106, "024931283632543395": 48, "024986973253222e": 14, "02504840e": 76, "0252525392177282e": 48, "025330291735910584": 103, "02538052e": 76, "025474574374679673": 48, "025607813233479115": 77, "0257310986172217e": 77, "025774846052082172": 83, "02605354778036957": 48, "0260991546942733e": 39, "026305373010949e": 48, "0263138": 24, "02642613e": 76, "02651352e": 76, "026686352908694286": 48, "026909264295318674": 85, "026991393488043674": 77, "027036794325895e": 110, "02717043861464373": 110, "027410842175647522": 48, "027487700817419e": 48, "0275036694747827e": 83, "02753241632854": 108, "0278865": 24, "0282924756612993e": 48, "028315523074329833": 48, "02840051e": 76, "028786277945169303": 110, "02881189922538989": 77, "02914557e": 76, "029378085186036483": 77, "0294866473975043e": 48, "02952541052556933": 83, "0297401890608049e": 48, "02974211765794692": 48, "03": [0, 55, 74, 76, 85, 89, 103], "030065910434030483": 83, "03019290019802654": 14, "03049776255121646": 115, "0305053745835908e": 39, "03064450321599e": 21, "030805": 24, "031006371846374843": 77, "0314919971368875e": 48, "0317430429749966e": 48, "032001079597646e": 39, "0320692": 24, "0321664556812178e": 38, "0322338191445864e": 77, "03235807e": 76, "032415672332935e": 48, "0327422": 24, "032920325740202e": 85, "03296601056961356": 85, "0330722476742483e": 77, "03307384e": 76, "033155171543706e": 48, "0332470429487766e": 48, "03394806e": 76, "0339567": 24, "03430579733377514": 83, "0344221": 24, "0344815": 24, "0346404566701807e": 48, "0349543337339e": 39, "03503614777663428": 77, "035057863467162e": 77, "0352011973243236e": 48, "03576187036516944": 85, "03591803e": 76, "036006096856494e": 84, "0360268233935707": 21, "036081925033399936": 76, "03616817538237669": 110, "036419664573553744": 76, "036882711834327e": 48, "0369993": 11, "0369995": 11, "037204475895342": 77, "037214875409793e": 77, "03732654e": 76, "0377694309296993e": 48, "03798027450044472": 84, "0382184": 24, "0383145887165588e": 21, "038578886687444205": 110, "03867743230879999": 81, "0391573266438795e": 83, "0395349049971244e": 48, "03971919917200752": 38, "0398083805786505e": 48, "039923161314892e": 38, "0399256": 11, "04": [20, 38, 63, 76], "0401019232202664e": 39, "0401499370989996e": 84, "04060539621517081": 83, "04091961212851145": 84, "04114423163307e": 48, "04148923094504725": 38, "04165481393936405": 85, "04182651492449851": 84, "0421362": 24, "042529997182085555": 106, "042700555926476e": 39, "0427258429588364": 109, "04295345e": 76, "04304343e": 76, "04322115e": 76, "04333400e": 76, "043410523067387e": 39, "04378948e": 76, "0440850635700838e": 48, "0441077852041042e": 48, "04445533e": 76, "04452565e": 76, "0448318988346594e": 48, "0456629339641523e": 77, "0458612819916141e": 48, "0460550393450818e": 21, "046167213571327394": 21, "04632471e": 76, "04640769538476629": 70, "0464076953847663": 70, "046559115980648e": 21, "047585281273978e": 48, "0476407832691725e": 48, "047831452027208586": 77, "0479234984947036e": 48, "04829608173279241": 77, "0483439025780243e": 80, "04844263e": 76, "04897732520653299": 106, "04901124e": 76, "0493228166340138e": 38, "05": [0, 6, 7, 14, 16, 18, 19, 21, 22, 38, 39, 42, 43, 48, 55, 58, 63, 76, 77, 78, 81, 83, 84, 85, 91, 106, 108, 110, 113, 114, 115, 119, 120, 121], "05017114e": 76, "0502118520135045e": 38, "050345607732123e": 48, "05056242146491291": 83, "05066057588433867": 103, "05070470124514234": 21, "05072273e": 76, "050808588781358e": 48, "051314861515644065": 85, "051348606027109": 48, "05147539e": 76, "0518600005494": 103, "0522261642833315e": 48, "05225642481371e": 48, "05261663e": 76, "05275907536407711": 0, "053091225238836e": 77, "053108611128666e": 48, "053506705294776e": 21, "0540162324290298e": 48, "0543115072670755e": 83, "054442809432531e": 48, "05473882e": 76, "054825222580238e": 48, "055292332179194e": 39, "05615795e": 76, "056463652129439e": 63, "056463652131211e": 63, "0565477676177174e": 48, "056816693425752e": 48, "056875365473615e": 80, "057699886783593e": 48, "05785815e": 76, "0583026929967562e": 48, "058629781795617e": 48, "05886485e": 76, "0596323": 24, "05969811055414346": 70, "059962390744391e": 77, "06": [14, 21, 38, 39, 48, 63, 76, 77, 80, 81, 83, 84, 85, 91, 110, 115], "060046408245776e": 48, "060188892523224485": 70, "060197008583245344": 70, "060197013034633824": 70, "060197013034635836": 70, "06041231e": 76, "060421857389568e": 77, "0604307395852179e": 48, "060638566853438e": 48, "0612004296881115e": 77, "06173136e": 76, "0622350370452333e": 48, "063041512342942e": 48, "0635527812072218e": 48, "06372227e": 76, "06389531595609149": 83, "064378099172241e": 38, "064615076190458e": 48, "06479309e": 76, "064853508518673e": 84, "06562033841012835": 76, "0658429693010248e": 38, "0665306914429312e": 39, "0666980034078006e": 83, "06672284005037e": 38, "0667455150851594": 85, "06687894741407885": 109, "0668842887956969e": 48, "067774904721001e": 48, "06815699e": 76, "06822155257283025": 81, "06852126e": 76, "0686992932246637e": 48, "06871088406211634": 76, "068880327604704e": 48, "06924526e": 76, "069312434365388e": 83, "06934594552348519": 110, "06941104395496575": 76, "069583126200502e": 83, "06967160e": 76, "06978025266715498": 115, "07": [7, 14, 16, 18, 19, 21, 38, 39, 48, 62, 63, 76, 77, 80, 81, 83, 84, 85, 91, 110, 115], "07053899603094559": 21, "070630129019807e": 77, "0708562851366104e": 38, "0708899746727645": 63, "07092390115693398": 38, "070986974525415e": 48, "0710721": 11, "07109762e": 76, "071677623287515e": 48, "071689163842817e": 48, "0718173613822063e": 38, "07213824e": 76, "072583399247498e": 48, "0728344173676783e": 38, "0733771059172907e": 48, "0736797747190897e": 83, "0752031579870268e": 48, "0754609239472857e": 48, "07563602425424365": 91, "0757704058947594e": 83, "0761093388261588e": 48, "07623135e": 76, "076773468362999e": 110, "07679038814089593": 83, "0769147677204536e": 48, "076925": 11, "0770325975566385e": 48, "07795637134606523": 23, "078183206613188e": 39, "07867885e": 76, "079255401192082e": 77, "0793232627093197e": 76, "0798810040492257e": 77, "0799504864837223e": 77, "08": [14, 21, 38, 39, 48, 63, 77, 80, 81, 83, 84, 85, 91, 110, 115], "0800838951175999e": 77, "080363248841906e": 84, "0816681711721685e": 76, "0817466086216224e": 48, "0817889380369626e": 21, "082154822634119e": 48, "082394154349409e": 48, "08243": 108, "0825055598480013e": 21, "082967519865576": 48, "08333333333333291": 76, "08333333333333307": 76, "08335855188353485": 84, "0835868976279029e": 48, "0837253706493335e": 77, "0838424427178355e": 110, "08403599336703109": 38, "08513052e": 76, "0851966894625983e": 48, "085203085594984e": 48, "0855068742571685e": 38, "085538139091289e": 83, "0861189697629896e": 39, "08658007e": 76, "08686469e": 76, "0871232244493624e": 39, "087166664294984e": 48, "08733": [119, 120], "08754364e": 76, "087705232362404e": 48, "088329890520765e": 48, "08848867205207e": 83, "088744055878357e": 48, "08916336e": 76, "089339379343828e": 48, "08966319416902137": 85, "09": [14, 21, 22, 38, 39, 48, 63, 77, 83, 84, 85, 91, 110, 115], "090080875328715e": 48, "090181364978928e": 48, "090382846710801e": 48, "0905496474263337e": 83, "091879177514704e": 48, "0920362969957407e": 48, "092098918926114e": 83, "0922993209640892e": 48, "09282291331667633": 83, "0938448611287085e": 39, "09384602e": 76, "093847080520956e": 38, "093854065721601e": 48, "09425754e": 76, "09561861e": 76, "0956401951870296e": 48, "095680466304": 103, "0956820207492046e": 84, "09574110e": 76, "095939553147835e": 84, "09680027e": 76, "0968159694944652e": 48, "096870591581053e": 39, "0971473988404078e": 84, "0972441079209224e": 84, "0972832118768083e": 21, "097469463440972e": 39, "097522274092465e": 48, "0985340239060944e": 84, "09862199e": 76, "0986818950368167e": 48, "09976291818609e": 48, "0_i": 47, "0_j": 48, "0m": 23, "0x10627a8b0": 76, "0x1098e6250": 23, "0x10e19adf0": 39, "0x10e1f1570": 38, "0x1105e7790": 23, "0x113fac400": 23, "0x11e514450": 23, "1": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 95, 96, 97, 98, 99, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "10": [0, 1, 2, 4, 5, 12, 14, 18, 19, 21, 38, 39, 40, 42, 43, 44, 47, 48, 49, 50, 51, 55, 58, 62, 63, 64, 65, 66, 73, 77, 80, 81, 82, 83, 84, 85, 86, 91, 92, 96, 97, 105, 110, 115, 118, 120, 121], "100": [0, 16, 21, 35, 38, 39, 48, 64, 77, 83, 84, 85, 89, 108, 110, 113, 114, 115, 116, 118, 121], "1000": [18, 19, 48, 64, 81, 82, 85, 112], "10000": [33, 35, 48, 50, 51], "100000": 11, "1001": 48, "100139362740519e": 77, "1002": 48, "1003": 48, "1004": 48, "1005": 48, "1006": 48, "10065": 118, "10066207816841645": 103, "1006946297625608e": 77, "1007": 48, "1008": 48, "1009": 48, "1009066348664202e": 48, "101": [38, 39, 48, 77, 83, 84, 85, 110], "1010": 48, "101080": 22, "1011": 48, "1012": 48, "1013": 48, "10132037983620129": 103, "10132093820158772": 103, "10132105302197744": 103, "1013211807015426": 103, "10132118307694862": 103, "1014": 48, "1014556933272555e": 48, "1015": 48, "1016": 48, "10163889987329": 108, "101670": 121, "1017": 48, "1018": 48, "1019": 48, "102": [38, 39, 48, 77, 83, 84, 85, 110], "1020": 48, "1021": 48, "1022": 48, "1023": 48, "10238050892831017": 85, "1024": 48, "1025": 48, "102520165565222e": 48, "1026": 48, "1027": 48, "1028": 48, "1029": 48, "103": [38, 39, 48, 77, 79, 83, 84, 85, 110], "1030": 48, "1031": 48, "1032": 48, "1033": 48, "1034": 48, "1034116": [65, 105], "1034909470615006e": 48, "1035": 48, "1036": 48, "1037": 48, "1038": 48, "103843864311542e": 48, "1039": 48, "104": [38, 39, 48, 77, 83, 84, 85, 110], "1040": 48, "1041": 48, "1042": 48, "1043": 48, "1044": 48, "1045": 48, "10455003e": 76, "10459024e": 76, "1046": 48, "1047": 48, "1048": 48, "1049": 48, "105": [38, 39, 48, 77, 83, 84, 85, 110], "1050": 48, "105030753170029e": 48, "1051": 48, "1052": 48, "1053": 48, "1053682271319834e": 48, "1054": 48, "1055": 48, "1056": 48, "1057": 48, "1058": 48, "1059": 48, "106": [38, 39, 48, 77, 83, 84, 85, 110], "1060": 48, "1061": 48, "1062": 48, "1063": 48, "1064": 48, "1065": 48, "1065178148932502e": 77, "1066": 48, "1067": 48, "10673815e": 76, "1068": 48, "1069": 48, "107": [38, 39, 48, 77, 83, 84, 85, 110], "1070": 48, "1071": 48, "1072": 48, "1072387109440784e": 48, "1072490070102246e": 48, "107279980834582e": 84, "1073": 48, "1073268104192383e": 77, "107346442453208e": 48, "1074": 48, "1075": 48, "1076": 48, "1077": 48, "1078": 48, "107896060669359e": 48, "1079": 48, "108": [38, 39, 48, 77, 83, 84, 85, 110], "1080": 48, "108072": 11, "1080931524797886e": 39, "1081": 48, "1082": 48, "1083": 48, "10835203e": 76, "1083784321502765": 48, "1084": 48, "1085": 48, "1085370483308783e": 38, "1086": 48, "1087": 48, "1088": 48, "1089": 48, "109": [38, 39, 48, 77, 83, 84, 85, 110], "1090": 48, "1091": 48, "1092": 48, "1093": 48, "1094": 48, "1095": 48, "10956896174344156": 81, "1096": [48, 67], "1097": 48, "1098": 48, "1099": 48, "109953797594372e": 91, "10999455e": 76, "11": [14, 15, 21, 38, 39, 48, 63, 77, 80, 81, 83, 84, 85, 91, 105, 108, 110, 111, 120], "110": [38, 39, 48, 77, 83, 84, 85, 110], "1100": 48, "1101": 48, "1102": 48, "1103": 48, "1104": 48, "1105": 48, "11052790e": 76, "1106": 48, "1107": 48, "1108": 48, "1109": 48, "110998": 11, "111": [38, 39, 48, 77, 83, 84, 85, 110], "1110": [39, 48], "1110696554423406e": 21, "1111": 48, "1111000000000000": 39, "1111111111111111111111111111111111111": 108, "11111111111111111111111111111111111111111111111111": 108, "11111111e": 76, "1112": 48, "1113": 48, "1114": 48, "1114129704008765e": 85, "1115": [15, 48], "1116": [15, 48], "1117": 48, "11175628e": 76, "1118": 48, "111842042028687e": 48, "1119": 48, "112": [38, 48, 77, 83, 84, 85, 110], "1120": 48, "11208822466509e": 115, "1120952556356904e": 48, "1121": 48, "1121913244649083e": 84, "1122": 48, "1123": 48, "1124": 48, "1125": [15, 48], "1126": [15, 48], "1127": 48, "11274286e": 76, "1128": 48, "1129": 48, "112971647279069e": 21, "113": [38, 48, 77, 83, 84, 85, 110], "1130": 48, "1131": 48, "1132": 48, "11326140e": 76, "1133": 48, "1134": 48, "1134450354834574e": 83, "1135": 48, "1136": 48, "1137": [48, 65, 105], "1138": 48, "1139": 48, "1139871745692467e": 48, "114": [38, 48, 77, 83, 84, 85, 110], "1140": 48, "1141": 48, "1142": 48, "114226012216223e": 48, "1142446998583604e": 48, "1143": 48, "114305718318": 103, "1143891383548138e": 83, "1144": 48, "1144795853898352e": 48, "1145": [15, 48], "1146": [15, 48], "1147": 48, "1148": 48, "1149": 48, "115": [38, 48, 77, 83, 84, 85, 110], "1150": 48, "1151": 48, "11519878e": 76, "1152": 48, "1153": 48, "1153690820937868e": 83, "1154": 48, "1155": 48, "1156": 48, "11562856971462293": 85, "1157": 48, "115776481808797e": 77, "1158": 48, "1159": 48, "116": [38, 48, 77, 83, 84, 85, 110], "1160": 48, "1160794010521349e": 38, "116080611146976e": 38, "1161": 48, "1162": 48, "1163": 48, "1164": 48, "1165": 48, "1166": 48, "1166378431252478e": 38, "1167": 48, "1168": 48, "1168648242240292e": 39, "1169": 48, "117": [38, 48, 77, 83, 84, 85, 110], "1170": 48, "1171": 48, "1172": 48, "1173": 48, "1174": 48, "1175": 48, "11756578349743e": 83, "1176": 48, "1177": 48, "11771686788897e": 48, "1178": 48, "117841478485599e": 48, "117889181764347": 63, "1179": 48, "118": [38, 48, 77, 83, 84, 85, 110], "1180": 48, "1181": 48, "1181654480683108e": 76, "1182": 48, "11825842e": 76, "1183": 48, "1184": [15, 48], "1185": 48, "1186": 48, "1187": 48, "1188": 48, "1188632171403412e": 48, "1189": [34, 48], "119": [15, 38, 48, 76, 77, 83, 84, 85, 110], "1190": 48, "1191": 48, "1192": 48, "1192648791598475e": 85, "1193": 48, "1194": 48, "11949520952077154": 83, "1195": 48, "1196": 48, "1197": [34, 48], "119777984490796e": 48, "1198": 48, "1199": 48, "12": [14, 15, 21, 38, 39, 47, 48, 58, 63, 71, 77, 80, 81, 83, 84, 85, 91, 110, 111, 118, 121], "120": [15, 38, 48, 76, 77, 83, 84, 85, 110], "1200": 48, "12007716e": 76, "1201": 48, "1202": 48, "120263469848235e": 83, "1203": 48, "1204": 48, "1205": 48, "1206": 48, "1207": 48, "1207666510181047e": 48, "1208": 48, "1209": 48, "121": [15, 38, 48, 77, 83, 84, 85, 110], "1210": 48, "1211": 48, "1212": 48, "121237851055954e": 48, "1213": 48, "1214": 48, "1214463186327964e": 48, "1215": 48, "1216": 48, "1217": 48, "1218": 48, "1219": 48, "12198974e": 76, "122": [38, 48, 77, 83, 84, 85, 110], "1220": 48, "1221": 48, "1222": 48, "1222150109879732e": 38, "1223": 48, "1224": 48, "1225": 48, "1226": 48, "1226736392497831e": 48, "1227": 48, "1227484703156794": 81, "1228": 48, "122841804400887e": 48, "1229": 48, "123": [38, 48, 77, 83, 84, 85, 110], "1230": 48, "1231": 48, "1232": 48, "1233": 48, "1234": [48, 51], "1235": 48, "1236": 48, "1236004679313227e": 85, "1237": 48, "1237307982241775e": 39, "1238": 48, "1239": 48, "124": [38, 48, 77, 83, 84, 85, 110], "1240": 48, "1241": 48, "1242": 48, "12424681e": 76, "1243": 48, "1244": 48, "1245": 48, "1246": 48, "1247": 48, "1248": [15, 48], "12481": 63, "124854140373132e": 48, "1249": 48, "125": [38, 48, 76, 77, 83, 84, 85, 108, 110], "1250": 48, "12502359e": 76, "125091180878627e": 48, "1251": 48, "12517048e": 76, "1252": 48, "1253": 48, "1254": 48, "1255": 48, "12559322630402628": 84, "1256": 48, "1257": 48, "1258": 48, "1259": 48, "126": [38, 48, 77, 83, 84, 85, 88, 110], "1260": 48, "12604564e": 76, "1261": 48, "12612009e": 76, "1262": 48, "1263": 48, "1264": 48, "1265": 48, "1266": 48, "1267": 48, "1268": 48, "1269": 48, "127": [38, 48, 77, 83, 84, 85, 88, 110], "1270": 48, "1271": 48, "1272": 48, "1273": 48, "1274": 48, "1275": 48, "1275873912643786e": 48, "1276": 48, "1277": 48, "12771016328862472": 85, "1278": 48, "127897790833774e": 14, "1279": 48, "128": [38, 48, 77, 83, 84, 85, 88, 110], "1280": [15, 48], "1281": 48, "1282": 48, "1282174616192325e": 84, "1283": 48, "128385842495598e": 38, "1284": 48, "1284839854964043e": 83, "1285": 48, "1286": 48, "1286760270120043e": 80, "1287": 48, "1288": 48, "1289": 48, "128937157167869": 48, "129": [38, 48, 77, 83, 84, 85, 110], "1290": 48, "1291": 48, "1292": 48, "1293": 48, "1294": 48, "1295": 48, "1295056655675621e": 48, "1296": 48, "1297": 48, "1298": 48, "1299": 48, "13": [14, 15, 21, 38, 39, 48, 58, 63, 77, 80, 81, 83, 84, 85, 91, 110], "130": [38, 48, 77, 83, 84, 85, 110], "1300": 48, "130024942605001e": 38, "1301": 48, "1302": 48, "13029395438108e": 39, "1303": [15, 48], "1304": [15, 48], "1305": 48, "1306": 48, "1307301970457093e": 48, "1308682699054299e": 38, "131": [38, 48, 77, 83, 84, 85, 110], "131205540534497e": 85, "1313": 15, "1313848999463623e": 77, "1314": 15, "1315250560178633e": 39, "1317": 15, "1318": 15, "132": [38, 48, 77, 83, 84, 85, 110], "132234434912973e": 48, "13234921846973": 108, "1324039739542678e": 77, "132574138753976e": 39, "133": [10, 38, 48, 77, 83, 84, 85], "13317477e": 76, "133217721853538e": 83, "133433727673969e": 48, "133572632812229e": 48, "133659": 103, "134": [38, 48, 77, 83, 84, 85], "1342275998439534e": 21, "134355059697924e": 48, "1344496302303633e": 48, "134630462323265e": 48, "135": [38, 48, 77, 83, 84], "1354489295710067e": 38, "136": [38, 48, 77, 83, 84], "1363485282619605e": 39, "136379583878647e": 48, "13658800e": 76, "137": [38, 48, 77, 83, 84, 108], "13724910e": 76, "1372528539117695e": 85, "138": [38, 48, 77, 83, 84], "138056451376175e": 38, "1383131459279391e": 48, "138764267205407e": 48, "138838983489785e": 77, "139": [38, 48, 77, 83, 84], "13912": 108, "13920": 120, "139264286603403e": 48, "139315271444288e": 48, "1393692959144535e": 84, "13946661e": 76, "1395336353993284e": 21, "13978020e": 76, "14": [14, 15, 21, 38, 39, 48, 58, 63, 77, 81, 83, 84, 85, 91, 110], "140": [38, 48, 77, 83], "140103064211771e": 83, "140448925990182e": 21, "1407461783284897e": 84, "141": [48, 77, 83], "1410181950377555e": 77, "14103409e": 76, "1415927925658518": 103, "1415940039476355": 103, "1415964586986087": 103, "141601546084023": 103, "142": [48, 77, 83], "14225686e": 76, "142329644220869e": 48, "142460317688723e": 84, "14247326e": 76, "1427436944464753": 103, "142946942564037e": 83, "142970521525175e": 48, "143": [48, 77, 83], "143020583810261e": 115, "143292625079821e": 63, "143292625081005e": 63, "1432953401769533e": 48, "14351": 118, "143647119956742e": 21, "14367": 85, "1436973564022294e": 48, "143924722657": 103, "144": [2, 15, 48, 77, 83], "14401498e": 76, "1441040745721575e": 48, "1449": 15, "145": [48, 77, 83], "1450": 15, "145066796269033e": 48, "145240338203072e": 48, "145481": 55, "145700503692558": 48, "1457055730195295e": 38, "1459": 15, "146": [48, 77, 83, 108], "1460": 15, "146247392809796e": 63, "146247392809993e": 63, "14630626612767e": 48, "14643913718708476": 85, "1466": 62, "14677104e": 76, "147": [48, 77, 83], "14706588838844e": 77, "14717576e": 76, "1473442451657755e": 48, "148": [48, 77, 83], "14800181052391703": 83, "1480141493998358e": 21, "1487": 15, "1488": 15, "149": [15, 48, 77, 83], "14929094e": 76, "1495557599322867e": 38, "14967093e": 76, "15": [12, 14, 15, 21, 38, 39, 48, 63, 77, 81, 83, 84, 85, 91, 110, 118, 121], "150": [48, 77, 83], "1501": 15, "150172892273618": 63, "1502": 15, "1502531949919336e": 48, "150513": 82, "1509993528977585e": 48, "151": [48, 77, 83], "151239212626374e": 77, "1517867052582708e": 85, "152": [48, 77, 83], "152210014075458e": 48, "1523": 15, "1524": 15, "152829147699458e": 77, "153": [15, 48, 77, 83], "153162058902952e": 48, "15375584e": 76, "154": [48, 77, 83], "1541705203835396e": 48, "154452942554955e": 85, "15456810e": 76, "154591039558322e": 84, "15479975e": 76, "155": [48, 77, 83], "1550389688295956e": 77, "155122265013862e": 48, "15531330e": 76, "155560354442374e": 48, "1557028423569548e": 83, "156": [48, 77, 83], "156042934268715e": 48, "1565545505886938e": 84, "156574908062523e": 21, "15669640292831505": 14, "15674039e": 76, "157": [15, 48, 77, 83], "157232828249118e": 39, "15777615782165e": 21, "1578660947274213e": 48, "158": [48, 77, 83], "1583250244863396e": 21, "15852596e": 76, "15866512e": 76, "1587": 15, "15873168e": 76, "1588": 15, "159": [48, 77, 83], "1592221117566454e": 48, "159226802859874e": 84, "159386196627027": 63, "159492305023506": 48, "15x15": 120, "16": [14, 21, 38, 39, 48, 63, 67, 76, 77, 81, 83, 84, 85, 88, 91, 103, 108, 110, 121], "160": [48, 77], "1601092281831917e": 77, "1604125116859914e": 48, "161": [48, 77], "1611": 15, "161194204007716e": 48, "1612": 15, "1614959668268976e": 110, "16192264e": 76, "162": [48, 77], "16212877624059613": 70, "16227787e": 76, "1627": 15, "162771755192398e": 48, "1628": 15, "163": [48, 77], "1631690327272827e": 38, "1632383109280923e": 48, "1637": 15, "1638": 15, "164": [48, 77], "164002978391902e": 38, "164087870949272e": 48, "164454944361612e": 48, "164814423789034e": 48, "165": [48, 77], "1652": 55, "1653": 15, "1654": 15, "1655": 108, "16551145145630625": 84, "166": [15, 48, 77], "1662768813921747e": 48, "1663": 15, "1664": 15, "166607912750655e": 77, "16666666666666605": 76, "167": [22, 48, 77], "16703366696607e": 48, "167355481226384e": 48, "167788478028774e": 84, "1678144767741274e": 110, "168": [48, 77], "1682609714160612e": 48, "168625673411682e": 48, "169": [48, 77], "16902662e": 76, "1690509397482924e": 39, "1695613375814276e": 77, "16992": 103, "17": [11, 14, 21, 38, 39, 48, 62, 63, 67, 76, 77, 81, 83, 84, 85, 91, 110, 121], "170": [48, 77], "17018031e": 76, "1702487976526392e": 48, "17034605e": 76, "170953469301783": 48, "171": [48, 77], "17104988e": 76, "171242169527921e": 38, "1715084968823221e": 110, "171539579468978e": 48, "171964483263876e": 48, "172": [15, 48, 77], "17266415e": 76, "173": [48, 77], "173145243523698e": 38, "1731567271102746e": 38, "173560312218078e": 48, "1736012556074996e": 48, "17363736e": 76, "174": [48, 77], "17444576e": 76, "17444909e": 76, "17484926e": 76, "175": [15, 48, 77], "1750190919030915e": 39, "17521646837299754": 63, "17521646837299884": 63, "175370751686568e": 48, "175796464246952e": 83, "176": [48, 77], "177": [15, 48, 77], "177084140486483e": 48, "1772580024815873e": 48, "1773703164820084e": 48, "177430": 77, "17787146650364064": 84, "177975481974726e": 77, "17798587e": 76, "178": [15, 48, 77], "178359635677189e": 48, "1786334543121044": 63, "17864942689424e": 48, "1787253881538545e": 21, "1789330005993": 103, "1789412583431215e": 77, "179": [15, 48, 77], "179038940459572e": 83, "179130114255554e": 91, "179276503633465e": 77, "1797414167033695e": 85, "18": [21, 38, 39, 48, 63, 77, 81, 83, 84, 85, 91, 110, 121], "180": [15, 48, 77], "180098524265756e": 77, "1801844070219922e": 83, "1803": 118, "18047030359955413": 70, "180523233901374e": 85, "180581597168563": 48, "1809640801639115e": 48, "181": [15, 48, 77], "181275397026587e": 84, "181384164542866e": 84, "18166648405462782": 85, "181919481874446e": 39, "18198359e": 76, "182": [15, 48, 77, 121], "18239326239495e": 38, "182464315033788e": 48, "18248908455579e": 48, "1825238009470132e": 48, "1829807261609314e": 83, "183": [15, 48, 77, 121], "183205809357812": 63, "18378034e": 76, "184": [15, 48, 77], "184518646616133e": 38, "1845355279110761e": 48, "1848020578536776e": 85, "185": [15, 48, 77], "18560": 120, "186": [15, 48, 77], "18615733922752714": 83, "186441139990981e": 48, "186544775446272e": 77, "1865510550568645e": 48, "1869": 108, "187": [15, 48, 76, 77], "18746770e": 76, "18773436e": 76, "1879032798831167e": 48, "188": [15, 48, 76, 77], "188084777154004e": 110, "1881411344828965": 70, "18829278797325397": 70, "18829323589939168": 70, "1882932359053115": 70, "18844850e": 76, "18854148259682096": 38, "18856044e": 76, "1886623726682327e": 84, "188762568457179": 48, "189": [15, 48, 77], "189004510910192e": 38, "1897989532745137e": 48, "19": [21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 91, 96, 110, 121], "190": [15, 48, 77], "190039": 82, "1905083170126303e": 48, "191": [15, 48, 76, 77], "19131": 108, "191412740756462e": 84, "1915768422357135e": 91, "19168273e": 76, "1917443509672249e": 48, "192": [15, 48, 76, 77], "192982629062653e": 48, "193": [15, 48, 77], "193017889368154e": 84, "1932267987396277e": 77, "1937717716297164e": 48, "194": [15, 48, 77], "19409740372493e": 48, "19414012e": 76, "194234877618424e": 48, "195": [15, 48, 77], "19502289e": 76, "1957737541722982": 76, "195795581626075": 48, "196": [15, 48, 77], "1960973838068986e": 84, "1961228339113691e": 84, "19613792e": 76, "1962463458313255e": 48, "196926650939042e": 48, "197": [15, 48, 77], "1973176149489421e": 77, "197377": 63, "19756385e": 76, "19794827628847217": 76, "198": [15, 34, 48, 77], "1985": [63, 118], "198666754833949e": 48, "1988": 81, "1989970409922486e": 48, "199": [48, 77], "1992": [65, 105], "19954481e": 76, "199662120862354e": 77, "1997": 5, "1_": [84, 109], "1_0": 76, "1_g": 98, "1_t": 9, "1d": [9, 10, 92, 97, 100], "1e": [1, 2, 5, 7, 14, 17, 18, 23, 47, 48, 49, 50, 51, 58, 62, 74, 81, 82, 83, 85, 106, 118], "1e10": 34, "1e16": [14, 23], "1e3": [0, 55], "1e4": 74, "1e5": 56, "1e6": [37, 39, 76], "1h": 31, "1j": [43, 44, 112], "2": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "20": [0, 4, 6, 21, 38, 39, 43, 47, 48, 51, 55, 63, 77, 81, 83, 84, 85, 91, 110, 115, 118, 119, 120, 121], "200": [47, 48, 49, 77, 81, 83, 84, 91, 103], "2000": [79, 83, 118], "20000": 63, "20003175e": 76, "2001": 63, "2002": [83, 119, 120], "2003": [24, 37], "2005": [60, 83], "2008": 34, "2009": [1, 34, 118], "201": [48, 77], "2010": [3, 7, 115], "201036798469105e": 48, "2014": 118, "2014276423085216e": 63, "20142764230892e": 63, "2016": 7, "20161982e": 76, "2017": 72, "20185210357764e": 83, "20185701640442e": 48, "201912825655762": 48, "202": [48, 77], "2021": 63, "2021410480985224e": 84, "2022": 115, "2024": 103, "20246277e": 76, "20278040915028e": 48, "20278558e": 76, "203": [48, 77], "2030804127834024e": 48, "203340764549582e": 38, "203514": 82, "203579372682444e": 38, "20360315e": 76, "2038392742993523e": 48, "204": [48, 77], "20406673503597e": 48, "20416263e": 76, "204170427930421e": 76, "2047": 118, "204721838426908e": 85, "2049856701270923e": 21, "205": [48, 77], "205086292997377e": 48, "205791598142954e": 80, "206": [48, 77], "20600328199376758": 115, "207": [48, 77], "20710259e": 76, "20729480460904": 48, "207304064918974e": 48, "2073949988316887e": 110, "207500945135074": 63, "2075884760544393e": 48, "207797463801602e": 39, "2078": 118, "2079866858818379e": 39, "208": [48, 77], "2081699355060198e": 21, "2083460448557326e": 48, "209": [48, 77], "209841802955364e": 38, "20x15": 120, "21": [21, 38, 39, 48, 63, 77, 81, 83, 84, 85, 91, 110], "210": [48, 74, 77], "2104017086174213e": 48, "210516495096637e": 38, "2105692323236415e": 115, "210914171261402e": 48, "211": [48, 77], "2113462356347832e": 48, "212": [48, 77], "212082195607958": 48, "212176823477045e": 83, "21240144e": 76, "2127666328263005e": 48, "2128164846413054e": 77, "213": [48, 77], "2131": 108, "2135": 103, "213965912854191e": 48, "214": [48, 77], "21422613e": 76, "21448524e": 76, "215": [48, 76, 77], "215611127226652e": 85, "2156997683938283e": 48, "2159255177579773": 81, "216": [48, 76], "216735134902866e": 48, "217": [48, 63, 76], "217767941323583e": 48, "218": [48, 76], "218066951299396e": 21, "218534373172311e": 48, "2189016160123873e": 91, "219": 48, "219015968824155e": 84, "21921127e": 76, "2198400344857686e": 48, "22": [15, 21, 38, 39, 48, 63, 77, 81, 83, 84, 85, 88, 91, 105, 108, 110, 121], "220": 48, "2200814982392898e": 84, "220089440885009e": 48, "22097120565400583": 85, "221": [48, 76], "2210233759685583e": 48, "221168193883369e": 38, "221342747583671e": 48, "221997": 11, "222": [48, 76], "22214879e": 76, "222468225080945e": 48, "22267988e": 76, "222905569117103e": 77, "223": [48, 76], "2231484138440663e": 77, "22396274e": 76, "224": [48, 76], "22447412e": 76, "224582426593474e": 84, "2248018497117877e": 48, "225": [48, 76], "2250262778494888e": 83, "22507913e": 76, "2251790106502366e": 48, "226": [48, 76], "22649519e": 76, "22665428481334e": 21, "226823012568907e": 48, "227": [48, 76], "2272633103295307e": 48, "2273752434466936e": 84, "22749714e": 76, "227869591181543e": 48, "228": [48, 106], "228111988549506e": 38, "2285806360324746e": 110, "22889271e": 76, "229": [15, 48], "22902254e": 76, "229139870798433e": 110, "2296515582177255e": 48, "23": [15, 21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 91, 108, 110, 121], "230": [15, 48], "230062807166462e": 77, "230426840079289e": 83, "23052075e": 76, "230671159880186e": 48, "230803664237456e": 48, "231": 48, "231350929217185e": 84, "23170524e": 76, "232": [15, 48, 121], "23222011e": 76, "2322662719717318e": 39, "2323126337494496e": 48, "2326351878065917e": 48, "23298683e": 76, "233": [48, 76, 108, 121], "23352169e": 76, "2336993139926075e": 84, "23383303211193643": 85, "234": [48, 121], "2340": 2, "234217088721998e": 48, "234404212444103e": 91, "2345400347809227e": 48, "234563447979218e": 84, "2346168202002402e": 38, "23468220e": 76, "234688003055873e": 85, "2347321588196066e": 48, "23493810e": 76, "235": 48, "23554864e": 76, "235558457392675e": 84, "235698848966272e": 38, "23588458e": 76, "236": 48, "236291551289979e": 21, "236763300738726e": 48, "236833127892583e": 48, "2368607926721074e": 48, "237": 48, "23725": 103, "23770460e": 76, "238": 48, "238341470639586e": 48, "239": [48, 121], "239553": 11, "24": [15, 20, 21, 26, 38, 39, 46, 48, 77, 81, 83, 84, 85, 91, 108, 110, 111, 121], "240": [48, 121], "240522273706592e": 115, "241": [15, 48, 121], "24123503e": 76, "241311930663991e": 48, "24131829e": 76, "242": [48, 121], "242147288218785e": 80, "2421516344125203e": 48, "2422464676774915e": 48, "2424762666737146e": 48, "243": [48, 121], "2434571319271414": 85, "2435260263769448": 103, "244": 48, "2441091631346645e": 84, "2443598039825515e": 48, "2444781742724771e": 38, "245": 48, "2452143271619613e": 77, "24523562e": 76, "245276461511493e": 14, "2453548564298967e": 77, "245672547037219e": 48, "246": 48, "24637796e": 76, "2464607544411714e": 48, "246753131763292e": 77, "247": 48, "24729233e": 76, "248": 48, "2481": [15, 23], "2481652589569217e": 48, "2482396528288556e": 38, "24856511e": 76, "249": 48, "2498065056018573e": 48, "249928356147489e": 48, "24999999999999872": 15, "24999999999999895": 15, "25": [5, 8, 15, 21, 33, 35, 38, 39, 48, 74, 77, 81, 83, 84, 85, 91, 108, 110], "250": 48, "251": 48, "25121637e": 76, "25169512e": 76, "2519326977123633e": 48, "2519635254418385e": 48, "2519836807584633e": 48, "252": 48, "252011089157917e": 38, "25270": 22, "252700643269565e": 48, "253": 48, "25322175e": 76, "2533077908298905e": 83, "253381714290263e": 39, "253472013629693e": 14, "254": 48, "2540629454140222e": 48, "25440808e": 76, "25464403586309087": 83, "254797290530035e": 48, "255": 48, "2550708572760314": 63, "2551405187698492e": 76, "255392285937742e": 48, "255794637574766e": 48, "256": 48, "2561165106050925e": 110, "2562185071306156e": 38, "256369048363894e": 21, "257": 48, "2570709463492e": 38, "2575225077989435e": 48, "2577596081448574e": 63, "2577596081450706e": 63, "257943509407669e": 48, "258": 48, "25805960e": 76, "2582316265611039": 81, "25880": 121, "259": 48, "2593": [15, 23], "2595516847568828e": 48, "2599270675412542e": 21, "26": [21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "260": 48, "26051191101564e": 83, "2607019225332258": 38, "26095800e": 76, "261": 48, "261614742574917e": 21, "2616944617779757e": 48, "262": [34, 48], "26215991e": 76, "262409501838687e": 38, "262631841144873e": 48, "263": 48, "2634537302230126e": 83, "2634965543630566": 85, "263818148215092e": 77, "264": 48, "264312876504194e": 83, "26447785e": 76, "2646": 1, "265": 48, "265668479439641e": 48, "2657": [15, 23], "266": 48, "2662179809977506e": 48, "266740360996993e": 48, "267": 48, "267202512577547e": 48, "2672170441103305e": 48, "2676": 82, "268": 48, "268299050241808e": 21, "268337563450049e": 39, "268589647581184e": 48, "268633095463936e": 48, "26887948e": 76, "269": 48, "269058169182027e": 39, "269303654021906e": 21, "269372850183054e": 48, "2693827572253804": 81, "269432811736806e": 38, "2695228725235466e": 48, "27": [21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "270": [1, 48], "27043380e": 76, "27068308e": 76, "2706849461530112e": 76, "271": 48, "2715327847755757e": 48, "27177453839846e": 48, "272": 48, "273": 48, "2732527515891966e": 84, "273342095791625e": 39, "2733861726088263e": 48, "27373870e": 76, "274": 48, "2741783209975205e": 84, "27470834e": 76, "275": 48, "2755319017225014e": 39, "276": 48, "27688891e": 76, "277": [48, 49], "277097975239419e": 48, "2773406359217477e": 48, "277596180054449e": 48, "2778255546402775e": 39, "278": 48, "278453604751753e": 38, "278591821354165": 116, "2787374496223366e": 77, "2787721169956141e": 21, "279": 48, "27921192240865e": 38, "279254503545001e": 48, "2792706014737654e": 48, "2794268561723303e": 38, "2799027620592605e": 38, "28": [6, 21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 110], "280": [48, 76], "280805241684333e": 48, "281": [48, 76], "281289667448394e": 48, "28179168e": 76, "282": [48, 76], "283": [48, 76], "28331106e": 76, "2833346908563155e": 48, "284": 48, "28400582e": 76, "284020343837986e": 21, "2848701103604469e": 48, "285": 48, "28519735e": 76, "285198733015978e": 39, "285261803064561e": 39, "286": [48, 115], "28614435e": 76, "28636674165708e": 77, "286505224127795e": 38, "286552374709478e": 48, "28660014129065e": 48, "287": 48, "2870559541431299e": 48, "287100776090044e": 77, "287219173082758e": 48, "28786796e": 76, "288": 48, "2885730706727105e": 48, "289": 48, "2892459896362076e": 48, "29": [15, 21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "290": 48, "290001584068349e": 91, "290161202956915e": 84, "290414512693373e": 48, "290854582112953e": 83, "291": 48, "2916858366707634e": 48, "292": 48, "29215552e": 76, "2924811429051319e": 39, "2926088093011324e": 38, "293": [15, 48, 73], "2931264312198394e": 110, "294": 48, "2941827144102711": 24, "2948886942405e": 48, "29497965478": 106, "295": [15, 48], "2950048008498856e": 38, "295673124262948e": 84, "29588309046074e": 48, "296": [15, 48], "2968072136285665e": 84, "297": 48, "297011872199007e": 84, "2970915870460925e": 48, "297230564528598e": 48, "297285709626039e": 48, "2975": 108, "297607436855025e": 84, "2977358264383947e": 81, "298": [15, 48], "2981497810886503e": 38, "298152195730669e": 38, "29849936e": 76, "2985082498568915e": 84, "298948379400162e": 48, "299": [15, 48], "29946292210663e": 84, "29948886e": 76, "2_": [53, 65, 79, 102, 103, 112], "2_g": 98, "2_t": 35, "2d": [1, 2, 6, 9, 10, 54, 61, 75, 76, 78, 108], "2e6": 76, "2i": 79, "2k": 79, "2kcg": [14, 21, 38, 39, 63, 77, 83, 84, 91, 110], "2klinearsolv": 84, "2l": [63, 83, 109], "2m": 64, "2me": 64, "2n": [47, 79], "2t": 65, "2ta": 27, "2y": 79, "3": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 33, 34, 35, 38, 39, 43, 44, 47, 48, 55, 58, 62, 63, 65, 66, 70, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 88, 89, 91, 92, 96, 97, 98, 101, 103, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "30": [0, 8, 15, 21, 38, 39, 48, 55, 77, 78, 81, 83, 84, 85, 88, 106, 110], "300": 48, "3004844587339315": 51, "3005077941994067e": 77, "300696790397088e": 48, "30074832e": 76, "301": [15, 48], "3011386494338866e": 48, "30138177e": 76, "301733059235292e": 21, "3017815971438115e": 39, "3019964774111864e": 84, "302": 48, "302264315279244e": 91, "3024": 2, "30269301e": 76, "302714596286466e": 91, "3027690999003073e": 48, "30288929e": 76, "302901864714575e": 48, "303": 48, "303008939522798e": 84, "304": 48, "30408705e": 76, "30491321882643e": 84, "3049853940133655e": 48, "305": [8, 48], "30505582847543e": 48, "3053572711634862": 63, "30571429e": 76, "306": 48, "30621953771071e": 48, "3063314760812088": 81, "3063771523894718": 103, "3064582455185055": 85, "3067389091504954e": 48, "307": 48, "3072059502377703e": 48, "307344765907323e": 48, "307914": 108, "308": 48, "30823297e": 76, "30844242e": 76, "308450122441681": 63, "30858827998629": 63, "30871063e": 76, "309": 48, "309714466727463e": 48, "30x15": 121, "31": [21, 38, 39, 46, 48, 77, 81, 83, 84, 85, 110, 118], "310": 48, "3100766606552662e": 84, "31053506e": 76, "310666361752348e": 84, "3106696194022617e": 39, "3106857175187873e": 84, "311": 48, "311366433940005e": 48, "3114977115398425": 81, "3117181936101093e": 48, "312": 48, "31221": 108, "3129272483674086e": 48, "313": 48, "3130323762459306": 103, "3130351788210097": 103, "3130352795102445": 103, "313035285298448": 103, "31340616e": 76, "3136911569268165e": 110, "314": 48, "31462394e": 76, "3146723828127433e": 48, "315": 48, "3151428740681202e": 48, "3151608293416554e": 48, "315483328987544e": 21, "316": 48, "31624": 55, "3165": 67, "31667701963457284": 63, "3168473544001734e": 91, "3169": 63, "317": 48, "317049926499982e": 48, "3173522661039894e": 48, "31746804233173e": 48, "318": 48, "3182030348089053": 63, "31830950064612856": 103, "318629762870249e": 38, "318747833945106e": 48, "318864756351678e": 21, "319": 48, "3190814707331234e": 48, "31923341e": 76, "319937819401064": 63, "31mout": 23, "32": [15, 21, 38, 39, 48, 70, 77, 81, 83, 84, 85, 103, 108, 110], "320": 48, "3206146501632598e": 77, "3209174368972213e": 48, "321": 48, "321603674500055e": 21, "3219928450516111": 63, "322": [48, 121], "32201279355107e": 48, "32202735e": 76, "322179773720817e": 48, "322459830883026e": 77, "32250970878595636": 84, "3229959045015813e": 48, "323": [48, 121], "323104671078037e": 91, "3231615399420306e": 77, "3231646014039378e": 48, "3232297657475674": 83, "323436571077358e": 39, "323767518440678e": 83, "324": [48, 121], "32474227e": 76, "324899": 108, "325": [48, 121], "3254161040503283e": 48, "32578507e": 76, "326": 48, "326212478264851e": 83, "326469974823354e": 39, "327": [48, 121], "328": 48, "328258117750278e": 48, "329": [48, 79], "3292323296073995e": 48, "32947036e": 76, "3294831066099182e": 83, "329876676381426e": 83, "33": [15, 21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "330": 48, "33003422e": 76, "330822151353692e": 21, "331": 48, "331199046890856e": 48, "3312170911824881e": 48, "332": 48, "33243573e": 76, "3329352056374313e": 77, "333": 48, "33316884e": 76, "3331948524618315e": 48, "33327795e": 76, "3334817867318555e": 48, "333629": 108, "334": 48, "335": 48, "3353005724913072e": 38, "3357651872399246e": 39, "336": 48, "336225652920626e": 21, "33643118654821e": 110, "337": [48, 79], "337032": 108, "3371286004187721e": 77, "337166663552109e": 48, "3375409442559155e": 48, "338": 48, "3387850598294922e": 21, "339": 48, "339109257479704e": 77, "3391246325802083": 70, "3393185901943043e": 48, "3399619828811302e": 85, "34": [15, 21, 38, 39, 48, 65, 76, 77, 81, 83, 84, 85, 105, 108, 110], "340": 48, "34032616e": 76, "34050016e": 76, "340930532910021e": 48, "341": 48, "341206724850267": 63, "341597055866288e": 48, "342": 48, "3420282227507415e": 48, "34223115e": 76, "3426066734105022": 24, "3427726344761957e": 83, "34281243904970204": 116, "343": 48, "343218821995261e": 48, "343368177647046e": 48, "343495008483151e": 39, "3438799364311088e": 48, "344": 48, "345": 48, "34503153e": 76, "345120350599629e": 48, "345286858268262e": 39, "34534586041024e": 115, "345465916709958e": 48, "34576114e": 76, "346": 48, "346108176246629e": 77, "34633": 103, "346737344250734e": 83, "34678613e": 76, "346953929262571e": 48, "347": 48, "34701629e": 76, "3473941636760532e": 48, "348": 48, "34808772e": 76, "34826800e": 76, "348315394185244e": 48, "348678469928673e": 84, "348913389301089e": 48, "349": [15, 48], "3491991701423895e": 77, "34920635e": 76, "34923238484178e": 84, "34927074006952e": 14, "34946": 55, "349635933929682e": 83, "3497617218728434e": 48, "35": [21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 88, 110], "350": 48, "3509906761220354e": 38, "351": [15, 48], "351116172274719e": 48, "3513876287934866e": 48, "352": [15, 48], "352057965747212e": 48, "352206186975885e": 84, "35262210e": 76, "353": 48, "3530": 103, "3538480472957442e": 77, "354": [15, 48], "354153128306859e": 38, "3542094017002498": 81, "3543762860934322e": 83, "3546130827285455e": 110, "35472110553465824": 70, "355": 48, "35572": 108, "3557276175693415e": 48, "355860847201666e": 48, "35591475658727e": 48, "356": 48, "3560043386452667": 63, "35640106452074e": 77, "35674978e": 76, "357": 48, "3578525479156935e": 48, "3579760806659418e": 48, "358": [15, 48], "35802220286281283": 70, "35820310983034076": 70, "35823429355522934": 70, "3582355218606291": 70, "35823552663766817": 70, "3582355266377316": 70, "359": 48, "36": [21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 110], "360": [15, 48], "360246232685851e": 48, "3602862850662532e": 48, "3606471893290851e": 38, "361": 48, "3616798896013715e": 48, "361744949075666e": 83, "362": 48, "36233754e": 76, "362346931360255e": 83, "3626009812903334e": 48, "3628551164479e": 48, "363": 48, "36321061217922673": 24, "363532192858452e": 77, "363564916457091e": 48, "36373064e": 76, "364": [15, 48], "3642450612653385": 51, "365": 48, "365027": 108, "3656976172150075e": 48, "3657345829248144e": 38, "366": 48, "366728196158377e": 48, "366915850720791": 85, "367": 48, "367143553464296e": 110, "3673484810309763e": 48, "367665665217973e": 48, "368": [15, 48], "3680394619761025e": 84, "368440535182689e": 48, "3685646891753328e": 48, "369": 48, "3690705057364501e": 84, "36909436e": 76, "3692871776920687e": 48, "3697248238186124e": 48, "3697656755438388e": 77, "37": [15, 21, 38, 39, 48, 77, 81, 83, 84, 85, 110], "370": 48, "370873187419749": 63, "3708929217814924e": 48, "371": 48, "371148838672236e": 83, "3712": 15, "3715758714675145e": 38, "37161190298016": 83, "372": 48, "37266314e": 76, "373": 48, "373443200688844e": 39, "374": 48, "374538642017021e": 48, "374778284382607e": 48, "375": 48, "3759257345796876e": 38, "37594503814550667": 91, "376": [15, 48], "376294787821936e": 38, "3767551669114815e": 38, "376893472830978e": 48, "377": 48, "3777": 118, "378": [15, 48], "37816023e": 76, "3782375941630555e": 39, "37848562e": 76, "378785299480245e": 48, "378919613280206e": 115, "379": 48, "3792358662036097e": 48, "379271921649497e": 21, "37948302e": 76, "37989344758183696": 63, "38": [15, 21, 38, 39, 48, 66, 76, 77, 81, 83, 84, 85, 110, 118], "380": 48, "380094733222787e": 48, "381": 48, "3815828217680814e": 48, "381900873406225e": 48, "382": 48, "382220957458106e": 48, "383": 48, "3834677872461816e": 77, "3834878735024925e": 21, "383681884834191e": 48, "38371554e": 76, "384": 48, "3840348380620775e": 48, "3841438305254625e": 48, "384548734703711e": 77, "385": 48, "385554547969789e": 70, "386": 48, "38610419e": 76, "386817": 108, "387": 48, "3876295931521244e": 48, "38776040e": 76, "38792594e": 76, "38794309e": 76, "388": 48, "38805825978633e": 48, "38832225e": 76, "389": 48, "389007726660379e": 39, "389791809557051e": 48, "3899902597436885e": 48, "39": [10, 15, 21, 38, 39, 48, 76, 77, 81, 83, 84, 85, 96, 110], "390": 48, "39022": 108, "390974803808109e": 21, "391": 48, "391198049839224e": 39, "392": 48, "3925445641690011e": 77, "392817659702821e": 83, "392841102594967e": 48, "393": 48, "3930402462834775e": 21, "39347185e": 76, "39376057770696e": 38, "3938574627111934e": 77, "394": 48, "39435204e": 76, "39436240052696975": 85, "39444956e": 76, "3945794772264324e": 48, "394595459241093e": 48, "39464": 22, "394917627264948e": 77, "395": 48, "395565888293071e": 48, "396": 48, "3960743990071075e": 48, "396161722131952e": 48, "39617577e": 76, "39634548e": 76, "3965604941420359e": 77, "39661832e": 76, "39682540e": 76, "397": 48, "3974281500533848e": 83, "398": [15, 48], "3980831180153135e": 84, "3984494259649153e": 48, "3986531644012435e": 48, "399": [47, 48], "3d": [1, 2, 6, 8, 10, 33, 34, 45, 46, 54, 61, 62, 75, 76, 77, 91, 108], "3ex": [68, 86], "4": [0, 5, 7, 8, 10, 11, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 32, 35, 38, 39, 40, 47, 48, 55, 58, 63, 65, 66, 67, 70, 75, 76, 77, 79, 80, 81, 83, 84, 85, 88, 91, 103, 104, 105, 106, 108, 110, 111, 114, 115, 119, 121], "40": [15, 21, 38, 39, 44, 48, 76, 77, 81, 83, 84, 85, 110], "400": [14, 15, 23, 48, 77, 84, 110, 121], "40022829e": 76, "40034946e": 76, "40049428e": 76, "400626986584763e": 48, "400764857435027e": 83, "4008290909238202e": 48, "400991083489937e": 83, "401": 48, "401402217647644e": 48, "40140778631308777": 85, "4014811948944164e": 38, "4017278282832695": 51, "401844236565597e": 83, "402": 48, "402546355528151e": 110, "402736540876462e": 48, "403": 48, "4032958225323066e": 77, "4034076037199853e": 21, "4036767277543443e": 85, "4038894412150484e": 14, "404": 48, "40405870e": 76, "404563477407528e": 39, "404629236349358e": 48, "405": 48, "405474055921332e": 48, "406": 48, "4060059508048765e": 83, "4069600824566074e": 48, "407": 48, "4074313271603042e": 38, "407631613365212e": 38, "408": [15, 48], "4086948298572175": 85, "40888996300808e": 48, "409": 48, "4093536347508964e": 48, "409720725027058e": 77, "41": [7, 15, 16, 18, 19, 21, 22, 38, 39, 48, 76, 77, 81, 83, 84, 85, 110], "410": 48, "41001347e": 76, "41019727e": 76, "410421337747203e": 48, "4105955877807516e": 77, "410634905428783e": 77, "41063837e": 76, "411": [15, 48], "4110760181683581e": 83, "411348617119802e": 38, "412": 48, "41280497087718e": 85, "413": 48, "413180318215851e": 85, "413218056147077e": 48, "413250840996505e": 48, "41373102e": 76, "4138806122733448e": 21, "414": 48, "4141783443506158e": 83, "415": 48, "4155225382861335e": 48, "416": 48, "4162233154336305e": 48, "416405828703569e": 48, "416415558992872e": 48, "416445795958666e": 48, "4164879047972151": 63, "417": 48, "41712": 22, "417356345279921e": 48, "417417797477789e": 77, "4179306418471782e": 48, "418": [15, 48], "41806640e": 76, "41888": 22, "419": 48, "41938614e": 76, "41983539667056e": 48, "42": [38, 39, 44, 48, 76, 77, 81, 83, 84, 85, 88, 108, 110], "420": [15, 48], "4201528772553445e": 83, "4203434724225752e": 48, "420344679804673e": 110, "420623649790215": 63, "42063694e": 76, "421": 48, "421373232085135e": 38, "422": [15, 48], "42272101e": 76, "423": [15, 48], "424": 48, "425": 48, "42512937e": 76, "42524628e": 76, "425660340878632e": 48, "426": 48, "4261": 67, "42626166e": 76, "426433": 11, "4265598564692116e": 48, "427": [15, 48], "42789858702521705": 83, "427937357377234e": 48, "428": 48, "428188197729902e": 48, "42879081e": 76, "428927912054942": 51, "4289867528748286e": 48, "429": 48, "42984684e": 76, "43": [15, 38, 39, 48, 76, 77, 81, 83, 84, 85, 110], "430": 48, "43052514e": 76, "43069641e": 76, "430869694970864e": 21, "431": [15, 48], "43101": 85, "431176412435023e": 48, "432": [15, 48], "4320678016306087e": 48, "4329112850224e": 48, "433": [15, 48], "433285162208582e": 85, "4333454712960677e": 83, "433399195958605e": 38, "433511536023269e": 48, "43369229e": 76, "434": [15, 48], "434050375655718e": 83, "43405785e": 76, "43407504e": 76, "434177851031543e": 39, "434918283921399e": 48, "435": [15, 48], "4350095376417224": 76, "4352415865436047e": 48, "43566373e": 76, "4357214966138604e": 48, "435860602860818e": 48, "4359368486597825e": 77, "436": 48, "436024997236365e": 38, "43606881e": 76, "4362081331413285e": 48, "437": 48, "4370135861158414e": 48, "437213226838117e": 81, "43752689e": 76, "43762385593524e": 83, "437683242447562e": 48, "438": [15, 48], "438531508912747e": 83, "439": 48, "4397588432028707e": 21, "439845270680144e": 83, "44": [38, 39, 48, 76, 77, 83, 84, 85, 109, 110], "440": 48, "4401297012520998e": 48, "440247699033299e": 77, "4406636395646856e": 21, "440755426206628e": 83, "441": 48, "4419425838448128": 85, "442": 48, "4428682911205534e": 48, "44287657201762e": 84, "44292495e": 76, "443": 48, "443176683566252e": 77, "443267856418581e": 48, "4438616067029353e": 83, "4439636276027798e": 84, "444": 48, "444456354642378e": 48, "44487840792874156": 22, "445": 48, "4456256298405192e": 106, "445632525789044e": 83, "445665566833718e": 48, "44579395591044385": 85, "446": 48, "4464326683285988e": 48, "446668911117099e": 77, "4468688322695497e": 48, "447": 48, "447257507807357e": 48, "448": 48, "4488933718100165e": 48, "448900761214395": 48, "449": 48, "449409064134838e": 48, "4499449472635008e": 115, "45": [0, 15, 38, 39, 48, 55, 76, 77, 83, 84, 85, 108, 110], "450": 48, "45079365e": 76, "450846077211376e": 38, "451": 48, "4510315297654936e": 48, "451059188051417e": 84, "451197927296262": 51, "451200504182154e": 84, "452": 48, "45204653576357e": 48, "452057792644223e": 48, "4521077551150774": 81, "452500262063084e": 110, "45261752e": 76, "453": 48, "453639114498798e": 80, "454": 48, "455": 48, "4552353471039174e": 48, "4553367053311626e": 83, "455420448937443e": 48, "45569001e": 76, "456": 48, "45688875e": 76, "457": 48, "45725397386741407": 63, "4577110083756291e": 48, "45788548e": 76, "457930495020287e": 48, "458": 48, "45816144e": 76, "4582431551015187e": 84, "458332278094533e": 48, "4586797121211655e": 48, "459": 48, "45942323241555755": 85, "46": [15, 38, 39, 48, 76, 77, 83, 84, 85, 110], "460": 48, "4601915644276485e": 48, "460411723147762e": 21, "460965197737416e": 21, "461": 48, "46135753154810366": 63, "46155": 11, "461725575996149e": 48, "461759935102699e": 48, "46177993e": 76, "462": 48, "4620885627225223e": 83, "46255587e": 76, "462716319985976e": 48, "46279346e": 76, "4628668141158896": 85, "463": 48, "4635327629266928": 83, "463813301050634e": 48, "4639916534842686e": 39, "464": 48, "46499": 108, "465": 48, "46500883e": 76, "4650109967698113e": 91, "46507838e": 76, "46524770740059607": 85, "4653646485897827e": 21, "465366340985294e": 84, "4659479241412505e": 48, "466": 48, "46644119e": 76, "4665823171826354e": 48, "466601479835905": 63, "467": 48, "46723283e": 76, "46730851e": 76, "467336896710443": 85, "46795278e": 76, "467966534081841e": 48, "468": 48, "46811512e": 76, "468997807675418e": 63, "46899780767589e": 63, "469": 48, "4690773015519533e": 48, "469188331783693e": 84, "4695364680635591": 85, "47": [15, 38, 39, 48, 76, 77, 83, 84, 85, 110], "470": 48, "47010385e": 76, "470109807340424e": 38, "4701460070543832e": 48, "4703083710549495e": 48, "4707544746314215e": 84, "4709697187344926": 51, "471": 48, "471088928931207e": 48, "4712663856765383e": 85, "471856140934451e": 21, "472": 48, "472437150273745e": 38, "472796970933967e": 38, "473": 48, "4731343120998787e": 39, "473282009251378e": 39, "4733739358435825e": 84, "4734783166392862": 85, "474": 48, "4743845285220006e": 84, "475": 48, "4755076200566074e": 48, "47570024488714774": 85, "476": 48, "47646191e": 76, "4764670997617955e": 84, "47657413e": 76, "4767521894186756e": 84, "47682560e": 76, "477": 48, "47711133461441e": 77, "477162405364844e": 48, "477239892603522e": 83, "477359372021876e": 77, "477929338164458e": 48, "478": 48, "4780177726832662e": 48, "47830128e": 76, "478486166301863e": 76, "478769914585975": 96, "4789746989912544e": 48, "47899861e": 76, "479": 48, "48": [15, 38, 39, 48, 76, 77, 83, 84, 85, 110], "480": 48, "4801364239010767e": 83, "48026033e": 76, "4805329017218728e": 48, "4807078864792983e": 77, "480955141420751e": 48, "481": 48, "4814120552195498": 63, "48148537704207556": 85, "48165354867617e": 38, "482": 48, "482969009887021e": 77, "483": 48, "48304492e": 76, "48344467893236e": 77, "48356595e": 76, "48361914822910884": 85, "483675900724828e": 48, "484": 48, "484232104216341e": 48, "484543077553714e": 83, "4846707043150605e": 38, "484893189401684e": 48, "485": 48, "485175829259252e": 48, "4856977656683026e": 83, "485707407198032e": 21, "486": 48, "487": 48, "4870126649549754e": 48, "488": [48, 108], "48834566896636394": 85, "489": [48, 80], "4892925374272817": 85, "4893134619348585e": 38, "489456390475471": 51, "4895424052178241e": 48, "4895926840424588e": 38, "489842622889638e": 39, "489875390949823e": 48, "49": [15, 38, 39, 48, 77, 83, 84, 85, 110], "490": 48, "4903482436642743e": 38, "490762189941222e": 110, "490829651617686e": 48, "490978217279541e": 48, "491": 48, "4914425821440525e": 21, "491900037077236e": 48, "492": 48, "492327674466352e": 83, "493": 48, "493285147546725e": 48, "493611611120771e": 21, "494": 48, "494419632350608e": 63, "494419632355397e": 63, "494706044701668e": 48, "4947798856049585": 85, "495": 48, "49537": 63, "49543876866909387": 85, "496": 48, "4960536837939773e": 48, "496062301292538e": 48, "4967019608058028": 106, "497": 48, "49706348e": 76, "497513684323499e": 48, "49754848815741554": 85, "4976441070032594e": 77, "498": 48, "498607421968768e": 48, "498665639019465e": 48, "499": 48, "499121459722843e": 38, "49944425e": 76, "49950661134314e": 39, "4996245073469755e": 21, "4999999999999974": 76, "499999999999999": 76, "5": [0, 2, 3, 4, 5, 6, 7, 8, 12, 14, 16, 18, 19, 21, 22, 33, 34, 35, 37, 38, 39, 43, 44, 47, 48, 55, 58, 62, 63, 70, 74, 76, 77, 78, 79, 80, 81, 83, 84, 85, 91, 92, 96, 97, 103, 108, 110, 112, 113, 114, 115, 116, 118, 119, 121], "50": [4, 15, 38, 39, 43, 47, 48, 76, 77, 78, 83, 84, 85, 108, 110, 121], "500": [22, 38, 39, 47, 48, 67, 69], "50000": 121, "500116704668778e": 38, "500146983254948e": 48, "5001615510129478": 85, "50016337e": 76, "500300016615044e": 48, "501": [48, 103], "5011576064000944e": 48, "501242952996463e": 48, "5017360038189035": 85, "502": 48, "5025194394231773": 85, "503": 48, "5035529105360154e": 21, "50358631e": 76, "5036485771930268": 85, "5036598509313203e": 77, "503722035383636e": 48, "504": 48, "5042919233686429e": 83, "50442051e": 76, "504556658304414e": 48, "505": 48, "505165174500594e": 48, "505801129603892e": 48, "506": 48, "506101484240894e": 48, "5061468121661231": 85, "5061729904018873": 85, "5061795188774857e": 84, "506336456463105e": 39, "506645750744541e": 83, "50682327e": 76, "506908218354334e": 77, "507": 48, "507026363501187e": 77, "5072557909082451": 51, "5073268053984769": 85, "50759366e": 76, "5077276185552118e": 48, "508": 48, "50835": 121, "5087570130423114e": 38, "509": 48, "509437297093308e": 48, "51": [15, 38, 39, 48, 77, 83, 84, 85, 110], "510": 48, "510174891172396e": 83, "5102925997151507e": 48, "510630588292769e": 48, "5107434241281112": 85, "511": 48, "5116661890005876e": 77, "5116759654916236e": 77, "512": 48, "5120662430136316e": 48, "512365097809827e": 38, "512762255199913e": 83, "513": 48, "5130": 1, "513125186992284e": 48, "514": 48, "5142198151406168e": 39, "51484360347095e": 77, "5149410265326552e": 77, "514952075563536e": 38, "515": 48, "5155162937637426e": 48, "515875985552747e": 83, "516": 48, "516488531819619e": 48, "5169033203791969e": 48, "517": 48, "517507837203933e": 38, "5175925277867024": 85, "517814835797797": 85, "518": 48, "518172758996038e": 81, "518845029685": 103, "519": 48, "5194838971425951e": 48, "519641137784833e": 77, "519795778795417e": 48, "519927604494129e": 83, "52": [15, 38, 39, 48, 77, 83, 84, 85, 110], "520": 48, "52020762e": 76, "52031126e": 76, "5203140321780024": 85, "5205257369901945e": 85, "5205385000271846": 85, "521": 48, "521850746831488e": 85, "522": 48, "5220696259625265e": 48, "5221995110452216e": 83, "5225189901845548": 39, "5225660928343492": 63, "523": 48, "523134156230471e": 48, "523391567900036e": 85, "524": 48, "52444753e": 76, "5246072026789994": 51, "5248845689998295e": 48, "525": 48, "525126310870192": 85, "52514963956198e": 77, "525322494411694e": 48, "5253742720462283": 85, "526": 48, "52636441e": 76, "527": 48, "5274388649262574e": 48, "52761847e": 76, "528": 48, "5284325663857127e": 38, "5287311423528937e": 48, "528784983278807e": 84, "529": 48, "52915266e": 76, "529219410275793e": 83, "5293620323637416e": 110, "53": [15, 38, 39, 48, 77, 83, 84, 85, 110], "530": 48, "5308251803984455e": 48, "531": 48, "531331856542912e": 48, "53171058e": 76, "532": 48, "532582373313546e": 48, "533": 48, "5330312963314107e": 83, "533439783514639e": 48, "5336307237815367e": 38, "533828877556726e": 39, "534": 48, "534231389431823e": 48, "534722304045493e": 48, "535": 48, "535091252152984e": 21, "535130653762732e": 48, "5355291443173377": 85, "5358648222961435": 85, "536": 48, "537": 48, "537520206911699e": 38, "538": 48, "5380346646190164e": 48, "5387047082349326e": 14, "539": 48, "539446673074001e": 48, "5394591497527506e": 48, "53986083e": 76, "54": [15, 38, 39, 48, 77, 83, 84, 85, 110], "540": 48, "5401318134887053": 85, "5402170320713728e": 21, "5403099220737555e": 48, "5404009954080077e": 39, "5406511938359275e": 48, "541": 48, "541248535042474": 66, "5414706478167903": 109, "5415208146951107": 51, "5419012858384775e": 38, "542": 48, "542308870311771e": 110, "54285092e": 76, "54285512e": 76, "543": 48, "5432729629011974e": 48, "543414380099021e": 38, "544": 48, "544138507039576e": 48, "544914608807454e": 21, "545": 48, "545348237201027e": 48, "545934144326212": 70, "546": 48, "5466647999174853e": 14, "547": 48, "548": 48, "54836020e": 76, "5483623051359466e": 77, "548906053040536e": 48, "549": 48, "54910123e": 76, "5491577368906567": 85, "54938804180488e": 48, "5494201485235853e": 77, "5497371221971641": 63, "55": [15, 38, 39, 48, 77, 83, 84, 85, 110], "550": 48, "5500272589058757e": 48, "550103983705584e": 77, "55036412e": 76, "550386658345834e": 48, "5505589020810506e": 48, "5505614653778326e": 83, "551": 48, "551059202512336e": 48, "5515751696702515e": 39, "551679871817023e": 21, "552": 48, "5524220444695906e": 48, "5526642024859626e": 48, "553": 48, "5530923154043247e": 39, "5533557874560733": 70, "553474254749655e": 84, "554": 48, "554507451136723e": 110, "554898004471734e": 48, "5549435215560785e": 48, "555": 48, "5558353127250373e": 84, "556": 48, "55632805e": 76, "556456083826408": 81, "556900320854911e": 21, "557": 48, "557134991353894e": 39, "5578609944298918": 51, "558": 48, "5582836045941635e": 84, "559": 48, "559079216644538": 106, "55921383e": 76, "5594603828450653e": 48, "55951239e": 76, "5598812394438284": 70, "56": [15, 38, 39, 48, 76, 77, 83, 84, 85, 108, 110], "560": 48, "560166670756158e": 48, "5602257500879521": 70, "560256843307439": 70, "5602573408039925": 70, "5602573409856751": 70, "56080516e": 76, "56081511e": 76, "5608872971611618e": 83, "560991376723551e": 48, "561": 48, "562": 48, "5621133631968823e": 48, "562143501253274e": 83, "562231195951519e": 48, "562283477553": 103, "56260820e": 76, "562726802531528e": 81, "5628463859901822": 51, "5629754213375459": 85, "563": [48, 118], "5632647714169753e": 21, "563856849971057e": 21, "564": 48, "564004473485101e": 110, "56415614e": 76, "56429751911025e": 14, "564771675466398e": 48, "56493273089252e": 48, "56496750e": 76, "565": 48, "565176484800696e": 38, "566": 48, "56608104233486e": 48, "5661479443423707e": 21, "56649872e": 76, "567": 48, "567307643680844e": 48, "5676517902103517e": 77, "567924758563753e": 48, "568": 48, "568405424836091e": 48, "568521269026107e": 80, "569": 48, "56906916257152e": 39, "56969874e": 76, "57": [15, 38, 39, 48, 77, 83, 84, 85, 110], "570": 48, "57030863e": 76, "5704465018664383e": 48, "571": 48, "5716200429890295e": 48, "571878152242305": 81, "572": 48, "573": 48, "5734139487420609": 51, "573761126328625e": 77, "573997293731076e": 48, "574": 48, "5742754062254667e": 81, "5742937204616135e": 48, "57468": 85, "574788922438835e": 85, "5748227292996332e": 48, "575": 48, "575167343788693e": 48, "575570123894986e": 38, "576": 48, "5765781335362466e": 48, "576679382931486e": 77, "5767269529447143e": 39, "577": 48, "57756990e": 76, "57787847517935e": 48, "578": 48, "578665958559203": 81, "579": 48, "5794": 108, "579604898714938e": 48, "58": [15, 38, 39, 48, 77, 83, 84, 85, 110], "580": 48, "58003557e": 76, "5802375801172456e": 38, "5808008395922967e": 77, "58097206e": 76, "581": [48, 65, 105], "5811845708510086e": 48, "582": 48, "58213525e": 76, "582460201324559e": 38, "582662659968393e": 48, "583": 48, "583071243121153e": 21, "583408203975017e": 84, "58369941e": 76, "583722552709536e": 48, "5838745132739276e": 48, "58390741e": 76, "584": 48, "585": 48, "585181086951244e": 48, "5856683186987715e": 84, "586": 48, "5860896744937203e": 48, "5865698784456028e": 48, "586717823012357e": 39, "587": 48, "5871503487735405e": 39, "58730159e": 76, "587397638255004e": 48, "58768986e": 76, "587918844820352e": 48, "587947333438726": 51, "588": 48, "5887660933918107e": 48, "589": 48, "5892049430232538": 63, "59": [15, 38, 39, 48, 77, 83, 84, 85, 110], "590": 48, "5904892246529526e": 48, "591": 48, "5917993944175503e": 77, "591872379395178e": 84, "592": 48, "592476359799728e": 110, "59289558e": 76, "593": 48, "593513625303192e": 48, "5937316187541024": 85, "594": 48, "5941864085865704e": 85, "595": 48, "595331929789697e": 48, "59571120e": 76, "5957238886105585e": 48, "596": 48, "596224550241645e": 48, "596431939936081e": 48, "597": 48, "5975760598785288e": 77, "598": 48, "598344404734277e": 48, "598740783185647e": 85, "598901": 108, "599": 48, "5e4": 74, "5em": 67, "6": [6, 14, 20, 21, 23, 34, 35, 38, 39, 40, 44, 47, 48, 58, 63, 70, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 88, 91, 103, 108, 110, 111, 112, 115, 118, 119], "60": [15, 38, 39, 48, 77, 80, 83, 84, 85, 110], "600": 48, "600238226564242e": 84, "601": 48, "6012589050513686": 51, "601489072773148e": 85, "601828051333945e": 48, "602": [48, 51], "60229352735577e": 48, "602422442088333e": 48, "60282600161908": 84, "603": 48, "6032113977710588e": 48, "603559520622855e": 48, "60372397e": 76, "603811278315021e": 83, "604": 48, "60426442e": 76, "6044659570055255e": 48, "605": 48, "6056347314808085e": 83, "605938808689276e": 48, "606": 48, "6060002408897057e": 83, "60625432046131e": 48, "6063431809725343e": 77, "606634889215221e": 84, "607": 48, "608": 48, "608253257484617e": 48, "608585554903282e": 38, "608662509689858e": 63, "608662509690732e": 63, "608671743947409e": 48, "6087661839121814e": 83, "609": 48, "609184832867373e": 48, "61": [15, 38, 39, 48, 63, 76, 77, 83, 84, 85, 110], "610": 48, "6103899203810424e": 77, "6105980661982686e": 48, "6106915979525028e": 48, "611": 48, "6118242125742767e": 48, "612": 48, "612651755831167e": 115, "613": [48, 65, 105], "613209024712011": 51, "61373488e": 76, "614": 48, "614005175441214e": 48, "615": 48, "6157121995544838e": 48, "6158399286683607e": 48, "616": 48, "617": 48, "617057007632026e": 48, "618": [48, 84], "6181368587478409": 81, "6184608892926782e": 48, "6186157638062868e": 77, "619": 48, "619150701221642e": 84, "62": [15, 38, 39, 48, 76, 77, 83, 84, 85, 110], "620": 48, "6209503608706": 106, "621": 48, "621197626469506e": 48, "621371600445931e": 48, "621944158161234e": 85, "622": 48, "6221154116416602e": 48, "623": 48, "623330289029238e": 77, "623457969303073e": 21, "62349341e": 76, "62363235e": 76, "623692451270183e": 48, "6237346368041081": 51, "62395745e": 76, "624": 48, "625": 48, "625231949488012e": 38, "625250580230092e": 21, "625257042884367e": 48, "625532145380201e": 48, "6255450647095954e": 48, "62576805756395e": 77, "625844366509997e": 38, "626": 48, "62624400e": 76, "626339847741722e": 39, "6264019576717673e": 48, "626576251343839e": 48, "627": 48, "627318460818314e": 39, "628": 48, "6283104727068234e": 48, "629": 48, "6298437650060975e": 77, "63": [38, 39, 48, 77, 83, 84, 85, 110], "630": 48, "63003331e": 76, "63058946e": 76, "631": 48, "6310815021997474e": 48, "631558620476784e": 48, "631718022580567e": 48, "632": 48, "632571239832806e": 48, "632783294297951e": 76, "632846787976177": 51, "633": 48, "633510002398147e": 83, "634": 48, "63404990e": 76, "63421433e": 76, "63463979e": 76, "6349639615833913e": 63, "6349639615837404e": 63, "63499185e": 76, "635": 48, "6351941581130006e": 110, "6357941765358154e": 77, "635817941556107e": 83, "636": 48, "63605128e": 76, "63609902e": 76, "6361360075956317e": 84, "6365287726637194e": 48, "636716541085751e": 84, "637": 48, "638": 48, "638073035428225e": 48, "6382200125324833e": 48, "638759920077451e": 48, "639": 48, "6397979567617955": 81, "63998469e": 76, "64": [38, 39, 48, 77, 83, 84, 85, 103, 108, 110], "640": 48, "6406169867162068": 51, "641": 48, "6410069935234305e": 48, "64119171e": 76, "6417376943526984e": 83, "641909355189532e": 48, "642": 48, "642007949458749e": 38, "642173216119092e": 83, "642257486881171": 85, "6425609626973794e": 48, "643": 48, "644": 48, "6441": 82, "6442510876374e": 14, "64428432e": 76, "645": 48, "645976545818384e": 48, "646": 48, "64691905e": 76, "647": 48, "6470601443285914e": 48, "6471585575973842": 51, "647555890574575e": 85, "648": 48, "6480462116": 103, "6481898622687437e": 48, "648411830187808e": 91, "648429": 11, "648471378408556e": 48, "6487932956182e": 91, "64890769583779e": 77, "64893941e": 76, "649": 48, "6491627752406698e": 38, "649248757945668e": 21, "6495382905210624e": 63, "6495382905217427e": 63, "64960": 120, "649936644773816e": 83, "64996884e": 76, "65": [38, 39, 48, 77, 83, 84, 85, 108, 110], "650": 48, "65063196e": 76, "65080144e": 76, "6508451302837636": 85, "6508768563981184e": 81, "650993786938887e": 48, "651": [34, 48], "6518312303347455e": 48, "65192015e": 76, "652": 48, "65287341e": 76, "653": [48, 103], "65359773e": 76, "6538034423658578e": 48, "653829414967276e": 110, "654": 48, "6542": 55, "654555424792147e": 48, "6546782690498294e": 48, "655": 48, "655442290755333e": 83, "655581574754693e": 48, "656": 48, "65603819e": 76, "657": 48, "658": 48, "6581012527089735e": 38, "6586430081313818e": 48, "659": 48, "65974500215008e": 48, "65982942e": 76, "66": [38, 39, 48, 77, 83, 84, 85, 110], "660": 48, "660895805916253e": 48, "661": 48, "661041373862165e": 48, "6612253938470774e": 38, "6612431188635448": 85, "6614054364313376e": 85, "662": 48, "662800535624399e": 76, "6628845003728513e": 39, "663": 48, "6631659914077622e": 48, "66339889e": 76, "663515736510115e": 83, "6638671781442963e": 48, "664": 48, "665": 48, "665234128197446e": 48, "66524538e": 76, "665576913766078e": 48, "665986": 11, "666": 48, "66666667e": 76, "667": 48, "66713703e": 76, "667577486274225e": 48, "6676902403680316e": 48, "668": 48, "66811773135793e": 81, "6687818393381375e": 39, "669": 48, "669498526513673e": 48, "67": [38, 39, 48, 67, 77, 83, 84, 85, 110], "670": 48, "6702589191564442e": 83, "671": 48, "6711501048505505e": 48, "67167662e": 76, "67198008e": 76, "672": [34, 48], "67211217e": 76, "6729157958379345e": 77, "673": 48, "6739930930784257e": 48, "674": 48, "6741148658528377e": 38, "6741498611348335e": 77, "67482292575374e": 48, "675": 48, "675150622392839e": 84, "6757907507089311": 47, "676": [15, 48], "676841913354895e": 48, "676916499495534e": 48, "6769198434232496e": 48, "677": [48, 50], "677150772435917e": 83, "67777778e": 76, "678": 48, "679": 48, "67903582e": 76, "6792916466594385": 85, "679373377825878e": 48, "67975607e": 76, "68": [38, 39, 48, 77, 83, 84, 85, 106, 110], "680": 48, "680141277151451e": 48, "68033669500627e": 48, "680627093063401e": 83, "68063272e": 76, "680695703563715e": 48, "681": 48, "681809447713418e": 84, "68199683202876e": 21, "682": 48, "68249759e": 76, "683": 48, "6831414237428172e": 38, "6831750753998855e": 48, "68347587e": 76, "683545153190316e": 21, "683864703529036e": 21, "6839353293700188e": 48, "684": 48, "68410527e": 76, "684111020480524e": 48, "684180647878445e": 48, "684247238336984e": 48, "685": 48, "6852922515902963e": 39, "6858896178966197e": 77, "686": 48, "6868": 108, "687": 48, "6870458184868213e": 48, "68751578e": 76, "6878351817818138e": 77, "68795971e": 76, "687990675892528e": 39, "688": 48, "688103183630812e": 48, "688325338782177e": 48, "6886868114230752e": 39, "689": 48, "68927196e": 76, "689323984540329e": 77, "689450198832634e": 48, "69": [38, 39, 48, 77, 83, 84, 85, 110], "690": 48, "690000195677042e": 48, "6904904901851865e": 39, "6906601592346228e": 84, "6907017058419193": 83, "69071121203703e": 21, "690793878626996e": 38, "691": 48, "6916369308059416": 63, "6917198260952784": 106, "6917198310276222": 106, "69173": 106, "691753672693137": 48, "692": [15, 48], "6924136155685763": 84, "69243": 108, "692753635219975e": 39, "692782966558739": 109, "693": 48, "694": 48, "6944302008399361e": 48, "695": 48, "6956794549358718e": 48, "696": 48, "696078939451168e": 48, "69680035e": 76, "6969039222829026e": 84, "697": 48, "6972458570229063e": 77, "6973127929579093e": 48, "697808726680315e": 39, "697838018857469e": 38, "69786564e": 76, "698": 48, "699": 48, "699126451692341e": 48, "69926933e": 76, "699296772001884e": 48, "699680596318089e": 48, "699684507733734e": 77, "7": [14, 21, 35, 38, 39, 46, 48, 63, 70, 76, 77, 78, 80, 81, 83, 84, 85, 89, 91, 103, 108, 110, 115, 118, 119, 120], "70": [38, 39, 48, 77, 83, 84, 85, 110], "700": 48, "7002013246680412e": 48, "7002654479423812e": 48, "700585818353643e": 48, "70080819e": 76, "701": 48, "70113937456317e": 48, "7017039819867693e": 83, "702": [15, 48], "7024528929213845e": 63, "7024528929217936e": 63, "7029432662800023e": 21, "703": 48, "703743410155175": 84, "7037791513458759": 81, "70398951e": 76, "704": [15, 48], "70420964e": 76, "705": 48, "705147728680346e": 83, "705590068837061e": 48, "706": 48, "706142233350342e": 48, "707": 48, "70764226689353e": 48, "708": 48, "70825255e": 76, "709": 48, "70940954e": 76, "71": [7, 38, 39, 48, 77, 83, 84, 85, 110], "710": 48, "71013089e": 76, "710547350811855e": 48, "71076796e": 76, "711": 48, "7110717192708665e": 77, "7111": 108, "7112644172491408e": 39, "7113164141127007e": 84, "71189422260038e": 48, "712": 48, "712084773076043e": 48, "712122195637226e": 21, "713": 48, "7131383862528984e": 110, "714": 48, "714008685346325e": 39, "71424066226727e": 48, "71428571e": 76, "714443543707031e": 85, "715": 48, "7152486742642178e": 85, "7156173727118508e": 21, "716": 48, "7165": 120, "7166280513160016": 85, "7166339448727095e": 48, "7166986112072625e": 48, "717": 48, "7171682211344884e": 77, "717233033971179e": 48, "71729962e": 76, "717550736731428e": 48, "7175647832785993": 85, "7178089277347426e": 84, "717901707233367e": 84, "718": 48, "7180346107522614e": 48, "7186175188828105e": 83, "7187250629149218": 110, "719": 48, "71933788e": 76, "719342925671642e": 21, "72": [38, 39, 48, 77, 83, 84, 85, 110], "720": 48, "720041089537788e": 21, "720277923249615e": 38, "7209573497964506e": 48, "720980706824187e": 84, "721": [15, 48], "7213242109925206e": 48, "7213460179708344e": 38, "722": [15, 48], "72201759e": 76, "72225926431386e": 48, "7228777450440256e": 77, "723": 48, "72312767e": 76, "723697328914656e": 88, "72386981e": 76, "723886147227309e": 48, "724": [15, 48], "724302716978304e": 77, "724587470723463e": 76, "724628290895526e": 48, "7246579670980434e": 48, "724988999951798": 108, "725": 48, "72535447e": 76, "725869823910824e": 21, "726": 48, "727": 48, "727365380912911e": 48, "72752866e": 76, "728": 48, "728141472198836e": 48, "7284332019465428e": 21, "7287649730899981": 85, "728766308288062e": 77, "728979506876731e": 48, "729": 48, "72977177e": 76, "73": [38, 39, 48, 67, 77, 83, 84, 85, 110], "730": 48, "731": [48, 108], "7312763680539798e": 77, "73128488e": 76, "7314307180782044e": 48, "732": [15, 48], "732708184104283e": 77, "7327143565346456e": 48, "733": [15, 48], "733231842002208e": 48, "734": [15, 48], "734367123883148e": 48, "7343762731976364e": 48, "7344838517945644e": 48, "734723475976807e": 76, "734731997623074e": 110, "73491052e": 76, "735": 48, "736": 48, "737": 48, "737097636041496e": 48, "73742": 108, "7378817215805044e": 48, "738": 48, "73825402e": 76, "738830450395595e": 39, "738961003585408e": 77, "739": 48, "739208989947937": 96, "739227318013271e": 21, "73992768129038e": 84, "74": [38, 39, 48, 77, 83, 84, 85, 110], "740": 48, "740471030598783e": 84, "740608113448077e": 83, "7408466320005056e": 48, "741": 48, "7419678304298498e": 48, "742": 48, "742066724764176e": 84, "74252409e": 76, "7427297251387739e": 81, "743": 48, "743046389129484e": 48, "743140248349613e": 77, "744": 48, "744154957255584e": 48, "744737564650945e": 91, "7449312891786651e": 48, "745": 48, "745338379740247e": 48, "746": 48, "746744809618211e": 39, "746988905891274e": 77, "747": 48, "747900914242783e": 48, "748": 48, "749": 48, "749865690221446e": 48, "75": [4, 5, 38, 39, 48, 77, 83, 84, 85, 110], "750": 48, "75053646e": 76, "7508301157973897e": 48, "751": 48, "751962927012061e": 48, "75197995777629e": 39, "752": 48, "7521849481247322e": 38, "75245664e": 76, "752780375295666e": 39, "753": 48, "75317153e": 76, "753411194334228e": 48, "754": 48, "7545438069154676e": 48, "75456096e": 76, "75494085e": 76, "755": 48, "755550557607619e": 48, "756": 48, "756363093945918e": 39, "757": 48, "75704791e": 76, "7572110993411804e": 48, "7572606786524734": 63, "7576109259346328e": 83, "758": 48, "75810": 22, "7584322804740845e": 48, "7584634540051176e": 39, "7585": [15, 23], "7585371463963765e": 48, "759": 48, "75906396e": 76, "7592339454418208e": 48, "759305": 108, "7594147515889492e": 84, "759467113806551e": 48, "759621420205265e": 48, "759857607305391e": 21, "76": [38, 39, 48, 77, 83, 84, 85, 110], "760": 48, "761": 48, "761187835174138e": 48, "761498064357539e": 48, "76156412171864e": 84, "762": 48, "762472029308668": 106, "7626795389173299": 85, "763": 48, "763102225221426e": 91, "763145703838146e": 84, "7636030771172796e": 48, "764": 48, "76404167e": 76, "7642322774630083e": 38, "7644792731733917e": 85, "76492876e": 76, "765": 48, "7651544666507633": 63, "76577828e": 76, "766": 48, "766234453974245e": 48, "76652503e": 76, "767": 48, "76707234881702e": 14, "767206937178794": 85, "767520055746677e": 85, "768": 48, "7683333378414936e": 39, "7684222523134668e": 38, "76870051e": 76, "769": 48, "769239199303403e": 48, "769265216173688e": 48, "7695164087369264e": 21, "769929839694957e": 48, "76999840e": 76, "77": [3, 34, 38, 39, 48, 77, 83, 84, 85, 110], "770": 48, "770924855637088e": 21, "771": 48, "7710075176906277e": 83, "7713073556456228e": 48, "772": 48, "7722502220772948e": 48, "77256": 108, "772794989088839e": 38, "773": 48, "7731": 15, "77344724236405e": 84, "77372": 83, "774": 48, "774391311803948e": 48, "775": 48, "775007175940237e": 77, "77504466331908e": 85, "7753694641455832": 81, "775540546546009e": 38, "776": 48, "7760220119243286e": 48, "776226828362013e": 48, "7766245457196026e": 39, "777": 48, "777798497624973e": 77, "778": 48, "77881017e": 76, "7789183584086092e": 21, "779": 48, "779080482626883e": 48, "7794": 108, "779751500223053e": 77, "77984470e": 76, "78": [38, 39, 48, 77, 83, 84, 85, 110], "780": 48, "780006401287035e": 48, "780076662289142e": 48, "7804": 103, "7805639122786753e": 38, "781": [15, 48], "781165681543078e": 81, "78117344e": 76, "78149891668565e": 48, "78173444e": 76, "782": [15, 48], "782339415582198e": 48, "782735872932991e": 39, "783": 48, "7830345946531105e": 48, "784": 48, "784093837858262e": 14, "7843641797264262e": 83, "784415947420096e": 84, "785": [15, 48], "785161810542102e": 38, "786": [15, 48], "7865073893161035e": 48, "786976191748678e": 38, "787": [15, 48], "787694209726307e": 48, "787969": 63, "788": [15, 48], "7881729236898696e": 48, "788196322974487e": 85, "78832306e": 76, "78842972e": 76, "789": 48, "79": [38, 39, 48, 76, 77, 83, 84, 85, 110, 121], "790": [15, 48], "790269700494604e": 48, "790475207480968e": 48, "7907346494362": 103, "7907524159514695e": 38, "7908391227508154e": 48, "7909557666719956e": 91, "791": 48, "791426428112145e": 84, "7917614043413373": 85, "792": [15, 48], "79291620834115e": 48, "79295905709389e": 48, "793": 48, "793885733041347e": 48, "794": [15, 48], "794336920486616e": 38, "7946": 108, "7948968585386554e": 38, "795": [15, 48], "7956752990680338": 48, "796": [15, 48], "796780788441403e": 84, "7969387311232552e": 48, "797": 48, "79711823e": 76, "7976717886559725e": 48, "7976737044634345e": 110, "7978921512925343e": 39, "798": 48, "7986443153856565e": 48, "799": 48, "799050078373569e": 21, "799562528394244e": 48, "8": [0, 6, 14, 17, 18, 20, 21, 23, 35, 38, 39, 40, 43, 47, 48, 49, 50, 51, 55, 63, 74, 76, 77, 80, 81, 83, 84, 85, 91, 103, 108, 110, 111, 115, 118, 120, 121], "80": [12, 38, 39, 48, 61, 76, 77, 83, 84, 85, 110, 121], "800": 48, "800144212122158e": 21, "80040475e": 76, "801": 48, "801171468526849e": 110, "801601436853941e": 77, "80161548e": 76, "802": 48, "803": 48, "803081364117639e": 48, "80317454445105e": 21, "803815226253984e": 80, "803880010352867e": 39, "804": 48, "804375697464095e": 48, "804802927823209e": 48, "805": 48, "805223866626383e": 48, "806": 48, "806223719927285e": 39, "807": 48, "807312175357524e": 38, "80767289e": 76, "807873306659926e": 48, "8079": 108, "808": 48, "808004503566659e": 39, "808420681895741e": 48, "808562920476457e": 110, "809": 48, "809551240211115e": 48, "809635276715441e": 39, "8097411151552746e": 85, "809913282837525e": 48, "8099867233648296e": 77, "81": [38, 39, 48, 77, 83, 84, 85, 88, 110, 118, 121], "810": 48, "81073187e": 76, "81090441e": 76, "811": 48, "811130133982537e": 48, "811443703268355e": 48, "81158": 55, "812": 48, "813": 48, "81353155e": 76, "8136342454941936e": 77, "8137155683617994": 63, "81378865681444e": 21, "814": 48, "814250201672253e": 48, "814352226978437": 48, "814693618811081e": 48, "815": 48, "815059836395663e": 84, "815400446992752e": 77, "815523139527669": 63, "8157865664294941e": 48, "81598785e": 76, "816": 48, "816032266267512e": 48, "816107452862656": 63, "8163916471489756e": 76, "81662709e": 76, "817": [38, 48, 63], "818": 48, "81873156e": 76, "818875607297657e": 48, "819": 48, "819016339413737e": 48, "819628984093517e": 48, "82": [38, 39, 48, 77, 83, 84, 85, 110, 121], "820": 48, "820233339776046e": 83, "820313116279269e": 48, "821": 48, "821704532877173e": 48, "8219274917452472e": 115, "8219711666983736e": 48, "822": 48, "822718134465209e": 48, "82286681e": 76, "823": 48, "824": 48, "8241752321672265e": 48, "82433803e": 76, "825": 48, "825107658370613e": 84, "825268149129328": 63, "826": 48, "826066219922423e": 22, "827": 48, "8270138008684686e": 48, "82734566e": 76, "8274090136404232e": 110, "82746652e": 76, "82748412619732e": 38, "8277013571751647e": 83, "827828516101396e": 48, "828": 48, "8287315681024927": 76, "829": 48, "829576433200149e": 83, "8299448851004696e": 48, "83": [38, 39, 48, 64, 76, 77, 83, 84, 85, 110, 121], "830": 48, "831": [15, 48], "831487175497996e": 21, "831823159500368e": 48, "832": [15, 48], "83208976e": 76, "8322959488698175e": 39, "83273533446297e": 48, "8329546473066384e": 83, "833": 48, "833058036442482e": 48, "833505": 55, "83371241e": 76, "834": 48, "83425690e": 76, "83432663e": 76, "8347": 55, "8347328893352373e": 21, "834859801621774e": 39, "835": 48, "8350121151719e": 48, "835720635727922e": 48, "836": 48, "8366450976608554e": 48, "837": 48, "837272655973157e": 39, "83757108e": 76, "837711441740575e": 80, "838": 48, "838580733733294e": 83, "838939606568788e": 77, "839": 48, "83915107e": 76, "839255645128162e": 48, "839699790942513e": 48, "839719533898004e": 39, "8399139114264356e": 84, "8399291261764208": 81, "84": [38, 39, 48, 76, 77, 83, 84, 85, 110, 121], "840": 48, "8402817281136264e": 83, "84032156624639e": 110, "841": [15, 48], "8410815328296521e": 48, "841635350994417e": 110, "842": [15, 48], "8420679358690194e": 81, "84206913e": 76, "84225488e": 76, "842446272464718e": 83, "842549500246563e": 38, "843": [15, 48], "843210063311627e": 39, "843722051876385e": 38, "84372631e": 76, "84374591e": 76, "843765347749792e": 48, "844": 48, "8442136050473178e": 48, "844363215072037e": 84, "845": 48, "845707984343893e": 38, "8457974486872674e": 48, "846": 48, "847": [15, 48], "8470072131627334e": 39, "8473523199457097e": 48, "848": 48, "848906170432547e": 39, "848960189067868e": 48, "849": [15, 48], "8490570129782678e": 48, "849634445174112e": 48, "8499": 108, "85": [38, 39, 48, 64, 67, 77, 83, 84, 85, 110], "850": 48, "851": [15, 48], "851545498892002e": 77, "852": [15, 48], "852581180555624e": 48, "852603540368492e": 48, "8529520194379847e": 77, "853": 48, "8531733433027572e": 84, "853903943783089e": 48, "854": 48, "854864477206059e": 39, "855": [15, 48], "8554370838921893e": 48, "856": [15, 48], "8560022007212456e": 21, "856060475024522e": 48, "857": [15, 48], "8570338555821037e": 77, "85716531130414": 109, "857209320503717e": 48, "85722573273506e": 76, "85774805e": 76, "858": 48, "858427395202262e": 39, "85855089e": 76, "8585936041936275e": 48, "858810500619308e": 48, "859": 48, "859291633883496e": 39, "8595864561205e": 48, "86": [38, 39, 48, 77, 83, 84, 85, 88, 110, 121], "860": [15, 48], "8600961405495873e": 38, "8601107411343858": 38, "860373752689044e": 21, "861": 48, "86183": 108, "862": 48, "862620510921282e": 48, "863": 48, "86300492e": 76, "86388258e": 76, "863948847061986e": 77, "864": 48, "864238910083353e": 48, "86452589e": 76, "865": 48, "8652485005035088e": 85, "86533680e": 76, "865492736595519e": 48, "866": 48, "86639575404848e": 48, "8667288714541835e": 48, "8667683822022966e": 77, "867": 48, "8670169881917116e": 38, "86702751e": 76, "8675": 103, "868": [15, 48], "868009654151403e": 38, "8682450158839147e": 83, "868479538952061e": 77, "869": [15, 48], "86914888e": 76, "86919182473513e": 48, "869305704937035e": 48, "869604494897487": 96, "869604494897613": 96, "8699045759210372e": 48, "87": [15, 38, 39, 48, 77, 83, 84, 85, 88, 110, 121], "870": [15, 48], "871": 48, "8712721072740293e": 48, "8718180412834895e": 83, "871898147569218e": 81, "872": 48, "873": [15, 48], "8730870487562395e": 48, "873362458040147e": 83, "8738829146687e": 48, "874": [15, 48], "874839108748753e": 85, "875": 48, "87581228e": 76, "875922769123227e": 77, "875970509314675e": 48, "876": 48, "87607688173": 103, "8761613319769555e": 48, "877": 48, "878": 48, "8783881773434057e": 77, "878469863474089e": 48, "878772652942522e": 48, "879": 48, "8791455960178543e": 83, "879557505654338e": 39, "88": [38, 39, 48, 77, 83, 84, 85, 110, 121], "880": 48, "8800": 83, "881": 48, "8810774163262462e": 38, "881284405121477e": 48, "88135271e": 76, "8813631598073865e": 84, "881460064130038e": 38, "882": 48, "882693116313131e": 48, "8828617717299907e": 21, "883": [15, 48], "883349334841279e": 77, "8834840815121618e": 77, "883740543600398e": 77, "884": [15, 48], "8842911389966815e": 38, "8844848973340027e": 48, "885": 48, "885234043123831e": 48, "886": 48, "886127569791755e": 48, "88629681e": 76, "886769225627127e": 48, "887": 48, "88721028e": 76, "88735676e": 76, "888": [15, 48], "8887460381335765e": 48, "889": [15, 48], "88970269e": 76, "88978359878323e": 38, "89": [15, 38, 39, 48, 67, 77, 83, 84, 85, 110], "890": 48, "891": 48, "89120579e": 76, "892": [15, 48], "8920270278170713e": 39, "8927334941616805e": 48, "892738714831724e": 48, "89298369e": 76, "893": [15, 48], "8932471540524113e": 84, "8936006388080886e": 84, "89366049152692e": 48, "893754159196256e": 84, "893809231662017e": 84, "894": 48, "89402445e": 76, "894284530766281e": 48, "895": [15, 48], "895082990001085e": 48, "895117949331075e": 38, "895246097319718e": 48, "89537770278461e": 38, "895578637246975e": 48, "8959534368152337e": 48, "896": [15, 48], "89614036e": 76, "896183345282972e": 48, "8968751750592811e": 21, "8969487973256553e": 84, "897": [15, 48], "897243155937746e": 83, "89738810e": 76, "897947407783534e": 38, "898": 48, "898245924316241": 48, "89843315e": 76, "89867542e": 76, "89899776e": 76, "899": [15, 48], "8991802786965872e": 48, "8993136308760616e": 83, "8993384818526183e": 83, "8993719203166894e": 48, "8pt": [68, 86], "9": [11, 14, 20, 21, 22, 38, 39, 48, 49, 63, 70, 76, 77, 80, 81, 83, 84, 85, 91, 96, 108, 110, 115, 119, 120], "90": [8, 33, 38, 39, 48, 77, 83, 84, 85, 110], "900": [15, 48], "90065562e": 76, "901": 48, "9014339920530615e": 77, "90154230e": 76, "901698813932705e": 84, "90189805e": 76, "901962432716777e": 84, "902": 48, "902098365795283e": 110, "903": [15, 48], "904": [15, 48], "90411155e": 76, "90491892e": 76, "905": [15, 48], "905310270171566e": 48, "905832846232014e": 38, "906": [48, 62], "906013148755061e": 48, "906326312921551e": 48, "906366472377646e": 84, "907": [15, 48], "9074917935399035e": 48, "908": [15, 48], "9081958235744878e": 76, "908274034770534e": 77, "909": 48, "90956415e": 76, "909777725564041e": 48, "9098392782407787e": 84, "91": [15, 38, 39, 48, 77, 83, 84, 85, 110], "910": [15, 48], "910553350297205e": 84, "910736871305874e": 48, "91093951e": 76, "911": [15, 48], "911054003166856e": 21, "9111102267874765e": 84, "91127060197297e": 48, "911627307675492e": 83, "912": [15, 48], "912312105164127e": 84, "91241542438913e": 48, "912723274012169e": 48, "91277642e": 76, "913": 48, "91358750e": 76, "91395496e": 76, "914": [15, 48], "9148388881307": 106, "91494610e": 76, "915": [15, 48], "916": [15, 48], "916054099105543": 83, "916228055046904e": 48, "916429066888301e": 48, "916509026727359e": 48, "917": 48, "9171532625864503e": 83, "91715363e": 76, "917196533515586e": 48, "917762596575794e": 48, "918": 48, "919": [15, 48], "9191003622777862e": 48, "91997734e": 76, "92": [38, 39, 48, 77, 83, 84, 85, 110], "920": [15, 48], "921": 48, "921050242517416e": 48, "92176436e": 76, "922": 48, "92210142e": 76, "9223651737713088e": 48, "922398248092175e": 77, "922852287859793e": 91, "923": [15, 48], "923091927017895e": 48, "923210728679797e": 38, "9236962476294173e": 38, "924": [15, 48], "924816628679435e": 39, "924873078433326e": 48, "92491430e": 76, "925": [15, 48], "9256370039507276e": 48, "926": [15, 48], "926136120458067e": 83, "927": [15, 48], "927885769755884e": 48, "928": [15, 48], "928987954782195e": 48, "929": 48, "93": [38, 39, 48, 64, 77, 82, 83, 84, 85, 110], "930": [15, 48], "930665437036461e": 48, "93070738e": 76, "931": [15, 48], "931123263847961e": 48, "931468204779986e": 77, "932": [15, 48], "933": [15, 48], "9332724156658535e": 48, "9335995335618345e": 110, "93362996e": 76, "9339708712085968e": 48, "934": [15, 48], "934064265915551e": 48, "935": [15, 48], "935021255196529e": 85, "9357393093205987e": 0, "93598132e": 76, "936": [15, 48], "9361261198607126e": 85, "93682062e": 76, "936851784211181e": 85, "937": [15, 48], "937354550595616e": 48, "937851126352127e": 85, "938": [15, 48], "938071598197655e": 39, "939": [15, 48], "939175519254065e": 38, "93996993e": 76, "94": [10, 38, 39, 48, 77, 83, 84, 85, 110], "940": [15, 48], "940263595804976e": 48, "94066240e": 76, "941": 48, "941213925298137e": 48, "942": 48, "94214740e": 76, "943": [15, 48], "943188092742385e": 48, "94354799e": 76, "944": [15, 48], "94415843236444e": 48, "944919015348104e": 21, "945": 48, "9458345541265103e": 48, "946": 48, "946429851535188e": 48, "946813128879899e": 48, "946966834883386e": 48, "947": 48, "947450640234518e": 83, "947907556144233e": 77, "947936681618856e": 21, "948": 48, "948041627199521e": 48, "949": 48, "949029909160572e": 76, "949144841480855e": 48, "949180345347565e": 84, "94941910837591e": 63, "9494191083769515e": 63, "94990000871906": 108, "94996": 108, "95": [38, 39, 48, 77, 83, 84, 85, 110], "950": 48, "951": 48, "95165835e": 76, "9518262989727527e": 48, "952": 48, "9521726909671235e": 81, "9523838011088034e": 77, "952462288277464e": 48, "9525636053155235e": 83, "953": [15, 48], "953692726428111e": 48, "954": [15, 48], "9544491028129525e": 38, "954524988040401e": 38, "954697536053879e": 38, "95474670424036e": 91, "9548447702941143e": 48, "955": 48, "95521894798872e": 83, "955252997009489e": 84, "955476591094188e": 77, "95574753e": 76, "955800959468116e": 48, "956": 48, "9564189071730394e": 84, "956788505148804e": 84, "9568529356807312e": 48, "956866708243204e": 84, "957": [15, 48], "95764202e": 76, "958": [15, 48], "9583025195272006e": 38, "959": 48, "959344694700465e": 77, "959623023960013e": 77, "9597946922305316e": 39, "959861760711115e": 48, "96": [38, 39, 48, 77, 83, 84, 85, 110], "960": 48, "961": [15, 48], "9610069089538715e": 48, "961192049799402e": 77, "961884348845643": 48, "962": [15, 48], "96251115e": 76, "962748743407555e": 77, "963": 48, "9632743582485296e": 48, "9634170326035583": 63, "964": 48, "9642431866822456e": 48, "964343029696402e": 48, "9647467894946307e": 48, "965": [15, 48], "966": [15, 48], "966100092889627e": 21, "966964255127543e": 48, "967": 48, "967068": 0, "967218775966888e": 48, "967816862422729e": 39, "968": 48, "968077666128608e": 115, "9685811568329946e": 84, "969": [15, 48], "969780154913643e": 83, "9697904651577707e": 48, "97": [38, 39, 48, 77, 83, 84, 85, 110], "970": [15, 48], "970987207121857e": 48, "971": 48, "971758200881336e": 48, "972": 48, "972737960055163e": 48, "9729411628519188e": 48, "973": [15, 48], "973280254961185e": 83, "973354406340976e": 63, "973354406342218e": 63, "9739": 108, "974": [15, 48], "9748427980784953e": 48, "975": 48, "97567491e": 76, "97584791e": 76, "975969": 0, "976": 48, "976297565594483e": 48, "97669399e": 76, "977": [15, 48], "97722": 0, "977370313033323e": 48, "97741566142421e": 39, "977742889310151e": 48, "978": [15, 48], "97870424274299e": 38, "9788107295271953e": 77, "978929725877198e": 48, "979": 48, "97906": 0, "97926710e": 76, "97947213e": 76, "979591970780287e": 77, "9796612659768636e": 48, "98": [38, 39, 48, 77, 83, 84, 85, 110], "980": 48, "9800158109928926e": 38, "981": 48, "982": 48, "982218896602567e": 48, "98236278e": 76, "982454733891398e": 84, "9824564452862945e": 84, "983": 48, "983324464566756": 66, "983673740203993e": 77, "984": 48, "984159946931181e": 77, "984341809190383e": 39, "984965667907077e": 39, "984998868844853e": 48, "985": 48, "98506472e": 76, "9851038052631544e": 38, "986": 48, "987": 48, "98757560193412e": 48, "98786396035351e": 48, "988": 48, "988324878740525e": 48, "989": 48, "989051753270956e": 38, "98985998902484e": 48, "99": [38, 39, 48, 77, 83, 84, 85, 110], "990": 48, "990325906208014e": 48, "990504227339704e": 85, "990897": 0, "991": 48, "99116470e": 76, "99169": 118, "991707474082117e": 48, "992": 48, "992503628103648e": 48, "99252178e": 76, "99272270e": 76, "99280637e": 76, "992946960452109e": 48, "993": 48, "993479432078307e": 48, "994": 48, "995": 48, "995153767474519e": 48, "996": 48, "9961888045847009": 63, "9962490853348933": 63, "996273588831822e": 84, "9967208414315467": 63, "99673786e": 76, "996951398840777": 63, "997": 48, "9970667770354094": 63, "99724319e": 76, "997501320449961e": 48, "99761394e": 76, "998": 48, "998043749532844e": 48, "9982691913414063": 22, "9987999466057931": 63, "999": 48, "999105875654954": 85, "999690227069886e": 21, "9999999999993516": 109, "999999999999967": 55, "9999999999999964": 76, "9999999999999974": 57, "9999999999999993": [81, 96], "A": [0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 47, 48, 49, 50, 51, 52, 53, 54, 56, 60, 61, 62, 63, 64, 65, 69, 72, 73, 75, 76, 78, 79, 81, 83, 84, 85, 91, 99, 101, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115, 118, 121], "As": [1, 12, 13, 38, 64, 89, 90, 105, 113], "At": 40, "But": [3, 6, 24, 31, 33, 34, 35, 47, 54, 58, 76, 79, 86, 100, 101, 104, 105, 106], "By": [8, 31, 32, 37, 47, 50, 51, 53, 54, 63, 65, 76, 79, 83, 84, 100, 101, 103, 105, 108, 109, 112, 113], "For": [1, 2, 5, 9, 12, 13, 23, 27, 28, 29, 30, 32, 33, 36, 40, 41, 46, 51, 53, 55, 58, 64, 65, 66, 69, 76, 77, 79, 81, 83, 84, 90, 97, 98, 99, 100, 102, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115], "If": [1, 4, 6, 8, 9, 10, 12, 20, 24, 25, 26, 27, 28, 29, 32, 33, 36, 41, 46, 47, 48, 49, 51, 53, 54, 56, 63, 64, 66, 69, 76, 79, 81, 83, 84, 91, 97, 98, 100, 101, 103, 104, 105, 109, 111, 112, 113, 121], "In": [0, 6, 8, 9, 11, 12, 20, 23, 27, 28, 29, 32, 33, 34, 35, 36, 40, 46, 49, 50, 51, 52, 54, 56, 57, 58, 61, 63, 69, 73, 76, 78, 79, 81, 83, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115, 119, 120, 121], "It": [0, 1, 5, 6, 9, 12, 15, 24, 25, 27, 28, 31, 32, 33, 34, 36, 37, 40, 47, 53, 54, 60, 63, 64, 73, 76, 77, 79, 82, 83, 84, 88, 91, 97, 98, 100, 109, 112, 115], "Its": [37, 83, 88, 100, 104, 105], "NOT": 103, "Of": 84, "On": [0, 8, 12, 13, 31, 34, 35, 36, 38, 41, 53, 63, 66, 83, 89, 100, 101, 103, 104, 106, 108, 112, 121], "One": [0, 6, 8, 24, 33, 36, 37, 41, 50, 51, 79, 83, 84, 100, 108, 110, 111, 112], "Such": [33, 40, 46, 109], "That": 12, "The": [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 21, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 39, 40, 41, 43, 49, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 65, 66, 69, 72, 73, 74, 75, 77, 78, 80, 82, 83, 84, 86, 87, 88, 89, 91, 98, 99, 101, 103, 106, 108, 109, 110, 111, 112, 113, 115, 121], "Their": 6, "Then": [0, 1, 8, 12, 13, 15, 20, 24, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 41, 46, 47, 48, 50, 53, 54, 63, 64, 65, 66, 79, 80, 81, 83, 84, 88, 92, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 112, 118], "There": [8, 9, 12, 27, 28, 29, 32, 33, 34, 39, 48, 50, 51, 64, 65, 66, 78, 79, 82, 83, 88, 89, 97, 99, 100, 101, 102, 104, 105, 106, 112], "These": [12, 24, 35, 51, 83, 88, 100, 109, 111], "To": [4, 8, 12, 13, 31, 33, 36, 40, 46, 47, 48, 51, 53, 55, 73, 76, 83, 84, 89, 100, 101, 103, 105, 106, 108, 109, 110, 111], "With": [6, 13, 20, 31, 32, 47, 51, 63, 67, 82, 88, 100, 106], "_": [1, 3, 6, 8, 9, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 41, 48, 51, 53, 55, 56, 61, 62, 64, 65, 66, 67, 69, 71, 75, 76, 79, 83, 84, 86, 87, 90, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 113], "_0": [34, 98, 99], "_1": 79, "_2": 105, "__call__": 76, "__dict__": [10, 76], "__doc__": 11, "__eq__": [10, 76], "__file__": 11, "__flags_doc__": [10, 76], "__getitem__": 76, "__getstate__": [10, 76], "__hash__": [10, 76], "__init__": [10, 42, 63, 66, 76, 109], "__loader__": 11, "__memory__": [10, 76], "__mul__": [10, 76], "__name__": 11, "__new__": [10, 76], "__package__": 11, "__pow__": [10, 76], "__setstate__": [10, 76], "__spec__": 11, "__special_treated_flags__": [10, 76], "__str__": [10, 76], "__timing__": [10, 76], "__version__": 46, "_a": [27, 28, 29, 34, 37, 49, 50, 51, 52, 54, 64, 65, 66, 83, 84, 101, 103, 105, 106, 108, 109], "_b": [47, 101, 103], "_c": [37, 47, 49, 51], "_d": 83, "_e": [34, 79, 87, 100], "_h": [12, 36], "_i": [9, 66, 104, 105, 108], "_k": 68, "_m": 106, "_mesh": 76, "_n": [12, 79, 90], "_q": [53, 54, 84], "_registered_draw_typ": 121, "_t": [9, 13, 34, 112], "_u": 101, "_v": [6, 8, 12, 27, 28, 29, 30, 31, 33, 35, 36, 53, 54, 75, 100, 101, 104], "_w": [6, 27, 30, 31, 41, 100, 101], "_x": 104, "a0": [106, 108], "a0inv": 109, "a2": 86, "a_": [13, 24, 51, 53, 63, 64, 73, 76, 105, 111], "a_0": [63, 64, 65], "a_2": 101, "a_h": [12, 83, 109], "a_i": [24, 105], "a_k": 102, "a_l": [63, 64, 65], "a_n": 79, "a_t": [13, 24], "ab": [112, 119, 120], "abbrevi": 20, "abl": 104, "about": [8, 20, 23, 40, 73, 100], "abov": [8, 9, 12, 28, 30, 31, 35, 36, 51, 53, 65, 75, 76, 86, 103, 105, 108, 109, 112], "absorb": [43, 45], "abstract": [41, 76, 101], "ac": 51, "academ": 63, "access": [2, 6, 40, 76], "accid": 86, "accord": [8, 15, 27, 31, 35, 104], "accordingli": [53, 73], "account": [35, 40], "accur": [0, 10, 76, 112], "accuraci": [12, 13, 33], "achiev": [0, 47], "aconv": 5, "across": [1, 8, 12, 34, 36, 42, 55, 79, 88, 100], "act": 21, "acta": 115, "action": [63, 81, 82, 105, 109], "activ": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 39, 46], "actual": [31, 36, 82, 100], "ad": [1, 2, 24, 40, 49, 58, 61, 64, 83, 88, 103, 106, 109, 118], "adapt": [2, 8, 73], "add": [0, 1, 2, 24, 30, 31, 37, 38, 39, 42, 46, 82, 83, 100, 106, 108, 109], "addcircl": 44, "addintegr": [0, 55], "addit": [1, 3, 37, 38, 39, 41, 46, 47, 63, 65, 108, 109, 110, 111, 113, 121], "addition": [28, 35, 46], "addrectangl": [44, 62], "addsurfac": 70, "adiff": 5, "adjac": [36, 108], "adjoint": [1, 2, 37, 48, 51, 105], "adjust": 111, "administr": 15, "advanc": [20, 26, 58, 65, 79], "advantag": [1, 4, 6, 12, 40, 49, 52, 108], "affin": [8, 9, 60], "aform": [113, 114, 115, 118], "after": [8, 10, 26, 48, 76, 84, 101, 105], "afw": 83, "again": [12, 30, 31, 46, 57, 83, 100, 101, 106, 109, 112], "against": 55, "agreement": 103, "aij": 23, "aim": [34, 40, 46], "ainv": [38, 39, 42, 84], "air": [34, 78, 83, 89], "aka": [31, 98, 110], "al": 64, "alanc": 37, "algebra": [88, 105], "algorithm": [8, 40, 45, 47, 49, 52, 63, 76, 84, 97], "align": [1, 8, 9, 12, 13, 28, 29, 31, 33, 34, 35, 36, 47, 48, 50, 51, 52, 53, 55, 56, 79, 81, 98, 100, 101, 104, 121], "all": [1, 2, 6, 8, 9, 10, 13, 15, 20, 24, 27, 29, 30, 31, 32, 33, 34, 36, 37, 39, 46, 49, 51, 52, 53, 54, 63, 65, 66, 69, 72, 73, 74, 76, 81, 83, 88, 90, 99, 100, 101, 102, 104, 105, 106, 108, 109, 110, 111, 112, 113, 115, 121], "all_dofs_togeth": 121, "allow": [0, 5, 7, 9, 12, 13, 32, 36, 37, 41, 47, 50, 51, 56, 58, 67, 76, 79, 89, 100, 113], "allreduc": [15, 24], "almost": 28, "along": [3, 8, 33, 50, 59], "alpha": [0, 1, 2, 3, 5, 6, 8, 9, 12, 13, 30, 33, 49, 50, 51, 53, 73, 79, 81, 83, 98, 99, 101, 102, 104, 106, 108, 121], "alpha_": [48, 50], "alpha_1": [8, 12, 28, 30, 98], "alpha_2": [8, 12, 28, 30], "alpha_d": 98, "alpha_i": 98, "alpha_j": 49, "alpha_t": 9, "alreadi": [1, 34, 41, 64, 65, 81, 109], "also": [5, 6, 7, 8, 9, 12, 15, 20, 27, 28, 29, 30, 31, 33, 34, 35, 36, 40, 41, 47, 48, 49, 50, 51, 53, 55, 56, 58, 63, 65, 72, 76, 79, 83, 84, 89, 98, 99, 100, 103, 104, 105, 106, 111, 112], "altern": [15, 46, 58, 99], "although": [38, 52, 77, 114], "alwai": [27, 33, 111], "am": 20, "amat": 118, "among": 84, "amount": 40, "amplitud": 112, "an": [0, 3, 4, 5, 8, 9, 10, 12, 13, 14, 27, 28, 29, 30, 31, 32, 33, 34, 37, 40, 41, 42, 47, 48, 49, 51, 54, 55, 56, 60, 61, 62, 64, 65, 67, 72, 75, 76, 79, 80, 81, 82, 83, 84, 87, 88, 91, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 112], "anaconda": 20, "analog": 8, "analyi": 87, "analysi": [6, 33, 46, 51, 63, 71, 79, 91, 103, 105, 106, 110], "analyz": [40, 56, 63, 64, 76, 83, 104, 105], "angl": [0, 8, 55, 104], "ani": [10, 27, 29, 34, 52, 54, 65, 76, 79, 83, 105, 106, 112], "anim": [43, 76, 108, 109], "anisotrop": [8, 71], "anl": 23, "anoth": [0, 49, 52, 62, 78, 83, 91, 112], "ansatz": 41, "anton": 63, "anymor": 121, "anywai": [58, 121], "ap": [49, 81, 118], "apart": 38, "apl": 118, "app": 121, "appear": 79, "append": [33, 35, 47, 48, 49, 50, 51, 80, 103, 108, 110, 115, 121], "appl": 20, "appli": [1, 4, 5, 8, 10, 12, 13, 34, 35, 36, 37, 46, 50, 51, 52, 54, 63, 64, 65, 69, 72, 74, 76, 79, 82, 83, 84, 90, 98, 99, 100, 101, 106, 108, 109, 112, 113, 114], "applic": [8, 12, 33, 35, 45, 46, 49, 63, 72, 81, 82, 121], "applym": [10, 76], "approach": [34, 40, 46], "approx": [4, 12, 13, 41, 47, 48, 51, 53, 54, 64, 65, 83, 87, 90, 106, 111, 112, 121], "approxiamt": 79, "approxim": [0, 12, 27, 33, 36, 40, 46, 47, 48, 51, 54, 62, 63, 65, 67, 76, 77, 79, 81, 83, 86, 97, 109, 111, 112, 113, 114, 115, 118], "apsc_loc": 23, "ar": [0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 15, 20, 21, 23, 24, 27, 28, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 82, 83, 84, 87, 88, 89, 90, 91, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112, 115, 121], "arbitrari": [12, 28, 30, 31, 48, 53, 55, 65, 67, 76, 83, 90, 100, 102, 105, 106], "arbitrarili": [3, 106], "arc": 8, "arcco": 47, "arcross": 89, "area": 8, "aren": 76, "arg": [10, 76, 86, 121], "arg0": [10, 76], "arg1": [10, 76], "arg2": [10, 76], "arg3": [10, 76], "argument": [6, 10, 32, 34, 47, 76, 80, 87, 104], "aris": [28, 52], "arithmet": 77, "arnold": [60, 61, 67, 83], "around": [31, 106], "arrai": [5, 6, 12, 23, 24, 38, 40, 52, 53, 55, 56, 57, 58, 60, 61, 62, 67, 68, 69, 73, 76, 81, 82, 84, 86, 89, 91, 98, 100, 102, 105, 111, 112, 121], "array_y_": 76, "artifici": [37, 64], "arxiv": [118, 119, 120], "asc": [10, 11], "ascher": 5, "ascii": 121, "ask": 15, "asm": [45, 46, 63, 65, 83, 105, 108, 109], "asmal": [115, 118], "assembl": [0, 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "assign": [15, 108, 110], "associ": [12, 34, 76, 110], "assum": [1, 8, 10, 12, 13, 24, 28, 29, 30, 31, 34, 49, 50, 52, 53, 55, 64, 65, 73, 76, 83, 90, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 111, 112], "assumpt": [12, 31, 33, 64, 65, 101], "ast": [27, 28, 29, 30, 31, 34, 41, 47, 48, 50, 51, 53, 75, 104, 105], "astrid": 62, "asvector": 42, "asymptot": [48, 106], "atan2": 0, "atild": 42, "atol": [14, 23], "atop": [31, 34, 36, 37, 41, 47, 65, 84, 86, 100, 104, 105, 108, 109, 112], "attain": 75, "attribut": [10, 76], "aubin": [8, 64, 65, 79], "augment": [45, 83], "author": [46, 72, 105], "automat": [8, 10, 70, 76], "autoscal": [4, 5, 119, 120], "autoupd": [10, 33, 35, 63, 66, 76, 83, 84, 109], "avail": [3, 5, 10, 11, 14, 20, 46, 51, 52, 72, 76, 84, 115], "averag": [10, 33, 34, 35, 42, 66, 76, 82, 109, 121], "averagingt": 42, "avoid": [12, 33, 40, 50, 71, 121], "await": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24], "ax": [8, 47], "axi": 112, "aziz": [31, 53], "b": [1, 4, 5, 6, 8, 13, 20, 22, 27, 28, 30, 31, 35, 37, 38, 39, 42, 47, 48, 49, 50, 51, 52, 53, 54, 56, 62, 68, 70, 73, 76, 78, 81, 82, 83, 84, 85, 86, 88, 89, 101, 102, 103, 108, 111, 112, 116, 119, 120, 121], "b64": 121, "b64encod": 121, "b_": [8, 13, 53, 111], "b_1": [91, 102, 111], "b_2": 91, "b_i": [51, 111], "b_k": 102, "b_l": 111, "b_n": 6, "b_r": 102, "b_t": 8, "b_x": 82, "b_y": 82, "babuska": 53, "babu\u0161ka": 31, "back": [8, 13, 15, 20, 37, 47, 51, 61, 84, 100, 102, 109, 112, 121], "backward": [51, 52, 63, 76, 112, 113], "bad": [6, 58, 69, 71, 83, 106], "badli": 98, "baker": 3, "balanc": [40, 55, 57, 58, 79], "ball": [99, 102], "banach": [27, 28, 99, 100], "banana": 20, "band": 40, "bar": [0, 55, 78, 89], "bare": 64, "barycentr": [9, 80], "base": [3, 15, 37, 40, 45, 63, 64, 65, 68, 76, 80, 100, 121], "base64": 121, "basematrix": [10, 42, 63, 76, 82, 96, 106, 108, 109], "basevector": [10, 76, 108], "basewebguiscen": [73, 96, 116, 121], "basi": [3, 4, 8, 9, 10, 12, 13, 34, 36, 38, 47, 49, 63, 66, 79, 83, 88, 100, 103, 105, 106, 110, 113, 114, 115, 118, 121], "basic": [45, 46, 58, 64, 108, 118], "bbboundari": [10, 76], "bbnd": [10, 37, 39, 76], "bboundari": [10, 37, 39, 76], "bc": [0, 37, 38, 39, 44, 55, 62, 70, 73, 108], "bcast": 20, "bddc": [45, 46, 74, 76, 77, 91], "bdm": [7, 34, 88], "bdm2": 88, "bdm_1": 88, "bdm_k": 88, "beauti": 58, "becom": [5, 12, 40, 51, 89, 103, 118], "been": [79, 100], "befor": [26, 37, 109], "begin": [1, 5, 6, 8, 9, 12, 13, 24, 28, 29, 31, 33, 34, 35, 36, 37, 38, 40, 41, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 73, 78, 79, 81, 82, 83, 84, 86, 87, 89, 90, 91, 92, 97, 98, 99, 100, 101, 102, 104, 105, 106, 109, 111, 112, 113, 114, 115, 121], "behav": [91, 98], "behavior": 112, "behaviour": 109, "behind": [20, 46, 49], "being": [27, 76, 103, 108], "bel": [116, 121], "belong": [73, 100, 104, 108, 109], "below": [28, 33, 51], "bend": [62, 67, 69, 70], "best": [8, 28, 54, 67, 79, 110], "beta": [8, 13, 49, 53, 54, 69, 70, 79, 81, 84, 104], "beta_": 31, "beta_1": [30, 31, 53], "beta_2": [30, 31], "beta_h": 69, "beta_i": 104, "beta_k": 49, "better": [8, 33, 49, 66, 86, 103], "between": [0, 3, 8, 13, 14, 20, 21, 24, 33, 34, 38, 40, 41, 45, 46, 62, 73, 74, 76, 97, 100, 108], "bezier_point": 121, "bezierpnt": 121, "bf": [0, 1, 8, 34, 55], "bfa": 81, "bfa1": [22, 85], "bfb": [22, 81, 85], "bfc": [22, 85], "bfequ": 34, "bfnorm": 0, "bfpre": 121, "bfschur": [22, 81, 85], "bg": 51, "big": [13, 30, 40, 47, 51, 53, 54, 56, 64, 65, 66, 77, 79, 83, 97, 103, 105, 106, 108, 109, 112, 113, 115, 121], "bigcup": 109, "bigcup_": 36, "bilinear": [0, 2, 4, 6, 8, 12, 13, 27, 28, 29, 31, 34, 35, 37, 38, 39, 42, 48, 51, 53, 56, 62, 68, 73, 75, 78, 83, 84, 89, 97, 101, 104, 106, 109, 121], "bilinearform": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 17, 18, 19, 21, 22, 23, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "bin": [20, 46], "binari": [20, 121], "bind": [10, 108], "binder": 25, "bisect": [76, 108], "bit": [38, 109], "bitarrai": [10, 66, 73, 76, 108, 109], "bitmask": 78, "bjac": 105, "block": [4, 20, 39, 40, 53, 56, 64, 70, 81, 83, 89, 91, 103, 121], "blockmatrix": 84, "blockvector": [81, 84], "blow": 6, "blue": 112, "bmat": 82, "bnd": [1, 4, 5, 10, 34, 55, 57, 66, 70, 76, 121], "bnddof": [38, 39], "bndmass": 66, "bndpart": 55, "bodi": [59, 62, 74], "boffi": [46, 72], "bone": [15, 20], "bonu": 26, "book": [25, 46], "bool": [10, 76], "bornemann": 111, "boss": 20, "bot": [27, 29, 31, 32, 37, 38, 39, 49, 53, 83, 86, 100, 101], "bot_": [49, 79], "bot_c": 49, "bot_v": 53, "both": [0, 2, 12, 15, 24, 29, 30, 31, 32, 33, 41, 49, 54, 55, 61, 63, 64, 65, 68, 76, 78, 79, 83, 84, 87, 99, 100, 101, 103, 106, 112, 121], "bottl": 40, "bottom": [1, 2, 3, 10, 14, 34, 37, 38, 39, 44, 56, 62, 66, 73, 80, 82, 91, 100, 108, 109], "bottomo": 107, "bound": [8, 9, 12, 27, 28, 30, 31, 33, 34, 35, 36, 47, 48, 49, 51, 65, 83, 84, 97, 98, 99, 100, 101, 102, 103, 104, 106, 108, 109], "boundari": [1, 3, 4, 6, 7, 10, 12, 13, 18, 19, 22, 28, 30, 31, 33, 34, 36, 37, 38, 39, 40, 43, 45, 46, 51, 58, 65, 67, 70, 74, 75, 76, 78, 79, 80, 81, 82, 84, 85, 88, 89, 90, 91, 92, 95, 97, 99, 100, 101, 103, 104, 106, 108, 110, 113, 121], "boundarycf": [73, 76], "bounded": 109, "box": [8, 12, 22, 27, 28, 29, 31, 32, 53, 65, 74, 77, 78, 79, 81, 83, 85, 97, 99, 100, 101, 104, 105, 108, 121], "bp": 81, "braess": [34, 46, 64, 67, 79], "brambl": [8, 12, 36, 45, 46, 64, 79, 85, 101], "bramblepasciakcg": [22, 81, 85], "break": [38, 39, 46, 47, 48, 49, 50, 51, 81, 91], "brenner": [3, 46], "brezzi": [31, 34, 46, 53, 54, 61, 67, 72, 88], "bring": [37, 102], "broadcast": [15, 20], "browser": 46, "bsize": 23, "bstab": [119, 120], "btr": [116, 121], "bu": [4, 5, 6, 31, 84], "bubbl": [54, 58, 61, 76, 80, 118], "build": [8, 11, 12, 38, 42, 74, 108, 110], "buildrefinementtre": 76, "buildrenderdata": 121, "built": [24, 40, 55, 57, 76, 88], "builtin": [10, 76, 80], "bv": 53, "bvp": [1, 8], "bx": 82, "byte": 20, "c": [5, 8, 9, 10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 34, 37, 38, 42, 46, 47, 48, 49, 51, 53, 54, 56, 63, 64, 65, 67, 73, 77, 79, 81, 82, 83, 84, 85, 90, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 111, 112, 113], "c2l": 109, "c_": [1, 9, 13, 24, 51, 63, 64, 65, 70, 79, 83, 105, 106, 108, 109, 111], "c_0": [64, 90, 98, 100], "c_1": [33, 101, 103, 106, 109, 111], "c_2": [33, 101, 103, 106, 109], "c_3": 101, "c_4": 101, "c_f": [92, 102, 103], "c_i": 111, "c_j": 111, "c_l": [64, 111], "c_m": 111, "c_p": 102, "c_t": [9, 13, 24], "cach": 20, "cal": [8, 9, 12, 13, 33, 35, 36, 98, 99], "calcelementmatrix": 11, "calcul": [21, 34, 48, 50, 51, 63, 76, 88, 121], "calculu": [100, 102], "call": [1, 2, 8, 9, 12, 15, 20, 24, 27, 31, 32, 33, 39, 48, 51, 52, 55, 56, 58, 65, 76, 79, 83, 84, 90, 91, 97, 98, 99, 101, 103, 104, 105, 106, 109, 110, 111, 112, 119, 120, 121], "callback": [47, 49], "can": [0, 1, 2, 3, 6, 8, 10, 12, 13, 14, 15, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 60, 62, 63, 64, 65, 66, 69, 72, 73, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 89, 91, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 112, 113, 115, 121], "cancel": [47, 79, 80, 90, 98, 121], "candid": [6, 30, 53, 54, 87], "cannot": [3, 7, 12, 17, 35, 36, 51, 53, 76, 84, 98, 109], "canon": [27, 41, 62, 75, 83], "canva": 47, "cap": [9, 13, 29, 34, 36, 40, 83, 90, 99, 100, 101, 107, 109], "captur": 51, "care": [8, 33], "carefulli": 64, "cartesian": 76, "case": [2, 8, 9, 23, 28, 29, 30, 31, 35, 36, 51, 54, 58, 60, 67, 69, 73, 76, 78, 83, 99, 100, 101, 104, 105, 106, 115], "cauchi": [12, 27, 31, 32, 36, 64, 66, 97, 99, 100, 101, 102, 105], "cc": [5, 24, 38, 61, 73, 81, 82, 84, 91, 111, 121], "ccc": [24, 53, 68, 81, 84, 86, 91, 111], "cccc": 105, "ccccccl": 91, "ccccccll": 61, "ccccl": [53, 57, 58, 60, 84], "ccccll": [12, 40, 53, 55, 56, 57, 58, 62, 67, 69, 84, 86, 89], "cccll": 60, "cdof": 63, "cdot": [3, 6, 12, 13, 27, 28, 29, 30, 31, 32, 34, 37, 47, 48, 53, 54, 57, 62, 65, 66, 75, 76, 79, 87, 89, 97, 98, 99, 100, 101, 105, 113, 121], "cea": [8, 12, 64], "cell": [10, 20, 46, 58, 76, 88, 110, 119, 120, 121], "center": [0, 8, 55, 75, 99, 102], "cerbsim": 46, "certain": [8, 35, 108, 111, 112], "cf": [0, 4, 6, 10, 11, 18, 19, 33, 37, 38, 39, 55, 74, 76, 88, 121], "cfl": 112, "cfn": 44, "cg": [14, 18, 23, 49, 77, 81, 91], "cgsolver": [14, 18, 19, 21, 22, 38, 39, 42, 63, 74, 77, 82, 83, 84, 108, 110], "ch": [83, 109, 112], "ch_i": 109, "chain": [8, 13, 88, 100, 106], "challeng": 60, "chamfer": 74, "chang": [3, 10, 48, 73, 76, 81, 100], "chapter": [12, 28, 52, 99, 111], "character": [32, 50, 100], "charg": 76, "cheap": [4, 12, 39, 51, 63, 66, 77, 82, 86], "cheaper": [37, 82, 108], "cheapli": [49, 56, 83, 84, 91, 121], "cheat": [38, 62], "chebi": 47, "chebyiter": 47, "chebyshev": [45, 46, 49, 52], "check": [0, 6, 32, 34, 46, 76, 99, 108, 121], "check_unus": [38, 39, 66, 103], "chen": [25, 67], "choic": [6, 8, 64, 66, 75, 100, 104, 112], "choos": [0, 6, 8, 12, 28, 31, 32, 33, 36, 47, 48, 51, 53, 55, 73, 76, 83, 87, 97, 100, 103, 104, 108, 118, 121], "chosen": [3, 10, 28, 40, 50, 57, 83, 97, 106, 112], "christoph": [1, 7, 72], "ciarlet": 9, "circ": [0, 8, 9, 12, 13, 36, 55, 89, 100, 102, 106, 121], "circ1": 78, "circ2": 78, "circl": [0, 7, 16, 18, 19, 55, 78, 81, 82, 85, 89, 118, 121], "circo": [0, 55], "circul": 4, "circumst": 8, "cjac": 106, "cl": [6, 12, 36, 98, 102], "claim": [28, 30, 54, 64, 101, 106], "clamp": 3, "clark": 37, "class": [0, 9, 10, 14, 20, 21, 22, 23, 24, 42, 46, 55, 63, 65, 66, 76, 78, 101, 109, 111], "classic": [36, 45, 98, 111], "clean": 46, "clear": [53, 90, 108, 109], "clearli": [100, 103], "clement": 54, "clementin": 20, "click": 25, "client": [15, 42], "clip": [22, 55, 83, 85], "clock": 34, "close": [8, 9, 27, 28, 29, 31, 32, 33, 48, 53, 64, 76, 81, 89, 100, 101, 104], "closest": [0, 32], "closur": [27, 33, 41, 99, 100, 104], "cluster": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 40], "cl\u00e9ment": [65, 109], "cnt": [4, 5, 16, 108, 121], "co": [0, 32, 41, 47, 55, 76, 100, 102, 104, 121], "coars": [42, 63, 64, 66], "coarsebndmass": 66, "coarsedof": 66, "coarseext": 66, "coarsegrid": 108, "coarsemat": 63, "coarsepr": 63, "coarser": [40, 63, 64, 65, 121], "coarsest": [63, 65], "coasr": [63, 64], "cockburn": 1, "code": [26, 46, 50], "codimens": 76, "coeffici": [4, 24, 28, 34, 46, 51, 63, 64, 73, 75, 76, 79, 89, 100, 102, 103, 105, 106, 108, 109, 111, 112, 113, 121], "coefficientfunct": [5, 7, 10, 16, 44, 70, 76], "coerciv": [0, 2, 8, 12, 31, 36, 45, 46, 53, 101, 104], "coffe": 6, "cohen": 118, "coil": 82, "coincid": [8, 37, 98, 100, 111, 112], "col": [23, 42, 63], "col_pardof": 42, "colleagu": [25, 26], "collect": [15, 20, 53, 99], "color": [47, 112], "coloumb": 83, "column": [13, 49, 106, 112, 115], "com": 46, "combin": [2, 6, 8, 10, 15, 30, 35, 36, 49, 51, 58, 63, 64, 65, 76, 81, 100, 105, 110, 115, 121], "come": [6, 30, 40, 55, 76, 84, 86, 109, 121], "comm": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 42, 76], "comm_self": 23, "comm_world": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 42], "command": 46, "commerci": 46, "common": [0, 8, 9, 33, 34, 55, 100], "commun": [15, 20, 21, 24, 40, 76], "communiactor": 15, "commut": [12, 79, 83, 86, 90, 102], "comodi": 67, "comp": [0, 10, 37, 38, 39, 55, 76, 78], "compact": [27, 90, 98, 99, 100, 101], "compar": [1, 34, 40, 47, 54, 79, 110, 112], "comparison": [1, 50, 121], "compat": [76, 83], "compens": [1, 6, 87], "competit": 40, "compil": [70, 74], "complement": [27, 37, 38, 82, 83, 103], "complet": [27, 29, 31, 64, 65, 73, 76, 83, 99, 111, 121], "complex": [9, 10, 31, 40, 43, 44, 52, 63, 76, 77], "complic": 40, "compon": [0, 1, 3, 5, 7, 10, 12, 17, 18, 19, 33, 34, 37, 38, 39, 47, 51, 54, 55, 57, 58, 62, 64, 65, 67, 69, 70, 73, 74, 76, 83, 84, 86, 88, 89, 91, 108, 119, 120, 121], "composit": [100, 101, 118], "compound": [0, 55], "compoundfespac": [10, 76], "compoundfespaceallsam": [10, 76], "compress": [18, 19, 37, 38, 39, 52], "compressed_row_storag": 52, "compuat": 46, "comput": [0, 4, 23, 24, 25, 26, 28, 33, 35, 36, 40, 41, 46, 47, 49, 51, 55, 63, 70, 72, 73, 76, 77, 82, 84, 88, 91, 97, 98, 102, 103, 108, 110, 111, 113, 121], "computation": [51, 52], "concentr": 6, "concept": 112, "conclud": [31, 53, 100], "condens": [1, 76], "condit": [1, 4, 6, 10, 12, 13, 28, 31, 33, 34, 39, 41, 43, 45, 46, 47, 48, 51, 54, 58, 61, 63, 64, 65, 67, 70, 74, 76, 78, 84, 87, 89, 91, 100, 103, 104, 105, 106, 108, 109, 110, 111, 113, 114, 115], "condition": 65, "condtion": 73, "conduct": [78, 83, 109], "configur": 20, "conflict": 46, "conform": [3, 9, 45, 46, 54, 58, 62, 76, 79, 83, 91, 100, 106], "conjug": [21, 37, 38, 45, 46, 52, 77, 82, 84], "connect": [6, 9, 10, 15, 23, 24, 36, 39, 67, 83, 98, 102, 108, 109], "conserv": [45, 118, 121], "consid": [0, 3, 6, 8, 12, 28, 30, 31, 33, 34, 35, 41, 43, 53, 57, 76, 81, 82, 83, 84, 89, 100, 102, 106, 108, 112, 113, 114, 115], "consider": 51, "consist": [0, 2, 9, 12, 15, 21, 23, 25, 36, 37, 40, 41, 42, 45, 46, 60, 62, 64, 76, 82, 83, 84, 99, 100, 105, 111, 112, 121], "consisteni": 0, "consol": 46, "const": 34, "constant": [8, 9, 10, 12, 15, 28, 29, 30, 33, 34, 36, 40, 42, 51, 54, 56, 58, 59, 63, 64, 66, 67, 76, 78, 79, 88, 92, 98, 100, 101, 102, 103, 106, 108, 109, 111, 112, 121], "constantebematrix": [120, 121], "consti": 108, "constrain": [1, 34, 37, 83, 91], "constrainst": 45, "constraint": [12, 34, 38, 46, 54, 58, 69, 79, 83, 84, 105], "construct": [8, 9, 29, 30, 33, 34, 49, 53, 54, 77, 79, 82, 83, 87, 97, 100, 104, 105], "constructor": 10, "consult": [75, 110], "contact": [0, 55], "contactboundari": [0, 55], "contain": [8, 9, 23, 27, 33, 34, 36, 41, 49, 76, 83, 88, 98, 99, 104, 109, 112], "content": 121, "context": 6, "continu": [0, 2, 6, 7, 8, 12, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 53, 54, 55, 56, 58, 60, 61, 64, 65, 68, 76, 78, 79, 83, 84, 87, 88, 89, 90, 91, 92, 95, 97, 99, 100, 101, 102, 103, 104, 108, 109, 121], "contourf": 112, "contract": 28, "contradict": [29, 31, 32, 101], "contrast": [31, 36, 51, 61, 76, 112, 114], "contribut": [13, 24, 33, 34, 36, 98, 104, 108], "control": [33, 74, 76, 106], "conv": [5, 17], "convect": [28, 45, 46], "conveni": [9, 46, 55, 65, 73, 81], "convent": 31, "converg": [8, 27, 28, 31, 32, 33, 38, 39, 41, 48, 50, 51, 56, 65, 75, 77, 79, 83, 99, 100, 101, 103, 110, 115], "convergenct": 51, "convers": [21, 24, 100, 121], "convert": [20, 23, 76], "convertl2oper": [10, 76], "convertoper": [17, 19, 116], "convex": [8, 32, 50, 65, 84, 104, 105], "convu": [5, 17], "convuhat": 17, "coo": [42, 76], "coordin": [0, 9, 76], "copi": [23, 63, 115], "core": 40, "corneliu": 63, "corner": [8, 33, 35, 104], "corollari": [101, 112], "correct": [34, 35, 38, 51, 63, 64, 65, 73, 82, 105, 106, 108, 109], "correspond": [2, 10, 12, 23, 33, 36, 41, 47, 51, 64, 65, 67, 71, 76, 80, 83, 91, 103, 105, 106, 108, 118], "cosen": 48, "cosh": 41, "cosin": 100, "cost": [34, 40, 82, 108], "could": [34, 51, 54, 82, 91, 112], "count": [15, 34], "counter": [34, 103], "coupl": [1, 10, 42, 53, 76, 111], "coupling_typ": [10, 18, 19, 76], "couplingmat": 42, "couplingtyp": [10, 18, 19, 76], "courant": 112, "cours": [25, 46, 72, 84], "covari": 116, "cover": 100, "cpp": [10, 80], "creat": [10, 14, 21, 23, 33, 46, 75, 76, 78, 80], "createaij": 23, "createblocksmooth": 110, "createcolvector": 42, "createdevicematrix": 120, "createdirectsolverclust": [10, 76], "createfromcoo": 42, "creategener": 23, "createi": 23, "creatematrix": [42, 63, 66], "createpetscmatrix": 14, "createrowvector": [42, 66, 84], "createsmooth": [47, 49, 51, 63, 64, 82, 106, 118], "createsmoothingblock": [10, 76], "createsparsematrix": 118, "createtranspos": [63, 66], "createvec": [14, 23], "createvector": [4, 5, 21, 38, 39, 42, 47, 48, 49, 50, 51, 63, 81, 108, 109, 115, 116, 118], "createvvector": 42, "creation": 80, "crime": 12, "critic": [84, 103], "cross": [70, 116], "crouzeix": [12, 111], "csg": [70, 77, 83], "csg2d": [37, 38, 39, 82], "csgeometri": [70, 83], "csr": [1, 2, 23, 52, 76], "csr_matrix": [1, 2, 52, 76], "ctautau": 70, "cube": 77, "cubic": [54, 58], "cumul": [21, 23, 24, 42, 66], "cup": [1, 9, 41, 55, 73, 90, 91, 100, 104, 106, 107], "cup_": [34, 100], "curl": [17, 34, 59, 62, 68, 69, 70, 83, 88, 116], "current": [0, 83], "curv": [0, 7, 8, 12, 16, 18, 19, 44, 55, 60, 64, 67, 70, 74, 76, 77, 78, 81, 85, 89, 121], "cut": [33, 100], "cycl": 33, "cyl": [7, 16, 22, 70, 74, 77, 81, 85], "cyl1": 8, "cyl2": 8, "cylbox": 74, "cylboxedg": 74, "cylind": [0, 8, 22, 70, 74, 77, 83, 85], "c\u00e9a": 28, "d": [0, 1, 2, 5, 6, 7, 8, 11, 12, 13, 14, 18, 19, 23, 27, 28, 29, 32, 34, 36, 37, 38, 39, 43, 44, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 72, 73, 74, 75, 76, 79, 80, 82, 83, 84, 88, 89, 90, 92, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 111, 112, 113, 115, 121], "d_": 63, "d_g": 98, "d_h": 83, "d_i": 90, "d_l": [63, 64, 65], "d_w": 98, "damp": [47, 48, 51, 64], "dampingp": [119, 120], "dampingu": [119, 120], "dash": 47, "data": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21, 22, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 67, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "date": 51, "dc": [58, 86, 91], "dd": [73, 108], "ddot": [105, 112, 114], "de": [52, 116], "debug": 20, "decad": [10, 40, 76], "decai": [8, 66, 108, 109], "decid": 103, "decis": 6, "declar": 76, "declaremathoper": [1, 4, 5, 6, 53, 62, 67, 68, 69, 78, 86, 87, 88, 89, 90, 91], "decod": 121, "decompos": [32, 38, 40, 53, 65, 66, 83, 101, 104, 105, 106, 108, 109], "decomposit": [37, 63, 65, 83, 90, 105, 106, 115], "decoupl": [62, 86, 111], "decreas": [8, 40, 48, 51], "def": [0, 2, 3, 7, 17, 18, 19, 33, 35, 42, 47, 49, 55, 58, 62, 63, 66, 67, 69, 70, 74, 81, 103, 109, 110, 112, 115, 121], "default": [0, 15, 20, 34, 35, 37, 38, 39, 55, 73, 76, 78, 89, 108], "defin": [0, 4, 8, 9, 10, 12, 13, 15, 20, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 41, 42, 47, 48, 50, 52, 53, 54, 55, 56, 58, 59, 62, 63, 64, 65, 66, 75, 76, 79, 84, 88, 90, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 109, 110, 112, 115, 120, 121], "definedon": [0, 2, 6, 7, 10, 12, 18, 19, 22, 34, 37, 38, 39, 55, 66, 73, 76, 81, 84, 85, 108], "defineproblem": 110, "definit": [0, 8, 9, 12, 27, 29, 30, 31, 32, 36, 37, 50, 51, 52, 56, 63, 65, 67, 77, 79, 81, 84, 90, 91, 98, 99, 103, 104, 105, 106, 109, 110, 112], "deflect": [67, 69], "deform": [0, 2, 12, 55, 56, 59, 66, 74, 76, 82, 103, 113, 114, 118, 121], "degener": 83, "degre": [9, 10, 34, 39, 47, 54, 62, 71, 73, 76, 77, 79, 80, 88, 91], "dehat": 116, "deletezeroel": 118, "deliv": 112, "delta": [1, 2, 12, 28, 31, 33, 36, 41, 43, 47, 53, 55, 57, 58, 64, 65, 69, 70, 76, 84, 98, 100, 104, 113, 114, 118], "delta_": [9, 27, 79], "delta_h": 121, "delta_t": 34, "demand": 98, "demkowicz": 79, "denomin": 80, "denot": [27, 79, 98, 99, 112], "dens": [24, 27, 42, 52, 79, 90, 97, 100, 101, 104, 109], "densiti": [8, 58, 83, 90, 100, 113], "dep": 20, "depend": [0, 1, 4, 8, 12, 20, 31, 33, 34, 51, 54, 62, 65, 75, 77, 79, 97, 102, 103, 106, 110, 112, 113, 119, 120], "deprec": [10, 76], "der": 118, "deriv": [1, 2, 3, 6, 8, 9, 12, 13, 33, 36, 40, 46, 53, 55, 57, 65, 70, 73, 76, 81, 84, 99, 100, 109, 112, 113, 114, 118], "describ": [12, 35, 46, 74, 82], "descriptor": [10, 76], "design": [35, 40, 60, 84, 121], "desir": 35, "dest": 20, "destin": 20, "det": [8, 13, 88, 100, 106, 112, 121], "detail": [6, 72, 79], "deterior": [106, 109], "determin": [13, 48, 81, 111], "detour": 83, "deuflhard": 111, "dev0": 20, "develop": [25, 64, 79, 83, 107], "df": 73, "dg": [3, 4, 5, 6, 10, 45, 46, 54, 67, 76, 121], "dgjump": [2, 6, 10, 76, 121], "dh": 116, "diag": [51, 63, 105, 112], "diagon": [4, 12, 21, 22, 49, 51, 52, 56, 81, 85, 91, 105, 118, 121], "diagonalmatrix": [42, 66, 108], "diagoniz": [48, 112], "diagram": [12, 79, 86], "diam": [8, 109], "diamet": 8, "diaz": 118, "dict": [10, 76], "did": [33, 46, 114], "diff": 5, "differ": [0, 1, 8, 9, 10, 21, 33, 45, 46, 47, 48, 54, 58, 63, 64, 65, 66, 73, 76, 77, 78, 79, 101, 109, 112, 113, 118], "differenti": [3, 27, 28, 46, 52, 65, 70, 76, 79, 84, 90, 98, 99, 100, 103, 105, 111, 112, 113], "differentialsymbol": [37, 39], "difficult": [5, 106], "difficulti": [26, 58], "diffus": [28, 45, 46], "dim": [0, 7, 8, 10, 12, 16, 18, 19, 24, 31, 33, 34, 35, 55, 57, 63, 66, 70, 76, 78, 81, 85, 89, 91, 103, 108, 109, 118, 121], "dimens": [8, 9, 34, 36, 63, 76, 88, 100, 109, 115], "dimension": [8, 10, 13, 27, 28, 29, 30, 31, 49, 52, 76, 79, 84, 87, 100, 101, 103, 106, 115], "dinv": 64, "dir": [11, 34, 42], "direct": [8, 34, 40, 50, 52, 53, 54, 63, 74, 77, 81, 85, 97, 100, 108, 109], "directli": [7, 8, 50, 76, 79, 84, 98], "dirichet": 108, "dirichlet": [0, 1, 3, 5, 7, 10, 11, 12, 13, 14, 18, 19, 21, 22, 28, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 55, 57, 58, 62, 63, 66, 67, 69, 70, 73, 74, 76, 77, 78, 79, 80, 81, 82, 84, 85, 89, 91, 92, 100, 103, 104, 106, 107, 108, 109, 113], "dirichlet_bbbnd": [10, 76], "dirichlet_bbnd": [10, 76], "dirk": 111, "disadvantag": [1, 91], "disc": 104, "discontinu": [1, 2, 3, 10, 18, 19, 34, 55, 56, 58, 72, 76, 78, 83, 89, 91, 119, 120], "discret": [0, 2, 4, 6, 7, 12, 31, 33, 34, 40, 46, 48, 52, 55, 56, 58, 59, 60, 62, 63, 64, 67, 69, 76, 83, 84, 86, 87, 91, 97, 105, 106, 109, 112, 113, 114, 115, 118], "discuss": [12, 28, 64, 73, 83], "disjoint": 41, "disp": [62, 67, 69, 70], "disp_dg": 3, "displac": [59, 60, 70, 74, 82], "dissect": 108, "dissert": [80, 88], "dist": [41, 109], "distanc": [8, 50], "distribut": [10, 21, 34, 42, 45, 46, 62, 76, 78, 98], "div": [1, 4, 5, 6, 12, 18, 19, 22, 27, 28, 34, 45, 46, 53, 54, 56, 57, 58, 62, 67, 68, 69, 70, 74, 76, 78, 81, 82, 83, 84, 85, 86, 87, 89, 91, 100, 121], "diverg": [7, 58, 60, 61, 83, 88, 90, 121], "divid": [8, 12, 24, 28, 64, 108], "divtol": [14, 23], "divvel": 7, "dl": [63, 65], "do": [1, 12, 25, 48, 51, 52, 55, 66, 73, 79, 82, 97, 106, 109, 112, 113], "doc": 52, "docinfo": 10, "docu": 75, "document": [10, 72, 75, 110], "doe": [0, 3, 8, 15, 21, 24, 31, 46, 48, 49, 64, 73, 76, 84, 97, 100, 102, 103, 108, 109, 118], "dof": [1, 10, 15, 21, 23, 24, 34, 37, 38, 39, 42, 62, 66, 76, 83, 88, 108, 109, 110, 118], "dof2proc": 42, "dofnr": [10, 76], "dofrang": [10, 76], "doi": [65, 105], "dom": [37, 38, 39, 76, 108], "dom1": [38, 39], "dom2": [38, 39], "domain": [8, 9, 10, 12, 24, 33, 37, 38, 39, 42, 43, 53, 55, 63, 65, 66, 73, 76, 82, 83, 97, 98, 99, 101, 103, 104, 113, 119, 120], "domaindof": 108, "domdof": 108, "domi": 108, "domin": [64, 65, 83, 101], "domtest": [37, 38, 39], "domtrial": [37, 38, 39], "don": [1, 2, 3, 38, 46, 57, 69, 73, 91, 105], "done": [12, 15, 21, 49, 65, 108, 121], "dormann": 37, "dot": [44, 112, 113, 114, 121], "dotimestep": 16, "doubl": [24, 84], "dougla": [24, 34, 88], "down": [37, 56], "downarrow": [68, 86], "download": 20, "dp": [37, 45, 46], "draft": 46, "draw": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 33, 34, 35, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "draw_surf": [83, 116, 121], "draw_vol": [74, 77, 116, 121], "drawback": [1, 2, 84], "drawgeo": 121, "driven": 46, "ds_i": 100, "dst": 20, "dt": [32, 65, 113, 115, 121], "dtype": [23, 121], "dual": [8, 9, 13, 27, 31, 34, 35, 39, 41, 55, 61, 62, 100, 104, 105], "dualiti": [28, 55], "dubin": 79, "due": [35, 36, 53, 64, 74, 83, 104, 109, 121], "dummyargu": [10, 76], "dummypardof": 42, "durian": 20, "dvert": [37, 39], "dx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 63, 64, 65, 66, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 113, 114, 115, 116, 118, 121], "dy": [79, 97, 102], "dynam": [12, 45], "e": [2, 3, 4, 6, 8, 9, 10, 12, 24, 27, 28, 29, 31, 32, 33, 34, 35, 36, 38, 39, 40, 47, 48, 49, 50, 51, 52, 53, 55, 58, 60, 61, 62, 64, 65, 66, 73, 74, 75, 76, 77, 79, 80, 83, 84, 87, 88, 90, 91, 97, 98, 99, 100, 101, 102, 103, 105, 109, 112, 113, 116, 118, 121], "e0": [106, 108], "e_": [66, 105], "e_0": [65, 100, 106], "e_1": 34, "e_2": 34, "e_3": 34, "e_i": [9, 34, 80, 105], "e_l": [64, 65, 66], "each": [8, 10, 12, 13, 15, 24, 33, 36, 51, 58, 63, 76, 82, 83, 91, 98, 99, 100, 108, 121], "earlier": 76, "eas": 55, "easi": [46, 60], "easili": [5, 24, 34, 41, 47, 48, 49, 64, 65, 79, 112], "easter": 26, "ecomposit": 37, "ed": 110, "edg": [0, 1, 2, 3, 6, 7, 8, 9, 10, 12, 16, 18, 19, 34, 35, 36, 54, 55, 59, 62, 68, 74, 76, 78, 79, 80, 81, 83, 85, 87, 88, 89, 91, 103, 108, 110], "edgedof": 76, "edof": 76, "effect": 83, "effici": [33, 34, 40, 49, 63, 79], "effort": 40, "ehat": 116, "ei": [10, 11, 22, 76, 85], "eid": 76, "eigen": [41, 47, 64, 100, 101, 103, 112], "eigenfunct": [41, 92, 96, 97, 103, 108], "eigenpair": 103, "eigensi": 121, "eigenspac": 103, "eigensystem": [27, 64, 115], "eigenvalu": [0, 41, 47, 48, 51, 64, 84, 92, 96, 97, 100, 103, 106, 108, 109, 110, 112], "eigenvalues_precondition": [42, 47, 63, 81, 83, 106, 108, 109, 110], "eigenvector": [48, 51, 103, 112], "eigh": 115, "either": [9, 20, 24, 35, 46, 76, 80, 87, 89, 121], "el": [8, 15, 37, 39, 108, 109], "elast": [60, 82], "elderberri": 20, "electr": [0, 76], "electromagnet": 61, "electrostat": 76, "element": [0, 1, 2, 3, 4, 6, 7, 24, 26, 27, 29, 33, 34, 36, 37, 38, 39, 40, 48, 52, 55, 56, 60, 61, 63, 64, 65, 67, 68, 69, 72, 73, 74, 75, 83, 84, 87, 91, 100, 101, 103, 105, 106, 108, 109, 112, 113, 118, 121], "element_boundari": [3, 4, 5, 6, 7, 17, 18, 19, 34, 62, 67, 69, 70, 91, 116, 121], "element_typ": [10, 76], "element_vb": [1, 121], "element_wis": [33, 35], "elementari": [100, 102], "elementid": [10, 11, 33, 35, 76], "elementrang": 76, "elements2d": 15, "elementtransform": 76, "elimin": [37, 52, 56, 76, 81], "eliminate_hidden": [18, 19], "eliminate_intern": 91, "ellipt": [24, 28, 37, 41, 45, 46, 53, 54, 64, 65, 76, 81, 91, 113], "els": [1, 12, 15, 18, 19, 20, 22, 27, 37, 38, 39, 41, 42, 63, 85, 108, 109, 119, 121], "elsewher": 76, "em": 79, "emb": [38, 39, 101, 105], "emb_p": [119, 120], "embed": [17, 22, 25, 83, 85, 100, 101, 105], "embeddedmatrix": 120, "embeddedtransposematrix": 120, "embu": 17, "embuhat": 17, "ement": 36, "emg": 83, "empti": [9, 105], "emptyset": [36, 100], "en": [14, 111], "enabl": [10, 46, 76], "encod": 121, "encodedata": 121, "end": [1, 5, 6, 8, 9, 12, 13, 16, 24, 28, 29, 31, 33, 34, 35, 36, 37, 38, 40, 41, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 73, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 90, 91, 92, 97, 98, 99, 100, 101, 102, 104, 105, 106, 109, 111, 112, 113, 114, 115, 121], "energi": [34, 35, 37, 45, 50, 51, 52, 66, 69, 70, 83, 118, 121], "enforc": [38, 40, 55, 61], "engin": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 34, 46], "enlarg": 109, "enough": [54, 61, 69, 76, 83, 106], "enrichr": 0, "ensur": 33, "enter": [25, 57, 76, 78], "entiti": 110, "entri": [1, 12, 13, 24, 51, 52, 61, 66, 76, 105], "enumer": [15, 23, 37, 38, 39, 42, 51, 80, 108], "enumerateglob": 23, "envelop": [119, 120], "environ": [46, 73], "envolv": 34, "ep": [5, 83, 106], "epsseminorm": 103, "epub": [65, 105], "eqnarrai": [8, 12, 24, 34, 36, 37, 41, 47, 50, 51, 53, 54, 58, 60, 62, 63, 64, 65, 66, 67, 78, 79, 83, 84, 86, 87, 89, 90, 92, 97, 99, 100, 105, 106, 109, 111, 112, 113, 114, 115, 121], "equ_clement_bh": 36, "equ_clement_bh2": 36, "equ_leadingcoef": 79, "equ_projbased2": 79, "equ_threeterm": 79, "equal": [33, 34, 40, 78, 79, 101, 105], "equat": [2, 8, 13, 28, 31, 33, 34, 36, 38, 40, 41, 48, 52, 54, 55, 56, 61, 65, 73, 74, 77, 78, 79, 81, 82, 84, 89, 91, 100, 101, 105, 111, 118, 119, 120], "equflux": 34, "equilibr": 46, "equilibrium": [12, 34, 74], "equip": [83, 90], "equival": [9, 13, 27, 28, 31, 33, 34, 45, 46, 50, 51, 60, 65, 84, 87, 88, 100, 102, 103, 104, 106, 112], "er": 34, "ern": 46, "eror": 45, "err": [21, 22, 47, 48, 49, 51, 80, 81, 85, 88, 115], "err0": [47, 48, 49, 50, 51, 81], "err2": [50, 51], "errest": [33, 35], "errestdu": 35, "errhist": [47, 48, 50, 51], "errlist": 80, "errmax": [33, 35], "error": [0, 12, 21, 28, 29, 31, 45, 47, 48, 49, 50, 51, 64, 67, 71, 75, 76, 83, 91, 105, 108, 109, 115], "essenti": [8, 34, 55, 57, 65, 89, 98, 100, 103, 108], "establish": [40, 79], "estim": [12, 28, 31, 37, 47, 49, 50, 61, 64, 66, 67, 81, 83, 86, 97, 100, 102, 105, 106, 108, 109], "et": [10, 64, 76], "et_quad": 10, "et_segm": 10, "et_tet": 10, "eta": [33, 34, 35, 36, 100], "eta_t": [33, 35, 36], "etautau": 70, "eu": 100, "euclidean": 24, "euklidean": [32, 48, 75], "euler": [4, 111, 115], "eval": [0, 92, 96, 97, 103], "evalu": [0, 8, 9, 11, 12, 22, 28, 34, 35, 38, 39, 66, 79, 81, 82, 84, 85, 98, 101, 104, 108, 115, 121], "evec": [0, 92, 96, 97, 103, 115], "evei": 6, "even": [4, 12, 40, 64, 78, 79, 121], "everi": [1, 2, 6, 12, 15, 20, 24, 27, 28, 30, 40, 42, 47, 48, 49, 51, 61, 65, 66, 83, 84, 86, 91, 97, 100, 103, 105, 108, 112, 115], "everyon": 20, "everyth": 100, "everywher": 28, "evp": 97, "exact": [12, 13, 34, 64, 111, 121], "exactli": [7, 12, 40, 41, 48, 55, 56, 79, 105, 106, 115], "exampl": [6, 9, 12, 20, 27, 28, 31, 34, 41, 45, 46, 52, 53, 63, 73, 76, 82, 83, 85, 98, 103, 108, 110, 113, 114], "excel": 25, "except": [37, 88, 121], "excercis": [48, 51], "exchang": [1, 20, 40], "exchangeproc": 42, "execut": [20, 46, 75], "exercis": [0, 3, 4, 10, 12, 26, 28, 31, 34, 41, 45, 46, 51, 73, 74, 77, 78, 79, 80, 81, 90, 100, 106, 108, 110], "exist": [1, 2, 8, 9, 12, 27, 28, 29, 30, 31, 32, 33, 34, 48, 50, 54, 66, 79, 83, 90, 99, 100, 101, 102, 103, 104, 105, 106], "exp": [4, 5, 6, 43, 44, 64, 75, 112, 113, 114, 115, 116, 118, 119, 121], "expand": [41, 47, 48, 64, 76, 81, 100, 102], "expans": [13, 41, 63, 112, 113], "expect": [20, 33, 106], "expens": [63, 109, 112], "experi": [25, 46, 47, 51, 72, 77, 78, 84, 97, 106, 108], "experiment": [25, 48], "explain": [46, 72, 75, 100], "explan": [46, 65], "explicit": [4, 5, 12, 30, 34, 45, 100, 109, 118, 121], "explicitli": [38, 49, 84, 105, 109], "explizit": 5, "explor": 110, "exponenti": [45, 79, 112], "exponentialpropag": 115, "export": 20, "express": [9, 10, 12, 30, 62, 65, 76, 102, 105, 113, 114], "exproc": 42, "ext": 66, "extend": [0, 10, 12, 14, 27, 41, 53, 73, 80, 90, 98, 100, 101, 102, 112], "extendrec": 66, "extens": [6, 12, 23, 27, 37, 45, 46, 63, 79, 91], "extent": 112, "extern": 46, "extra": [31, 55, 61, 79, 91, 110], "extract": [21, 101, 105], "extrem": [51, 110], "f": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 18, 21, 22, 23, 24, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 67, 69, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 100, 102, 104, 108, 110, 111, 112, 113, 114, 121], "f2": 86, "f_": [13, 24, 36, 53, 62, 80], "f_0": 53, "f_1": 31, "f_2": 31, "f_d": 73, "f_e": 62, "f_f": 73, "f_h": 12, "f_i": [13, 24], "f_j": [13, 76], "f_t": [13, 24, 34, 62], "fa": 110, "fac": 47, "face": [0, 1, 6, 7, 8, 9, 10, 12, 16, 18, 19, 22, 33, 34, 35, 54, 55, 62, 74, 76, 77, 78, 80, 81, 85, 88, 89, 91, 103, 108, 110, 118, 121], "facedof": 76, "facet": [1, 2, 3, 6, 62, 76, 108, 121], "facetfespac": [1, 5, 34, 91, 121], "facetspac": 3, "facetvari": 17, "fact": 48, "factor": [8, 28, 31, 48, 58, 63, 64, 66, 70, 76, 79, 80, 83, 84, 97, 98, 100, 101, 108, 109, 121], "fail": 58, "falk": [60, 83], "fall": 112, "fals": [0, 1, 4, 5, 10, 18, 19, 22, 38, 39, 47, 49, 55, 66, 74, 76, 77, 80, 82, 83, 85, 96, 97, 103, 109, 110, 116, 118, 119, 120, 121], "famili": [8, 88, 100], "famou": [84, 86], "far": 77, "fast": [5, 46, 50, 100, 103, 112], "faster": [8, 112, 121], "faustmann": 46, "fb": 51, "fbg": 51, "fc": 20, "fcen": 119, "fd": 73, "fdof": 76, "fe": [0, 1, 2, 4, 5, 6, 10, 11, 12, 14, 15, 21, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 63, 64, 66, 70, 73, 74, 76, 77, 78, 80, 82, 83, 89, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 118, 121], "feasibl": [34, 47, 63, 105, 108], "featur": [48, 55], "feb": 115, "fem": [10, 34, 46, 76, 79, 110], "fes1": [1, 5, 70], "fes2": [1, 5, 70], "fes3": 70, "fes_p": 121, "fes_pf": 121, "fes_pt": 121, "fes_tr": 121, "fes_u": 121, "fesc": 116, "fescurl": 17, "fesdom": 109, "fesflux": [33, 34, 35], "fesh1": 12, "fesi": [37, 38, 39], "fesl2": 12, "feslam": [38, 39, 55, 84], "feslami": [38, 39], "fesp": 56, "fespac": [3, 10, 12, 62, 67, 69, 70, 76, 121], "fespaceelementrang": [10, 76], "fesstress": 74, "festr": 116, "fesu": [55, 84], "fesvertex": [37, 39], "feti": [37, 45, 46], "few": [0, 25, 40, 46, 47, 48, 79, 83, 84, 97, 103], "ff": [73, 80], "fg": 51, "fictiti": [37, 83, 105], "field": [0, 6, 12, 28, 39, 40, 54, 57, 58, 62, 70, 73, 76, 82, 83, 84], "fig": [20, 47], "figsiz": 47, "figur": 112, "file": [76, 121], "filenam": 76, "filter": 86, "final": [6, 10, 28, 29, 47, 49, 53, 54, 66, 76, 79, 81, 97, 98, 99, 102, 105, 108, 109], "finalizeupd": [10, 76], "find": [0, 3, 4, 6, 8, 12, 13, 28, 30, 31, 34, 35, 40, 41, 47, 48, 53, 54, 55, 57, 58, 60, 61, 62, 64, 67, 69, 73, 74, 75, 76, 78, 82, 83, 84, 86, 89, 91, 100, 101, 102, 103, 104, 105, 108, 110, 112, 115], "fine": [83, 109], "finer": 118, "finest": 65, "finit": [0, 3, 6, 24, 26, 27, 28, 29, 31, 33, 34, 36, 37, 38, 48, 52, 55, 56, 60, 63, 64, 65, 68, 69, 72, 73, 74, 75, 83, 84, 87, 99, 100, 101, 103, 105, 106, 108, 109, 113, 118], "finitecyl": 70, "first": [6, 8, 10, 13, 25, 27, 29, 32, 33, 36, 38, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 61, 62, 64, 65, 70, 76, 79, 80, 81, 82, 83, 84, 88, 89, 91, 97, 98, 100, 101, 102, 103, 104, 106, 108, 109, 112, 114, 115, 118, 121], "fit": [14, 62], "fix": [0, 10, 26, 28, 32, 39, 40, 48, 58, 62, 74, 76, 79, 82, 97, 105, 106, 109], "flag": [6, 10, 46, 55, 76], "flat": 71, "flatten": 121, "flip": 100, "float": [39, 76], "float32": 121, "float64": 76, "flow": [6, 28, 84, 112], "fluid": [12, 58], "flux": [34, 35, 55, 57, 73, 78, 86, 88, 89], "fnum": 76, "focu": 98, "follow": [1, 6, 8, 10, 12, 31, 33, 34, 35, 36, 37, 40, 41, 46, 47, 51, 53, 54, 63, 64, 65, 66, 73, 79, 81, 83, 87, 88, 91, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 111, 121], "foral": [0, 8, 9, 12, 13, 27, 28, 29, 30, 31, 32, 34, 35, 36, 40, 41, 49, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 64, 65, 67, 69, 73, 75, 76, 79, 82, 83, 84, 86, 89, 90, 91, 92, 95, 98, 100, 101, 102, 103, 104, 105, 109, 112, 113, 114, 115], "forc": [12, 20, 27, 58, 74, 82], "forg": 20, "form": [0, 2, 4, 6, 8, 12, 13, 24, 27, 28, 29, 31, 34, 35, 37, 38, 39, 41, 42, 43, 46, 48, 51, 53, 54, 55, 56, 57, 58, 62, 68, 73, 75, 78, 79, 81, 82, 83, 84, 88, 89, 91, 97, 100, 101, 104, 105, 106, 109, 113, 121], "formal": [76, 87, 100, 112], "format": 52, "formul": [3, 8, 12, 28, 31, 34, 40, 45, 46, 56, 57, 61, 64, 67, 69, 84, 87, 91, 100, 101, 114], "formula": [47, 79, 90, 98, 112], "forth": 2, "fortin": [46, 61, 72], "forward": [51, 52, 63, 91, 112], "found": [6, 46, 50, 64, 72, 84, 101, 105, 110], "four": [33, 49], "fourier": [41, 65, 79, 100, 102, 104], "fourth": [45, 46], "frac": [0, 1, 2, 4, 5, 6, 8, 12, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 41, 42, 43, 47, 48, 49, 50, 51, 53, 54, 55, 56, 61, 64, 65, 66, 69, 73, 75, 76, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 90, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 118, 121], "fraction": [30, 84, 100], "frame": 59, "framework": [12, 24, 121], "free": [4, 7, 10, 15, 61, 62, 67, 69, 73, 76, 82, 83, 88, 119, 120, 121], "freedof": [0, 1, 2, 3, 5, 7, 10, 11, 12, 14, 18, 33, 34, 35, 37, 38, 39, 43, 44, 55, 56, 58, 62, 63, 67, 69, 73, 76, 78, 80, 82, 84, 89, 91, 92, 103, 106, 108, 109, 110, 118], "freedom": [9, 10, 34, 39, 54, 61, 62, 71, 73, 76, 77, 80, 88, 91], "frequenc": [43, 64, 65, 73], "friedrich": [45, 53, 101, 106, 109, 112], "from": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 96, 97, 98, 100, 101, 103, 105, 106, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121], "front": 48, "frontend": [46, 72], "fruit": 20, "ftau": 70, "fubini": [97, 100], "fulfil": [8, 28, 29, 31, 33, 36, 53, 100, 104], "full": [33, 48, 64, 65, 84, 100, 105], "fullb": [119, 120], "fulli": 3, "func": [75, 80, 88, 121], "func2": 121, "funcf": 76, "function": [0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 13, 14, 15, 20, 22, 27, 29, 32, 33, 34, 35, 36, 37, 38, 39, 41, 45, 46, 47, 49, 50, 53, 54, 55, 57, 58, 63, 64, 65, 66, 69, 73, 75, 78, 79, 83, 84, 85, 88, 89, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 110, 111, 113, 114, 115, 118, 121], "fundament": [28, 100, 102], "further": [30, 33, 72, 83, 100], "furthermor": [29, 31, 65, 79, 88, 100], "furthoer": 34, "fv": 55, "fvec": 82, "g": [4, 11, 13, 22, 24, 28, 31, 33, 38, 39, 41, 49, 51, 53, 54, 57, 62, 66, 73, 74, 79, 81, 82, 84, 85, 89, 91, 98, 100, 101, 104, 105, 110, 111, 112, 113, 118, 121], "g_e": 62, "g_i": 100, "g_k": 100, "g_n": 100, "g_t": 62, "g_v": 62, "galerkin": [3, 8, 12, 28, 31, 34, 35, 36, 54, 63, 65, 72, 76, 79, 105, 114, 115, 119, 120], "game": 87, "gamg": [14, 18, 19, 22, 23], "gamma": [0, 29, 41, 55, 61, 79, 81, 101, 106, 107, 111], "gamma1": 47, "gamma2": 47, "gamma_": [4, 6, 38, 40, 42, 90, 100], "gamma_1": [28, 30, 47, 51, 81, 105], "gamma_2": [28, 30, 47, 51, 81, 105], "gamma_b": 100, "gamma_d": [0, 2, 8, 12, 55, 56, 57, 73, 74, 83, 84, 89, 92, 95, 101, 103, 104, 107], "gamma_i": 100, "gamma_n": [57, 73, 74, 89, 91, 100, 104], "gamma_r": [43, 51, 73, 82, 104], "gammai": [0, 55], "gamman": 57, "gammao": [0, 55], "gaug": 83, "gauss": [0, 12, 63, 64, 76, 105, 110], "gaussian": 81, "gb": 40, "geever": 118, "gener": [8, 10, 12, 15, 23, 27, 28, 30, 31, 33, 46, 48, 51, 52, 53, 61, 64, 65, 73, 75, 76, 77, 79, 83, 84, 86, 97, 99, 101, 103, 111, 112, 115, 121], "generatemesh": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "genuin": 67, "geo": [37, 38, 39, 44, 62, 70, 74, 78, 82, 83, 108, 118, 121], "geom": 10, "geom2d": [5, 37, 38, 39, 42, 43, 44, 47, 62, 63, 67, 69, 82], "geom_fre": [116, 121], "geometr": [0, 9, 32, 40, 64, 65, 70, 76, 78, 108, 110], "geometri": [40, 75, 76, 77, 121], "geoparamcf": 76, "geq": [6, 8, 12, 27, 28, 30, 31, 32, 33, 34, 47, 51, 53, 54, 64, 66, 67, 79, 84, 87, 102, 104], "get": [0, 3, 10, 15, 24, 26, 33, 34, 36, 38, 40, 46, 47, 49, 51, 53, 54, 55, 56, 61, 64, 65, 66, 76, 79, 80, 84, 86, 88, 89, 90, 92, 100, 103, 105, 106, 109, 110, 121], "getarrai": 23, "getbbboundari": 76, "getbboundari": [39, 76], "getboundari": [0, 38, 39, 55, 76, 78, 108], "getcurveord": 76, "getdata": 121, "getdof": [10, 38, 39, 66, 76, 108, 118], "getdofnr": [10, 76, 88, 110], "getf": [10, 11, 76], "gethpelementlevel": 76, "getintegrationrul": 118, "getmateri": [0, 38, 39, 44, 55, 76, 78, 108], "getn": [15, 22, 33, 35, 76], "getoperatorinfo": [120, 121], "getord": [10, 76], "getparentel": 76, "getparentfac": 76, "getparentvertic": 76, "getpc": [14, 23], "getperiodicnodepair": 76, "getpmltrafo": 76, "getsubvector": 23, "gettrac": [10, 76], "gettracetran": [10, 76], "gettrafo": [11, 76], "gf": [0, 1, 17, 18, 19, 34, 76, 121], "gfaux": 17, "gfcoars": 108, "gfconst": 106, "gfdom": [108, 109], "gfe": 116, "gfequ": 34, "gfetr": 116, "gfflux": [33, 35], "gfh": 116, "gfhtr": 116, "gfi": 108, "gfl2": 15, "gflam": 84, "gfp": [22, 81, 85, 121], "gfpf": 121, "gfpt": 121, "gfsigma": [18, 19], "gfstab": [119, 120], "gfstress": 74, "gftot": [37, 38, 39], "gftr": 121, "gfu": [0, 1, 2, 4, 5, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 63, 64, 66, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 108, 110, 113, 114, 115, 118, 119, 120, 121], "gfu0": [82, 108], "gfu1": 82, "gfu2": 86, "gfucorr": 18, "gfudual": 35, "gfuhat": [0, 17, 18, 19], "gfuref": 115, "gfv": 114, "gfvi": 17, "gg": 56, "gil": 12, "girault": 101, "github": [10, 11, 25, 46, 80], "give": [2, 6, 12, 20, 24, 26, 30, 34, 46, 62, 73, 75, 79, 82, 84, 87, 88, 90, 91, 100, 103, 104, 110], "given": [4, 6, 10, 28, 30, 32, 34, 41, 46, 47, 49, 50, 51, 52, 53, 54, 58, 64, 65, 66, 73, 76, 83, 87, 100, 105, 106, 109, 112, 113], "global": [0, 8, 9, 10, 11, 13, 15, 23, 24, 34, 37, 38, 42, 76, 79, 108, 109, 121], "globalinterfacespac": 0, "globalord": [10, 76], "globcoupl": 42, "globcouplingmat": 42, "globinv": 42, "globnum": 23, "globschur": 42, "glu": [0, 7, 39], "glue": [33, 34, 35, 78, 89, 108, 118], "gmre": 84, "go": [6, 41, 53, 61, 83], "goal": [40, 46, 47, 51, 66, 83, 105], "goe": 21, "good": [3, 8, 28, 30, 33, 35, 47, 50, 52, 56, 60, 66, 81, 86, 106, 109], "gopalakrishnan": 1, "got": [15, 20, 25, 56], "gov": 23, "grad": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 21, 22, 23, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 58, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 118, 121], "grade": 118, "gradient": [0, 13, 21, 33, 36, 37, 38, 45, 46, 52, 62, 69, 73, 77, 78, 82, 83, 84, 88, 100, 103, 121], "gradu": 70, "gradv": 70, "gram": [49, 115], "grape": 20, "graph": [15, 40, 52, 99], "grate": 45, "graviti": 75, "green": 35, "grew": 37, "grid": [10, 15, 40, 42, 47, 63, 64, 76, 112], "grid_siz": [4, 6, 12, 78, 83, 88], "gridfunct": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "grote": 118, "group": [20, 76], "grow": [8, 48, 103, 109, 112], "gtild": 84, "guarante": [51, 53], "gudi": 3, "guermond": 46, "gui": [44, 62, 67, 69, 70, 74], "guid": [33, 121], "h": [0, 1, 2, 3, 5, 8, 12, 17, 22, 30, 34, 35, 36, 38, 40, 42, 45, 46, 48, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 73, 74, 75, 77, 78, 79, 83, 84, 85, 86, 89, 91, 92, 95, 97, 98, 99, 102, 103, 104, 107, 108, 109, 112, 113, 116, 120, 121], "h0": 8, "h1": [0, 3, 10, 11, 12, 14, 15, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 58, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 84, 85, 89, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 118], "h1amg": 85, "h1lumpingfespac": 118, "h1norm": [96, 97, 103], "h1seminorm": [92, 96, 103], "h_": [83, 103], "h_0": [30, 34, 35, 41, 53, 58, 76, 100, 101, 104, 106, 109, 113], "h_1": 108, "h_e": [12, 34, 36], "h_i": 109, "h_k": [65, 66], "h_l": [63, 64, 65, 66], "h_t": [8, 12, 34, 36], "h_x": 71, "h_y": 71, "ha": [0, 2, 9, 12, 13, 25, 28, 29, 31, 33, 34, 40, 41, 51, 53, 63, 79, 82, 84, 86, 88, 91, 97, 98, 99, 100, 101, 104, 105, 106, 108, 111, 112, 114], "haas": 24, "hackbusch": [63, 64], "had": 73, "halv": 48, "hand": [0, 8, 10, 11, 21, 23, 24, 31, 34, 35, 36, 37, 41, 53, 64, 65, 73, 76, 82, 83, 89, 91, 98, 101, 102, 104, 105, 112, 121], "happen": [1, 86, 105], "hard": [40, 43, 105], "harmonic_extens": 91, "harmonic_extension_tran": 91, "hat": [0, 9, 13, 30, 34, 36, 51, 63, 66, 76, 81, 82, 83, 100, 106, 112, 121], "have": [1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 20, 29, 31, 36, 37, 38, 39, 40, 41, 46, 47, 48, 49, 51, 53, 54, 55, 58, 59, 61, 62, 63, 64, 65, 66, 69, 73, 76, 77, 79, 81, 82, 83, 84, 89, 90, 97, 99, 100, 101, 103, 104, 105, 106, 108, 109, 112, 118], "hcurl": [17, 62, 69, 70, 83], "hcurldiv": [18, 19], "hd": 8, "hdg": [2, 5, 7], "hdiv": [3, 7, 17, 18, 19, 33, 34, 35, 57, 86, 88, 89, 91], "hdivdiv": [62, 67, 69], "hdivdivsurfac": 70, "header_vis": 47, "heat": [45, 46, 73, 76, 78, 112, 114], "height": [23, 47, 109], "hellan": 45, "helling": [45, 62], "hello": 20, "help": [10, 19, 21, 26, 76, 85, 103, 121], "here": [1, 2, 4, 6, 10, 12, 24, 34, 40, 49, 56, 72, 73, 76, 88, 91, 100, 102, 113], "hermann": 105, "hermit": 9, "herrmann": 45, "hess": [3, 34], "hessenn": 3, "hessian": [3, 50, 84], "hesthaven": 121, "heun": 111, "heurist": 35, "hexhedra": 9, "hh": 108, "hhj": [45, 67], "hht": 45, "hidden_dof": [10, 18, 19, 76], "hidealldof": [10, 76], "hierarch": [63, 76, 88], "hierarchi": 63, "high": [4, 10, 24, 42, 53, 64, 76, 78, 88, 115], "higher": [4, 5, 6, 8, 12, 73, 76, 91, 100, 118, 121], "highest_order_dc": 1, "highli": [0, 33, 64, 78, 121], "hilbert": [8, 12, 27, 28, 29, 30, 31, 32, 36, 53, 54, 56, 65, 76, 79, 83, 99, 100, 101, 104, 113], "hint": [30, 102, 108], "hinv": 42, "hist": [33, 35, 49], "histori": 112, "hline": 111, "hmat": 42, "hochbruck": 115, "hochsteg": 25, "hold": [1, 8, 9, 12, 27, 28, 29, 31, 32, 33, 34, 41, 49, 50, 51, 53, 63, 64, 65, 66, 74, 76, 79, 81, 83, 90, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 109, 112], "hole": [35, 118], "holidai": 26, "holmholtz": 43, "home": 25, "homogen": [0, 33, 36, 41, 55, 73, 76, 82, 115], "honeydew": 20, "hood": 58, "hook": 74, "hope": [33, 103], "how": [0, 10, 20, 23, 30, 40, 47, 48, 56, 73, 77, 102], "how_to": 75, "howev": [1, 6, 41, 48, 51, 56, 57, 63, 64, 65, 82, 83, 84, 100, 103, 109, 118], "howto_linalg": 75, "howto_numpi": 75, "hp": [8, 46, 76, 103], "hpp": [10, 80], "hpref": 8, "html": [14, 25, 52, 75], "http": [14, 20, 23, 25, 46, 52, 65, 75, 105, 111, 118, 119, 120], "hu": 67, "huang": 67, "hub": [14, 15, 20, 21, 22, 23, 24], "huge": 121, "hv": [42, 48, 51, 81, 116, 121], "hv1": 42, "hv1glob": 42, "hv2": [42, 48, 51], "hv2glob": 42, "hv3": 51, "hvstab": 119, "hx": 63, "hy": 109, "hybrid": [45, 46, 54, 67, 72], "hyperbol": [12, 114], "hypr": 20, "i": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112, 113, 114, 115, 118, 119, 120, 121], "i_": [8, 9, 88, 105], "i_1": [8, 41], "i_2": 41, "i_h": [8, 12, 36, 54, 67, 68, 75, 83, 86, 109], "i_m": 8, "i_p": 79, "i_t": [8, 9, 12, 88], "id": [0, 8, 10, 14, 15, 20, 21, 23, 24, 42, 55, 70, 74, 76, 79, 100, 101, 105], "idea": [34, 36, 37, 40, 49, 65, 81, 91, 100, 101, 115, 118], "ideal": [51, 105], "ident": [48, 79, 81, 115], "identif": [10, 76], "identifi": [21, 24, 34, 38, 98, 105], "identitymatrix": [38, 39, 47, 82], "idiag": 108, "idnr": 76, "ifem": [25, 46, 72], "iff": [90, 109, 112], "ifpo": [4, 5, 6, 108], "ignor": 100, "ii": [34, 51, 105], "iii": [34, 46], "ij": [9, 38, 40, 42, 51, 79, 90, 100, 111], "il": 111, "ill": 115, "imagin": 55, "imaginari": 112, "imex": 5, "immedi": [31, 37, 49, 84, 101, 105, 106, 108, 109, 111], "impi_rt": 20, "implement": [0, 4, 12, 20, 26, 39, 46, 63, 66, 88, 105, 109, 110, 112, 115], "impli": [29, 30, 31, 32, 53, 55, 79, 83, 84, 99, 100, 101, 105, 106], "implicit": [5, 115], "import": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 28, 33, 34, 35, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 99, 103, 106, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121], "improv": [28, 31, 33, 46, 49, 51, 64, 65, 83, 92, 100, 108, 110, 111, 115], "includ": [10, 20, 21, 61, 73, 76, 78, 100, 110, 113], "inclus": [12, 67], "incompress": [12, 28, 58, 84], "inconsist": [38, 39], "increas": [8, 49, 64, 70, 79, 103, 114], "increment": [47, 113], "ind": 23, "inde": [33, 41, 79, 100, 101, 104, 105], "indefinit": 81, "independ": [0, 9, 12, 27, 40, 64, 79, 83, 100, 102, 106, 109, 121], "index": [8, 47, 75, 98, 121], "indexset": 23, "indi": 42, "indic": [23, 76, 84, 106], "individu": [8, 10, 40, 65, 74, 76, 100, 105, 108, 111], "indj": 42, "induc": [0, 28, 84, 106], "induct": [47, 64], "inequ": [12, 27, 30, 33, 53, 54, 64, 66, 71, 79, 81, 83, 86, 100, 101, 105, 106, 109], "inexact": [12, 82], "inf": [6, 45, 46, 53, 54, 86], "inf_": [8, 12, 28, 30, 31, 32, 37, 41, 48, 53, 54, 62, 65, 67, 79, 83, 100, 101, 105, 106, 108, 109], "infimum": [28, 83, 106], "infinit": [76, 87, 100, 101], "inflow": [6, 16, 18, 19], "inform": [8, 40], "infti": [41, 65, 79, 90, 98, 99, 100, 101, 102, 104, 109, 112], "inherit": [10, 28, 54, 76, 112], "initi": [4, 33, 47, 48, 112, 113, 114, 115], "inject": [6, 30, 31, 100], "inlet": [7, 12, 16, 18, 19, 22, 81, 85], "inlin": [64, 115], "inner": [0, 3, 12, 21, 27, 28, 29, 33, 35, 36, 48, 49, 51, 52, 55, 64, 65, 76, 79, 80, 83, 84, 98, 99, 101, 102, 103], "inner_solv": 91, "innerdof": [38, 39], "innerproduct": [3, 7, 12, 18, 19, 21, 22, 23, 24, 35, 42, 47, 49, 50, 51, 56, 58, 62, 66, 67, 69, 70, 74, 75, 81, 82, 85, 103, 106, 108, 115], "input": [10, 21, 33, 42, 64, 76], "insert": [8, 12, 36, 47, 56, 79, 84, 105, 108, 113, 114], "insid": [76, 100, 108, 109], "inspect": [76, 121], "instabl": 8, "instal": [25, 72, 103], "instanc": [10, 20, 76, 80], "instationari": [45, 46], "instead": [28, 30, 33, 34, 46, 47, 54, 55, 56, 76, 81, 82, 83, 100], "institut": 46, "insul": 73, "insur": 31, "int": [3, 6, 10, 12, 27, 28, 34, 35, 39, 41, 42, 53, 55, 56, 57, 58, 60, 61, 62, 67, 69, 73, 76, 79, 83, 84, 86, 87, 89, 91, 99, 102, 107, 113, 114, 118], "int32": 23, "int_": [0, 1, 2, 3, 4, 6, 8, 11, 12, 13, 28, 29, 30, 34, 35, 36, 38, 40, 41, 42, 43, 51, 53, 54, 55, 56, 57, 62, 65, 67, 70, 73, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 87, 88, 89, 90, 91, 96, 97, 98, 99, 100, 101, 102, 104, 106, 107, 111, 112, 113, 121], "int_0": [30, 41, 65, 79, 92, 97, 98, 100, 111], "int_a": 112, "int_i": [41, 79], "int_t": [1, 2, 6, 11, 12, 13, 34, 36, 54, 62, 67, 79, 86, 87, 88, 91, 112, 121], "int_x": 102, "integ": [55, 99], "integr": [0, 1, 2, 6, 8, 10, 12, 13, 15, 24, 33, 34, 35, 36, 45, 54, 55, 57, 65, 75, 76, 79, 80, 84, 88, 89, 90, 91, 92, 96, 97, 98, 103, 106, 111, 113, 118, 121], "integrand": [13, 79], "integrationrul": [76, 118], "intelmpi": 20, "intent": 101, "inter": [38, 39], "interact": [37, 47], "interconnect": 38, "interest": [8, 35, 51, 56, 69, 100], "interfac": [15, 20, 23, 25, 33, 38, 40, 42, 44, 45, 46, 61, 76, 78, 90, 100, 107, 108], "interface_dof": [18, 19], "interior": [1, 2, 66, 104, 118], "interleav": 72, "intern": [1, 2, 38, 69, 73, 74, 76], "interplai": 64, "interpol": [8, 9, 12, 17, 33, 36, 46, 54, 56, 65, 66, 73, 74, 75, 76, 83, 86, 88, 109, 118, 121], "interpret": [24, 35, 51, 62, 84], "intersect": [46, 99, 101], "intertest": [38, 39], "intertri": [38, 39], "interv": [9, 10, 41, 47, 76, 98, 100, 102, 111, 112, 113], "intord": [0, 55], "intpoint": 121, "intrang": [10, 18, 19, 76, 109], "intro": 25, "introduc": [0, 1, 2, 39, 47, 51, 53, 55, 56, 57, 61, 64, 79, 83, 88, 89, 100, 104, 111, 112, 114], "introduct": 72, "intrul": 118, "intuit": 65, "inv": [1, 2, 5, 14, 21, 38, 39, 42, 44, 63, 66, 73, 74, 77, 83, 108, 110, 121], "inva": [18, 19], "inva0": 108, "invent": 37, "invers": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 17, 18, 22, 33, 34, 35, 37, 38, 39, 42, 43, 44, 51, 52, 55, 56, 57, 58, 62, 63, 65, 66, 67, 69, 73, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 91, 92, 96, 97, 105, 106, 108, 109, 113, 114, 115, 116, 118, 121], "invert": [4, 8, 9, 37, 39, 56, 63, 73, 83, 91, 100, 108, 109, 121], "investig": 97, "invglob": 42, "invi": [108, 109], "invloc": 42, "invm": [4, 10, 76], "invmass": 116, "invmassbnd": [38, 39], "invmassp": 121, "invmassu": 121, "involv": [1, 2, 28, 63, 83], "invp": [119, 120, 121], "invstok": 7, "invu": [119, 120, 121], "io": [14, 25, 46], "ip": 24, "ipynb": [25, 46], "ipyparallel": [14, 15, 16, 18, 19, 21, 22, 23, 24, 42], "ipywidget": 47, "irregular": 33, "is_complex": [10, 76], "iset": 23, "isinst": 121, "isomorph": [28, 53, 105], "isomorphismu": 105, "isotrop": [10, 76], "issu": [26, 46, 60, 78, 89, 115], "iter": [10, 14, 22, 28, 37, 38, 39, 49, 50, 63, 65, 70, 76, 81, 82, 83, 85, 91, 105, 108, 110], "iteraton": 21, "its": [3, 8, 9, 10, 15, 20, 23, 27, 28, 32, 33, 34, 36, 40, 41, 46, 48, 50, 51, 55, 64, 65, 76, 83, 88, 90, 99, 100, 103, 105, 109, 115], "itself": [8, 15, 20, 105, 114], "itutori": 11, "j": [1, 5, 9, 13, 20, 30, 31, 34, 37, 38, 39, 42, 46, 48, 49, 51, 64, 67, 76, 79, 80, 83, 88, 99, 100, 105, 108, 109, 111, 112, 113, 114, 115, 118, 119, 120, 121], "j_1": 8, "j_m": 8, "j_t": [9, 106], "j_v": 28, "jaboci": 76, "jac": [105, 106], "jacobi": [13, 21, 47, 63, 64, 65, 79, 82, 83, 106, 110], "jacobian": [88, 112, 121], "ji": [13, 105, 111], "jinchao": [65, 105], "joachim": [25, 45, 46, 72], "johnson": [45, 46], "joint": 70, "joli": 118, "jschoeberl": [25, 46], "jump": [2, 34, 36, 87, 91, 121], "jumpdn": 3, "jupyt": [20, 25, 46, 72], "jupyterhub": 46, "jupyterlite_ngsolv": 25, "just": [2, 24, 25, 39, 60, 66, 73, 76, 84, 87, 121], "k": [1, 8, 13, 27, 28, 34, 41, 47, 48, 49, 50, 51, 58, 65, 66, 67, 79, 81, 84, 86, 87, 88, 91, 98, 99, 100, 101, 102, 104, 112], "kapidani": [119, 120], "kappa": [47, 48, 49, 63, 65, 83, 84, 106, 108, 109], "kappa_a": 105, "karush": [53, 84], "keep": [0, 15, 39, 55, 112], "kei": 12, "kept": 15, "ker": [27, 29, 82, 83, 101], "kernel": [27, 36, 40, 53, 54, 67, 82, 100, 101, 103, 106], "keyboardinterrupt": [119, 121], "keyword": [10, 76], "kind": [20, 41, 47, 73, 75, 79, 83, 89, 113], "kink": 76, "kirchhoff": [3, 69], "kkt": [53, 84], "kl": [79, 104], "know": [15, 20, 23, 33, 40, 41, 66, 76, 101, 105, 118], "knowledg": 8, "known": [28, 33, 46, 49, 61, 67, 69, 76, 91, 112, 113], "korn": 71, "krendl": 67, "krylov": [14, 84], "krylovsolv": 18, "krylovspac": [14, 18, 19, 21, 22, 38, 39, 42, 63, 74, 77, 82, 83, 84, 85, 108, 110], "ksp": [14, 23], "ksp_monitor": 18, "ksp_rtol": 18, "ksp_type": 18, "kuhn": [53, 84], "kutta": [4, 5, 45], "kutta_method": 111, "kwarg": [10, 55, 76, 121], "kx": [41, 104], "l": [8, 11, 12, 14, 15, 18, 19, 23, 27, 28, 29, 31, 46, 47, 51, 53, 63, 64, 65, 66, 68, 77, 79, 83, 84, 86, 89, 99, 100, 101, 103, 104, 105, 109, 111, 112, 118], "l2": [1, 2, 4, 5, 6, 7, 10, 12, 15, 18, 19, 34, 56, 57, 58, 76, 81, 86, 89, 91, 108, 109, 121], "l2gammanorm": 103, "l2norm": [92, 96, 103], "l2space": [10, 76], "l_": [80, 98, 99, 100, 101, 104, 109], "l_1": [98, 99], "l_2": [1, 4, 6, 12, 27, 28, 30, 31, 33, 34, 36, 40, 41, 45, 46, 53, 55, 56, 57, 58, 60, 61, 62, 64, 65, 66, 67, 69, 71, 75, 76, 83, 86, 89, 90, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 108, 109, 113], "l_p": 99, "l_u": 29, "la": [10, 42, 47, 63, 76, 81, 83, 106, 108, 109, 110], "lab": [25, 46], "label": [34, 36, 47, 79, 108], "lagrang": [8, 9, 38, 55, 61, 82, 83, 84, 91, 105], "lagrangian": [9, 45, 53, 68], "lam": [33, 34, 35, 38, 39, 42, 47, 55, 56, 63, 74, 78, 81, 83, 84, 89, 103, 106, 109, 110, 115, 121], "lam1": 92, "lam2": 106, "lambda": [0, 1, 13, 27, 34, 35, 38, 41, 47, 48, 49, 51, 55, 56, 64, 74, 78, 84, 86, 89, 92, 96, 97, 100, 101, 103, 105, 112, 121], "lambda_": [28, 34, 40, 80, 108], "lambda_0": [80, 100], "lambda_1": [27, 28, 48, 80, 103], "lambda_2": [28, 48, 55, 80, 103], "lambda_i": [47, 48, 51, 64, 80, 112], "lambda_j": 48, "lambda_k": [41, 100], "lambda_l": 78, "lambda_n": [27, 48], "lambda_r": 78, "lambda_t": 34, "lami": 108, "lammax": [22, 63, 81, 83, 85, 106, 109, 110], "lammin": [22, 63, 81, 83, 85, 103, 106, 109, 110], "lanczo": 47, "langer": 24, "languag": [10, 20], "laplac": [41, 64, 108], "larg": [0, 1, 2, 8, 30, 31, 33, 51, 52, 56, 64, 76, 77, 78, 83, 108, 112, 118], "larger": [15, 37, 40, 55, 92, 100], "largest": [13, 33, 47, 48, 51, 97, 103, 106], "last": [8, 10, 34, 35, 40, 49, 51, 58, 76, 79, 84, 97, 109, 119, 120, 121], "latenc": 40, "later": [6, 12, 29, 48, 54, 76, 84, 100, 112], "latest": [14, 72, 75], "launch": 25, "launcher": [14, 15, 20, 21, 22, 23, 24], "law": 74, "lax": [28, 30, 53, 101, 104], "layer": [12, 108], "lazarov": 1, "lbb": [53, 61, 84, 87], "ldot": [8, 9, 13, 30, 34, 47, 48, 49, 50, 51, 63, 66, 76, 79, 90, 98, 100, 104, 105, 111, 112, 113, 115], "lead": [1, 2, 5, 6, 8, 9, 12, 28, 29, 39, 48, 52, 53, 56, 63, 67, 79, 81, 83, 84, 88, 89, 91, 97, 99, 105, 106, 108, 111, 112], "learn": [23, 25, 52], "least": [40, 50, 54, 110], "leav": 78, "lebesgu": 27, "lectur": [6, 10, 25, 46, 65, 72, 81, 105], "left": [1, 2, 3, 5, 6, 8, 10, 12, 14, 24, 28, 30, 31, 34, 36, 38, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 61, 62, 64, 65, 66, 68, 69, 70, 73, 76, 78, 79, 80, 81, 82, 83, 84, 89, 90, 91, 92, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 110, 112, 113, 114, 121], "leftdomain": 44, "leftrightarrow": [13, 27], "legend": 47, "legendr": [76, 80], "lehrenfeld": [1, 7, 72], "lemma": [8, 9, 27, 28, 31, 32, 36, 37, 45, 63, 64, 65, 79, 83, 100, 101, 106, 108, 109, 112], "lemma_l2est": 79, "lemma_ortho": 79, "lemma_polext": 79, "lemma_sturmliouvil": 79, "len": [15, 35, 38, 39, 42, 50, 66, 108, 118, 121], "length": [33, 100], "leq": [1, 8, 9, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 47, 48, 49, 50, 51, 53, 54, 61, 64, 65, 66, 67, 71, 79, 80, 81, 83, 84, 86, 90, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 112, 115], "less": [34, 73, 98], "let": [8, 12, 20, 25, 27, 28, 29, 30, 31, 32, 33, 36, 37, 40, 41, 47, 48, 49, 50, 55, 62, 63, 64, 65, 66, 76, 79, 81, 82, 83, 84, 86, 92, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 109, 110, 113], "level": [8, 40, 46, 54, 64, 76, 103], "leveldof": 63, "lewi": 112, "lf": [1, 34], "lfequ": 34, "lgmap": 23, "lh": 79, "li": 100, "lib": 121, "libngpi": 76, "librari": [10, 11, 15, 23, 25, 72, 80, 121], "lift": 37, "like": [1, 2, 10, 15, 20, 24, 38, 40, 46, 50, 51, 61, 64, 73, 74, 76, 91, 109], "lim_": [2, 99, 112], "limit": [27, 41, 56, 69, 97, 100, 106], "linalg": 115, "line": [8, 9, 31, 32, 33, 35, 47, 50, 58, 87, 119, 120, 121], "linear": [1, 2, 5, 8, 9, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 35, 45, 47, 49, 50, 51, 52, 53, 56, 57, 63, 69, 70, 73, 77, 82, 84, 88, 89, 98, 100, 101, 102, 104, 105, 109, 111, 113, 114, 115, 121], "linearform": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 17, 18, 21, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 67, 69, 73, 74, 76, 77, 78, 80, 82, 83, 84, 86, 89, 91, 96, 108, 110, 113, 114], "linearli": 9, "linerar": 82, "linestyl": 47, "link": 64, "linspac": [47, 64, 112], "linux": 20, "lion": 105, "liouvil": 79, "lipschitz": [28, 92, 95, 97, 99, 100, 101], "list": [10, 15, 20, 23, 37, 38, 39, 42, 47, 76, 92, 96, 97, 103, 106, 108, 109, 110], "list_of_rung": 111, "lite": [25, 46], "literatur": [34, 36, 64, 65, 105, 121], "live": [55, 76], "ll": [33, 112], "lm": 111, "load": [10, 11, 40, 70, 74, 76, 80, 82], "loadstep": 70, "lobpcg": [92, 103], "loc": [98, 99], "local": [8, 9, 10, 12, 13, 15, 21, 23, 24, 33, 34, 35, 36, 37, 40, 42, 45, 46, 59, 63, 64, 66, 76, 77, 79, 81, 83, 98, 99, 100, 109, 121], "local2glob": 23, "local_mat": [23, 42], "local_vec": [24, 42], "localdof": 118, "localhcf": 76, "localip": 24, "localpr": 63, "locat": 9, "lock": [69, 83], "locmat": 23, "loembed": [10, 76], "log": [33, 47, 48, 49, 50, 51, 65, 80, 115], "logic": 108, "login": 46, "logo": 25, "long": [25, 40, 108], "longrightarrow": [68, 86], "look": [10, 11, 24, 50, 56, 73, 74, 76, 83, 100, 103], "loop": [1, 2, 6, 33, 51], "lospac": [10, 76], "lot": 8, "low": [64, 83, 115], "low_order_spac": [10, 76], "lower": [33, 34, 40, 51, 81, 86, 89, 103, 108, 109, 111], "lowest": [10, 67, 76, 87, 88, 91, 103], "lshape": 33, "lsrc": [119, 120], "lu": 101, "lucki": 86, "lump": [12, 45, 46], "lv": 101, "ly": 104, "m": [4, 5, 8, 9, 12, 13, 15, 18, 19, 20, 25, 27, 32, 46, 48, 51, 64, 70, 72, 79, 82, 83, 84, 96, 100, 103, 104, 105, 106, 108, 111, 112, 113, 114, 115, 118, 121], "m_": [51, 64, 105], "m_0": 64, "m_l": [63, 64, 66], "m_p": 121, "m_u": 121, "m_x": 108, "m_y": 108, "machin": 55, "maco": 20, "magic": [20, 67], "magnet": 83, "magnitud": [40, 109], "mai": [1, 2, 8, 9, 12, 20, 25, 33, 34, 40, 46, 58, 79, 83, 103], "mail": 46, "main": [8, 34], "maintain": 73, "major": 79, "make": [0, 8, 20, 62, 76, 78, 100, 112], "makebfi": 11, "makechamf": 74, "makestructured2dmesh": [92, 96, 97], "maketwomaterialmesh": 33, "manag": [20, 46], "mani": [20, 23, 46, 56, 64, 76, 82, 84, 105], "manifold": 104, "manual": 39, "map": [0, 8, 13, 14, 23, 27, 28, 38, 47, 60, 67, 76, 83, 88, 92, 96, 97, 100, 101, 102, 104, 105, 106, 121], "mappedintegrationpoint": 76, "mapsto": [27, 30, 53, 63, 75, 82, 88, 97, 102, 105, 109], "maptoallel": 76, "marini": [34, 88], "mark": [73, 76, 108], "mark_surface_el": 76, "markers": [1, 2], "mask": [0, 39, 78, 109], "mass": [4, 10, 12, 46, 66, 76, 112, 113, 116, 121], "massbnd": [38, 39], "massint": [38, 39], "master": [1, 7, 15, 72], "master_nr": 76, "masterinvers": 18, "mat": [0, 1, 2, 3, 5, 6, 7, 10, 11, 12, 14, 17, 18, 19, 21, 22, 23, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "mat00": [38, 39, 108], "mat01": [38, 39, 108], "mat02": [38, 39, 108], "mat10": [38, 39, 108], "mat11": [38, 39, 108], "mat12": [38, 39, 108], "mat20": [38, 39, 108], "mat21": [38, 39, 108], "mat22": [38, 39, 108], "mata": [22, 85], "match": [76, 100, 111], "materi": [0, 10, 25, 33, 37, 38, 39, 45, 46, 55, 72, 74, 75, 76, 78, 108, 118], "materialcf": [0, 33, 34, 35, 76, 78, 83, 89], "math": [43, 44, 64, 83, 119], "mathbb": [6, 8, 9, 12, 13, 27, 28, 30, 31, 32, 35, 36, 37, 48, 49, 50, 51, 52, 58, 63, 66, 74, 75, 76, 78, 79, 81, 82, 84, 88, 92, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 111, 112, 115, 121], "mathbf": [41, 84, 106], "mathcal": [34, 49, 65, 68, 81, 105], "mathemat": [5, 34, 46, 76, 103], "mathematik": 83, "matlab": 25, "matplotlib": [1, 2, 33, 46, 47, 48, 49, 50, 51, 64, 80, 103, 112, 115], "matric": [10, 11, 12, 14, 38, 51, 52, 53, 60, 64, 66, 76, 77, 103, 105, 112, 113, 115, 121], "matrix": [0, 1, 3, 4, 8, 9, 10, 12, 13, 14, 21, 23, 27, 30, 37, 38, 39, 42, 48, 49, 50, 51, 52, 53, 56, 61, 62, 63, 64, 66, 67, 73, 76, 77, 81, 82, 83, 84, 91, 96, 105, 106, 108, 109, 111, 112, 113, 115, 118, 119, 120, 121], "matrixvalu": 74, "max": [4, 5, 6, 7, 12, 15, 16, 18, 19, 22, 28, 33, 34, 35, 42, 47, 48, 74, 78, 81, 84, 85, 89, 106, 108, 109, 119, 120], "max_": [8, 47, 48, 64, 99], "max_i": 48, "max_it": [14, 23], "maxh": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "maxim": [27, 64, 74, 103, 109], "maximum": [47, 48], "maxit": [18, 19, 22, 38, 39, 47, 49, 77, 81, 82, 83, 84, 85, 92, 96, 97, 103, 110, 121], "maxstep": [84, 91], "maxwel": [34, 119, 120], "mayb": 55, "mbox": [8, 9, 12, 13, 27, 28, 31, 33, 35, 53, 98, 100, 104], "mc": 23, "mdcomp": 108, "mean": [2, 12, 29, 31, 36, 40, 42, 46, 48, 51, 54, 56, 58, 60, 76, 79, 82, 84, 91, 100, 103, 105, 106, 108, 112, 113, 121], "meanvalmat": 96, "meanvalu": 96, "measur": [8, 12, 33, 50, 51, 92, 101, 104], "mechan": [34, 46, 56, 103], "mechanid": 45, "medium": 44, "melon": 20, "member": 15, "membership": 108, "membran": 70, "memori": [40, 77, 121], "mesh": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 16, 17, 18, 19, 21, 22, 23, 24, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 100, 103, 106, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "mesh_siz": [0, 1, 2, 3, 5, 7, 22, 85], "meshgrid": 112, "meshnod": 76, "meshnoderang": 76, "meshpoint": 76, "meshrot": [0, 55], "messag": 20, "methdo": 68, "method": [1, 4, 10, 28, 33, 34, 35, 37, 39, 40, 49, 56, 62, 64, 65, 66, 69, 70, 72, 74, 77, 79, 82, 84, 86, 89, 91, 101, 110, 115, 119, 120, 121], "meti": [15, 108], "mform": [113, 114, 115, 118], "mg": [45, 63], "mgprecondition": 63, "michael": 59, "mid": 12, "middl": 33, "midpoint": 9, "might": 46, "milgram": [28, 30, 53, 101, 104], "milk": 6, "mimic": [6, 87], "min": [4, 5, 6, 7, 8, 12, 15, 16, 18, 19, 22, 28, 33, 34, 42, 47, 74, 78, 81, 84, 85, 89, 103, 106, 119, 120], "min_": [30, 34, 47, 49, 50, 53, 65, 84, 86], "min_u": 84, "mindlin": 45, "mini": 20, "miniconda3": 20, "minim": [29, 32, 34, 37, 40, 41, 45, 46, 47, 48, 50, 60, 65, 79, 83, 100, 105, 112], "minimum": [32, 50, 103, 105], "minion_nr": 76, "minr": 84, "minut": [25, 46], "minv": 118, "minva": 118, "miracl": 49, "misfit": 48, "miss": [3, 40, 87], "mix": [12, 34, 56, 61, 67, 70, 72, 81, 83, 86, 87, 91], "mixedflux": 89, "mixedtemp": 89, "mixtur": 79, "ml": [45, 63], "mlextens": 66, "mlprecondition": 63, "mlpreconditioner2": 63, "mm": 105, "mmat": 118, "mmax": 115, "modal": 76, "mode": [62, 76], "model": [6, 58, 69, 70, 73, 76, 77, 78, 118], "moder": 99, "modern": 40, "modif": [12, 79], "modifi": [0, 41, 47, 48], "modul": [10, 14, 20, 76, 85], "mollifi": 90, "moment": 67, "momentum": 58, "mone": 66, "monoton": [48, 51], "more": [0, 1, 2, 5, 10, 31, 40, 49, 52, 63, 65, 66, 73, 76, 79, 98, 99, 100, 101, 105, 106, 109, 110, 112, 118, 121], "morlei": [9, 12, 67], "most": [47, 58, 99, 108, 111, 119, 120, 121], "motiv": [62, 100, 112], "motor": 0, "move": [8, 14, 50, 73, 89], "moveto": [0, 8, 33, 34, 35, 55, 78, 89, 108, 118, 121], "mp": 121, "mpi": [10, 14, 15, 16, 18, 19, 21, 22, 23, 24, 42, 46, 76], "mpi4pi": [15, 16, 18, 19, 21, 22, 24, 42, 46], "mpi_comm": 76, "mpiaij": 23, "mpich": 20, "mpienginesetlaunch": [14, 15, 20, 21, 22, 23, 24], "msg_level": [16, 18, 19], "msm": 105, "msmall": 115, "mstabinv": [119, 120], "mstar": [5, 113, 114, 115], "mstarinv": [113, 114, 115], "mtcurl": 17, "mu": [27, 34, 38, 39, 53, 55, 56, 74, 83, 84, 121], "mu0": 115, "mu_": [38, 40], "much": [33, 37, 40, 50, 67], "mulder": 118, "mult": [42, 63, 109], "multi": [10, 13, 40, 46, 63, 76, 98, 108], "multidim": [76, 108, 109, 121], "multigrid": [40, 76, 83, 84], "multilevel": [46, 83], "multipl": [34, 49, 51, 63, 79, 105], "multiplel": 15, "multipli": [1, 2, 6, 13, 52, 55, 57, 63, 66, 76, 80, 81, 82, 100, 105, 113], "multiscal": 83, "multistep": 45, "multivector": [42, 82, 108, 115], "multtran": 109, "must": [12, 33, 34, 40, 41, 53, 54, 58, 60, 77, 83, 84, 103, 111, 115], "mv": 108, "mx": [37, 38, 39, 66, 108], "my": [10, 37, 38, 39, 108], "myassembl": 11, "myassemblematrix": 11, "myassemblevector": 11, "mydiffop": 10, "myelement": 10, "myfe": 10, "myfespac": 10, "myhighorderfespac": 80, "myhoel": 80, "myhof": 80, "myhofespac": 80, "mylap": 11, "mylaplac": 11, "mymatrix": 11, "mymodul": 10, "myneumannintegr": 11, "mysourc": 11, "n": [0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 31, 32, 34, 36, 40, 41, 43, 44, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 62, 63, 65, 67, 68, 69, 70, 73, 74, 76, 78, 79, 80, 82, 84, 87, 89, 90, 91, 96, 97, 98, 99, 100, 103, 104, 106, 108, 112, 113, 115, 116, 118, 121], "n2p": [14, 22], "n4ngla18embeddingtranspose": 120, "n4ngla9embedding": 120, "n6ngcomp9applymass": 120, "n_": [8, 63, 108], "n_1": [2, 36, 55, 91], "n_2": [36, 55, 91], "n_cut": 108, "n_e": 3, "n_i": [47, 90, 100, 105], "n_j": [90, 100], "n_l": [63, 66, 121], "n_r": 121, "n_t": [9, 13, 106], "nabla": [0, 1, 2, 3, 5, 6, 11, 12, 13, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 49, 50, 51, 53, 55, 56, 57, 58, 61, 62, 65, 67, 68, 69, 70, 73, 74, 75, 76, 78, 79, 83, 84, 86, 87, 88, 89, 90, 92, 95, 96, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 112, 113, 121], "nabla_": 100, "nabla_h": 6, "nabla_t": [62, 67], "nabla_x": 100, "naiver": 101, "name": [0, 7, 8, 10, 12, 13, 16, 18, 19, 22, 31, 33, 34, 35, 38, 55, 62, 67, 69, 74, 75, 76, 77, 78, 81, 85, 89, 101, 103, 105, 106, 108, 118, 120], "nameerror": 120, "nano": 46, "natur": [8, 24, 36, 46, 55, 57, 58, 62, 89, 110], "navier": [5, 12, 58], "navierstok": [16, 18, 19], "navierstokessimpl": [16, 18, 19], "navstok": 16, "nb": 108, "nbel": 108, "nbextens": 46, "nc": [12, 109], "nd": 101, "ndarrai": 76, "ndglob": 23, "ndof": [1, 2, 10, 11, 15, 18, 19, 22, 23, 33, 35, 38, 39, 42, 55, 58, 62, 63, 67, 76, 77, 80, 83, 85, 88, 108, 109, 121], "ndof_p": 121, "ndof_u": 121, "ndofglob": [10, 15, 22, 23, 76], "ndofloc": 42, "ndom": 108, "ne": 76, "nearest": 8, "nearli": [62, 112], "necessari": [31, 37, 101, 112], "necessarili": [27, 28, 30, 61, 98, 105], "neck": 40, "nedelec": [62, 68, 69, 83], "nedg": [76, 88], "need": [1, 3, 8, 10, 20, 24, 34, 36, 38, 47, 48, 49, 50, 51, 53, 54, 57, 58, 70, 76, 83, 88, 90, 91, 97, 98, 101, 103, 109, 111, 113, 114, 121], "neg": [6, 29, 49, 50, 56, 84, 101, 104, 108, 112], "neglect": 8, "neighbor": [12, 33, 36, 40, 108], "neighbour": [38, 39, 79], "neighbourhood": [98, 100], "nel": 70, "nepomnyashchikh": 105, "neq": [9, 27, 28, 29, 31, 32, 36, 49, 51, 52, 79, 84, 98, 100, 101, 103, 109], "ner": 118, "nest": [63, 64, 65, 66, 83, 108, 109], "netgen": [0, 5, 7, 8, 12, 15, 16, 18, 19, 22, 33, 34, 35, 37, 38, 39, 42, 43, 44, 46, 47, 55, 62, 63, 67, 69, 70, 72, 74, 76, 77, 78, 81, 82, 83, 85, 89, 103, 108, 113, 114, 115, 118, 121], "network": [40, 52], "neumann": [1, 28, 40, 41, 55, 57, 73, 78, 89, 91, 104, 108], "neunteufel": [59, 70], "never": [24, 46], "new": [3, 4, 10, 25, 33, 36, 46, 47, 48, 49, 55, 56, 57, 61, 76, 91, 112, 113, 121], "new_scen": 121, "newer": 67, "newlin": 33, "newton": 70, "newtonminim": 70, "newtonsolv": 70, "next": [2, 6, 8, 10, 14, 23, 28, 29, 38, 48, 50, 53, 65, 66, 79, 81, 82, 87, 88, 98, 100, 105, 113], "nface": 76, "nfacet": 76, "ng": [10, 11, 72, 121], "ngcomp": [10, 76], "ngexcept": 58, "nglob": 23, "ngmesh": [14, 15, 22, 23, 76, 77, 83], "ngmg": [10, 76], "ngradu": 70, "ngradv": 70, "ngs2petsc": [14, 22], "ngs_element": 76, "ngs_object": [10, 76], "ngsglobal": [16, 18, 19], "ngshub_xx": 46, "ngsolv": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 117, 118, 119, 120, 121], "ngspetsc": [14, 18, 19, 22], "ngstd": [10, 76], "ni": [10, 13, 76], "nice": 108, "nitsch": [8, 45, 46, 64, 65, 79], "nn": [3, 67], "nnode": 76, "nodal": [8, 9, 10, 12, 13, 36, 66, 68, 76, 100, 109, 118, 121], "nodal_p1": 121, "node": [9, 10, 15, 33, 40, 73, 76, 110, 118], "node_typ": 76, "nodeid": [10, 76, 88], "non": [0, 1, 2, 5, 8, 9, 10, 28, 29, 31, 36, 37, 38, 53, 54, 56, 60, 62, 63, 64, 70, 73, 76, 84, 90, 91, 99, 100, 101, 104, 105, 106, 108, 109, 111, 112, 121], "nonassembl": [4, 5], "nonconform": 12, "none": [10, 15, 18, 20, 37, 38, 39, 47, 49, 63, 70, 76, 84, 108, 109, 121], "nonlinear": 45, "norm": [0, 1, 6, 12, 16, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 45, 46, 48, 50, 51, 52, 53, 62, 64, 65, 66, 74, 76, 79, 83, 84, 85, 87, 90, 96, 97, 99, 100, 104, 108, 109, 115, 116], "norm_natur": [14, 23], "normal": [0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 18, 19, 33, 34, 36, 40, 54, 55, 57, 60, 67, 69, 70, 84, 88, 89, 91, 92, 96, 97, 100, 103, 111, 116, 121], "normalfacetfespac": 3, "normtyp": [14, 23], "notat": [2, 28, 55, 63, 65, 73, 105, 109], "note": [3, 6, 20, 25, 31, 34, 35, 46, 48, 65, 72, 76, 79, 89, 103], "notebook": [20, 25, 26, 46, 72], "noth": [25, 73], "now": [0, 1, 5, 6, 8, 10, 12, 23, 24, 27, 28, 29, 30, 31, 34, 36, 41, 47, 51, 53, 54, 55, 56, 57, 62, 64, 65, 66, 69, 73, 76, 80, 81, 82, 83, 84, 86, 89, 90, 91, 100, 101, 103, 104, 106, 109, 110, 111, 112, 114], "np": [23, 47, 76, 112, 121], "nr": [15, 24, 37, 38, 39, 42, 108, 109], "nsurf": 70, "nsurfmat": 70, "nt": [62, 67], "nu": [7, 16, 18, 19, 74, 83], "null": [53, 82, 83, 99, 106], "num": [0, 15, 88, 92, 96, 97, 103, 108, 121], "number": [8, 10, 13, 15, 20, 23, 24, 36, 38, 40, 45, 46, 47, 48, 49, 63, 64, 65, 76, 77, 79, 83, 84, 91, 99, 103, 105, 106, 108, 109, 110], "numberspac": 34, "numer": [5, 12, 13, 36, 46, 49, 52, 58, 76, 83, 84, 86, 103, 106, 111, 112, 115, 121], "numerisch": 83, "numpd": [46, 65], "numpi": [23, 46, 47, 64, 75, 76, 112], "numset": [108, 118], "nv": [10, 76, 109], "nx": [92, 96, 97], "ny": [92, 96, 97], "nze": [1, 2, 118], "o": [1, 52, 63, 79, 81, 83, 84, 101, 105, 106, 108], "o_i": 99, "obj": 121, "object": [10, 14, 15, 20, 23, 76, 121], "observ": [0, 27, 31, 33, 34, 36, 47, 48, 50, 51, 55, 62, 69, 75, 79, 83, 84, 89, 90, 92, 98, 102, 103, 108, 109], "obtain": [8, 12, 13, 15, 27, 28, 31, 32, 33, 35, 38, 41, 47, 48, 49, 50, 51, 53, 55, 56, 58, 61, 63, 65, 66, 67, 76, 79, 81, 83, 84, 88, 89, 91, 97, 100, 101, 103, 106, 108, 109, 112, 113, 114, 115, 121], "obvious": 38, "occ": [0, 7, 8, 12, 16, 18, 19, 22, 33, 34, 35, 55, 74, 77, 78, 81, 85, 89, 103, 108, 113, 114, 115, 118, 121], "occgeometri": [0, 7, 8, 12, 16, 18, 19, 22, 33, 34, 35, 55, 74, 77, 78, 81, 85, 89, 103, 108, 118, 121], "occur": 33, "od": [112, 113, 114, 115, 121], "odd": 79, "off": [56, 100], "offer": 23, "offic": 25, "offici": 72, "often": [0, 12, 27, 28, 46, 47, 52, 63, 105, 108, 112, 113], "og": 121, "old": [33, 48, 113], "older": 67, "oldstyl": [10, 76], "omain": 37, "omega": [0, 1, 2, 4, 6, 8, 9, 11, 12, 13, 27, 28, 29, 30, 33, 34, 35, 36, 40, 41, 43, 44, 51, 53, 55, 57, 58, 61, 65, 66, 73, 74, 75, 76, 78, 84, 86, 87, 89, 90, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 112, 113, 114], "omega_": 36, "omega_1": [13, 41, 55, 90, 107], "omega_2": [41, 55, 90, 107], "omega_i": [0, 24, 38, 40, 42, 90, 100, 104, 108, 109], "omega_j": [40, 90, 100, 109], "omega_k": 13, "omega_m": 13, "omega_n": 90, "omega_t": [33, 36], "omega_v": 34, "omega_x": 36, "ompi": 20, "onc": [12, 73, 75], "one": [0, 1, 2, 6, 8, 9, 10, 12, 13, 15, 20, 24, 28, 30, 31, 32, 33, 34, 35, 36, 40, 42, 47, 48, 49, 50, 51, 52, 53, 54, 55, 58, 62, 63, 64, 65, 66, 76, 78, 79, 82, 83, 84, 88, 89, 99, 101, 104, 105, 106, 108, 109, 110, 111, 121], "ones": [9, 31, 64, 66, 73, 89, 108, 112], "onli": [0, 1, 6, 7, 8, 10, 12, 13, 15, 24, 27, 33, 36, 37, 38, 39, 40, 41, 48, 49, 52, 54, 55, 62, 64, 65, 70, 76, 79, 82, 83, 84, 98, 100, 106, 109, 112, 113, 121], "onlin": [25, 46], "onlyonc": 76, "onstraint": 37, "onto": [12, 27, 29, 31, 33, 38, 45, 46, 53, 66, 79, 83, 100, 101, 102, 105, 109], "oosterle": 63, "opcurl": [62, 69], "opdiv": [1, 4, 5, 6, 53, 62, 67, 68, 69, 78, 86, 87, 88, 89, 90, 91], "opdiv_t": 62, "open": [20, 46, 76, 98, 99, 102, 109], "openmpi": 20, "openmpi4": 20, "oper": [1, 2, 3, 4, 8, 9, 10, 12, 15, 20, 21, 27, 28, 30, 31, 32, 34, 40, 41, 42, 45, 48, 51, 52, 53, 56, 61, 63, 64, 65, 66, 76, 79, 83, 84, 86, 90, 96, 97, 98, 101, 103, 104, 105, 108, 109, 118], "operatornam": [7, 8, 12, 13, 24, 27, 28, 29, 31, 34, 38, 41, 46, 49, 53, 54, 56, 57, 58, 59, 61, 62, 63, 66, 74, 76, 82, 83, 84, 86, 89, 97, 98, 100, 101, 103, 104, 105, 106, 108, 109, 112, 121], "oplu": 31, "opposit": [21, 29, 33, 51, 55, 57], "opt": [47, 48, 50], "optic": 46, "optim": [8, 28, 31, 34, 40, 47, 50, 51, 63, 66, 77, 79, 81, 83, 86], "option": [3, 10, 52, 55, 76, 103], "order": [0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 33, 34, 35, 40, 43, 44, 47, 48, 49, 50, 51, 55, 56, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 96, 97, 99, 100, 101, 103, 105, 106, 108, 109, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121], "order_equ": 34, "order_polici": [10, 76], "orderfac": 55, "orderinn": [17, 18, 19], "ordinari": [111, 112, 113], "org": [20, 52, 65, 72, 75, 105, 111, 118, 119, 120], "orient": 3, "origin": [28, 30, 31, 32, 34, 37, 49], "orthobrick": 83, "orthogon": [4, 8, 12, 27, 28, 29, 31, 32, 34, 36, 41, 49, 53, 54, 64, 65, 83, 86, 100, 101, 102, 103, 105, 115, 121], "orthonorm": [41, 47, 100, 101], "oscil": [0, 12, 64, 112], "ostermann": 115, "ot": 60, "other": [0, 2, 4, 5, 6, 10, 15, 21, 23, 24, 31, 37, 46, 51, 52, 54, 55, 63, 65, 73, 75, 76, 80, 83, 103, 105, 106, 108, 121], "otherp": 15, "otherwis": [29, 34, 58, 98, 106], "our": [10, 23, 46, 53, 54, 80, 100], "out": [1, 3, 6, 25, 37, 47, 64, 77, 80, 83, 90, 98, 121], "outer": [0, 33, 35, 55, 74, 77, 89, 100], "outflow": [6, 16, 18, 19, 57], "outlet": [7, 12, 16, 18, 19, 22, 81, 85], "outlin": [65, 109], "outperform": 1, "output": [21, 42, 64], "outsid": [100, 108, 109, 112], "outward": 6, "over": [0, 2, 6, 8, 12, 13, 21, 27, 33, 47, 49, 54, 55, 65, 76, 83, 87, 88, 108, 109, 112, 113, 121], "overal": 63, "overcom": [33, 91], "overestim": [33, 34], "overlap": [36, 37, 63, 73, 83, 90, 100], "overlin": [9, 12, 27, 29, 36, 40, 84, 90, 92, 97, 98, 99, 100, 102, 107, 108, 109], "overload": [10, 76], "overrid": 76, "overwritten": [10, 76], "own": [10, 20, 46, 80], "p": [0, 1, 2, 7, 8, 9, 10, 22, 27, 32, 33, 34, 42, 47, 49, 50, 51, 53, 54, 56, 58, 75, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 91, 99, 101, 102, 103, 106, 109, 110, 118, 121], "p2": 86, "p209": 115, "p2n": 14, "p_": [32, 33, 49, 63, 65, 66, 79, 100, 101, 121], "p_0": [12, 34, 49, 79, 88], "p_1": [49, 76, 79, 105, 113], "p_2": [79, 105], "p_h": [33, 54, 56, 86], "p_i": [49, 76, 79, 80, 105, 113], "p_j": [49, 76, 79, 80, 113], "p_k": [49, 79, 86], "p_l": [63, 64, 65, 66, 79], "p_m": [79, 105], "p_n": [49, 76, 79, 113], "p_su": 32, "p_sv": 32, "packag": [10, 25, 46, 108, 121], "pad": [73, 108], "page": [7, 47, 49, 80, 88], "pai": 109, "paid": 109, "pair": [54, 55, 58, 100, 112], "pajetrac": 118, "pap": [49, 81], "paper": 1, "papu": 81, "parabol": [12, 111, 113], "parallel": [11, 14, 15, 20, 23, 24, 40, 76, 77, 121], "parallel_statu": 42, "paralleldof": [10, 23, 24, 42, 76], "parallelmatrix": 42, "parallelogram": [48, 60], "parallelprecondition": 42, "parallelpreconditioner2": 42, "paramet": [0, 1, 2, 8, 10, 38, 45, 46, 47, 50, 51, 61, 69, 70, 76, 82, 84, 91, 97, 99], "parameter": 100, "paramt": [56, 91], "pardof": [15, 23, 42], "parent": [76, 105], "parsev": 100, "part": [0, 1, 2, 6, 12, 15, 34, 36, 55, 57, 61, 65, 73, 79, 83, 84, 86, 89, 90, 91, 98, 99, 101, 104, 105, 106, 112, 113, 121], "part_graph": 108, "partial": [0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 28, 33, 34, 36, 38, 40, 41, 42, 43, 46, 52, 53, 54, 55, 56, 57, 58, 61, 62, 65, 66, 67, 73, 75, 76, 78, 79, 84, 87, 89, 90, 92, 95, 97, 98, 99, 100, 101, 103, 104, 108, 113, 114, 118, 121], "partial_": 79, "partial_i": [71, 104], "partial_n": [0, 3, 34, 65, 73, 75], "partial_q": 53, "partial_t": 67, "partial_v": 53, "partial_x": [71, 97, 104], "particip": 20, "particular": [15, 40, 51, 58, 65, 88, 103], "partit": [15, 40, 100, 109], "partner": 15, "pasciak": [45, 46, 83, 85], "pass": [20, 113], "patch": [33, 34, 36, 83, 109], "patchwisesolv": 34, "path": [20, 25, 46, 108], "pattern": [10, 76], "pc": [18, 19], "pc_type": [18, 19], "pde": [5, 24, 28, 31, 40, 100, 105], "pdf": 118, "peak": [116, 121], "pechstein": [62, 67, 71], "pen": 0, "penalti": [0, 1, 2, 34, 39, 69, 76], "per": [8, 29, 39, 40, 42, 46, 49, 64, 78, 83, 88, 110, 121], "perform": [0, 1, 12, 15, 24, 40, 47, 64, 77, 100, 103, 106, 108, 113, 115, 118], "period": [0, 76], "permeabl": 83, "person": 20, "perturb": 64, "petsc": [22, 45, 46], "petsc4pi": [14, 20, 22, 23], "petsc_configure_opt": 20, "petscpc": [18, 19, 22], "pf": 121, "phase": 82, "phat": 121, "phd": 72, "phenomena": 76, "phi": [8, 9, 13, 83, 88, 97, 102, 104, 106, 112], "phi_t": [8, 9, 12, 13, 106], "phi_v": 34, "physic": [10, 40, 76, 82, 84, 88, 91], "pi": [8, 12, 41, 43, 44, 66, 73, 80, 100, 101, 102, 104, 119], "pi_": [1, 38, 47, 56, 64, 65, 66, 79, 83], "pi_0": [65, 66], "pi_h": [36, 109], "pi_i": 38, "pi_l": [65, 66], "pi_n": 79, "pi_p": 79, "pick": [28, 29, 31], "pickl": 20, "piec": [15, 58, 100], "piecewis": [6, 33, 36, 100, 108, 109], "pillwein": 34, "pinvit": [0, 92, 96, 97, 121], "piola": 121, "pip": [46, 103], "pip3": [20, 46], "pl": 118, "place": 109, "placement": 76, "plai": [86, 87, 103], "plan": 97, "plane": 70, "plate": 3, "platform": 20, "pleas": [26, 46], "plot": [33, 47, 48, 49, 50, 51, 64, 75, 80, 103, 110, 115], "plt": [1, 2, 33, 47, 48, 49, 50, 51, 64, 80, 103, 112, 115], "plu": [39, 49, 62, 64, 82], "plug": [89, 91], "pm": 84, "pmax": [37, 38, 39, 82], "pmin": [37, 38, 39, 82], "pml": 76, "pmltrafo": 76, "pnt": [70, 83], "po": [0, 55], "poincar\u00e9": [45, 101, 106], "point": [0, 6, 8, 9, 10, 12, 13, 20, 26, 28, 32, 35, 38, 48, 56, 58, 76, 80, 81, 82, 83, 91, 98, 101, 103, 105, 108, 111, 118], "poisson": [1, 2, 33, 36, 45, 46, 53, 73], "pol": 47, "polygon": 104, "polyhedr": 83, "polynomi": [1, 2, 6, 8, 9, 10, 13, 34, 36, 49, 75, 76, 80, 88, 91, 99, 100, 101, 103, 110], "pop": 49, "popular": 6, "portabl": 23, "pose": [28, 31, 34, 40, 41, 55, 100, 104], "posit": [0, 6, 27, 30, 34, 37, 50, 51, 52, 63, 64, 67, 69, 77, 78, 81, 84, 91, 92, 99, 101, 103, 104, 108, 112], "possibl": [6, 15, 25, 26, 28, 33, 34, 46, 100, 101, 108, 110], "possibli": 67, "post": 63, "posteriori": 67, "postprocess": 34, "postsmooth": 63, "potenti": [76, 83], "power": [46, 48], "pp": 118, "pp43": 65, "practic": [40, 51, 108], "prager": 67, "pre": [0, 14, 18, 19, 20, 21, 22, 38, 39, 42, 46, 47, 49, 51, 63, 74, 76, 77, 81, 83, 84, 85, 91, 92, 96, 97, 103, 108, 109, 110], "pre2": 108, "pre3": 108, "prea": [22, 81, 82, 85], "prea1": [22, 85], "preambl": 45, "prec": [53, 54, 62, 90, 108, 109], "preceq": [1, 8, 12, 35, 36, 53, 64, 65, 66, 79, 83, 86, 100, 101, 104, 106, 108], "precis": [1, 2, 47, 55, 99, 100, 104], "precondit": [21, 40, 45, 46, 47, 49, 77, 81, 82, 105, 109, 110], "precondition": [10, 18, 19, 21, 22, 40, 45, 46, 47, 49, 52, 64, 74, 76, 77, 81, 83, 84, 85, 91, 108], "preconditiong": 105, "precontion": 108, "predefin": 76, "prefer": 13, "preform": [51, 64], "prelam": 84, "prepar": 25, "preschur": [22, 81, 85], "prescrib": [41, 82, 89], "present": [63, 64, 82, 101], "preserv": [12, 54, 79, 88, 100, 112], "presmooth": 63, "press": [25, 63], "pressur": [7, 12, 56, 58, 83, 84], "previou": [47, 49, 83, 99, 106, 109], "primal": [8, 35, 37, 38, 39, 57, 61, 62, 70, 91], "primari": 0, "prime": [13, 30, 36, 79, 88, 92, 97, 98, 100, 102, 106, 112], "principl": 27, "print": [0, 1, 2, 4, 10, 15, 18, 20, 21, 22, 23, 24, 38, 39, 42, 46, 47, 48, 50, 51, 55, 62, 63, 66, 67, 70, 76, 77, 78, 80, 81, 82, 83, 85, 86, 88, 92, 96, 97, 103, 106, 108, 109, 110, 115, 116, 118, 119, 120, 121], "printmast": 16, "printrat": [0, 14, 18, 19, 21, 22, 38, 39, 42, 63, 77, 82, 83, 84, 85, 91, 92, 96, 97, 103, 110], "prior": 103, "prism": 9, "prismat": 71, "problem": [8, 10, 12, 13, 21, 33, 34, 35, 37, 38, 39, 40, 41, 47, 50, 51, 60, 63, 64, 65, 67, 69, 70, 75, 76, 77, 79, 80, 81, 82, 86, 89, 91, 97, 100, 101, 103, 104, 105, 106, 107, 108, 109, 112], "proc": 20, "proc2dof": [15, 42], "proce": [34, 46, 51, 65, 106, 109, 114], "procedur": [27, 79], "process": [15, 20, 24], "processor": [15, 20, 24, 40, 121], "prod_": 104, "produc": [49, 79], "product": [12, 21, 27, 28, 29, 38, 48, 49, 50, 51, 52, 53, 64, 65, 76, 79, 83, 98, 99, 100, 101, 102, 103, 105, 109], "productmatrix": 120, "productspac": [10, 76], "profit": 77, "program": 110, "programm": 46, "progress": [60, 62], "proj": [82, 108], "project": [1, 8, 9, 12, 28, 29, 37, 45, 46, 56, 64, 65, 83, 86, 108, 109, 113, 118], "projectedmg": 63, "projector": [32, 45, 56, 64, 83, 84, 103, 105, 108, 109, 118], "projinv": 82, "projpr": 82, "prol": [63, 66], "prolong": [10, 63, 65, 66, 76, 83, 109], "prolongationop": 109, "promin": 111, "proof": [6, 8, 12, 27, 28, 29, 31, 32, 34, 36, 37, 53, 54, 64, 65, 66, 75, 79, 81, 83, 84, 87, 88, 90, 92, 98, 99, 100, 101, 104, 105, 106, 108, 109, 112], "propag": [48, 51, 105, 118], "proper": [8, 41, 76, 79, 88], "properli": 48, "properti": [10, 12, 28, 33, 45, 46, 50, 63, 65, 67, 76, 79, 83, 86, 89, 90, 109], "proport": 48, "prove": [8, 12, 29, 30, 32, 34, 48, 51, 53, 64, 65, 79, 83, 84, 90, 97, 99, 100, 101, 102, 103, 104, 105, 106, 109], "proven": [1, 12, 34, 47, 48, 51, 64, 65, 66, 79, 81, 83, 87, 90, 97, 100, 103, 106], "provid": [0, 3, 8, 10, 12, 13, 14, 15, 20, 29, 34, 35, 48, 51, 61, 63, 65, 72, 76, 79, 83, 84, 103, 108, 109, 121], "proxi": [3, 10, 76], "pru": 81, "psc": [14, 22, 23], "psc_f": 14, "psc_mat": 14, "psc_u": 14, "psi": [9, 64, 83], "psi_": [8, 9, 64], "psi_1": 9, "psi_h": 83, "psi_i": [13, 34, 83, 109], "psi_j": 9, "psi_l": 64, "psi_n": 9, "psi_t": [8, 9], "ptau": 70, "pu": [32, 109], "pull": [13, 100, 102, 112], "pure": [21, 24], "purpos": 23, "put": [11, 65, 115], "pv": 32, "pwd": 46, "px": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 42], "pxc": 66, "py": [46, 72, 98, 121], "pybind11_builtin": [10, 76], "pybind11_object": [10, 76], "pycapsul": [10, 76], "pymeti": 108, "pyngcor": [10, 76], "pypi": 20, "pyplot": [1, 2, 33, 47, 48, 49, 50, 51, 64, 80, 103, 112, 115], "pyramid": 9, "pythagora": 102, "pythagorean": 65, "python": [10, 14, 15, 20, 23, 25, 46, 72, 76, 108, 121], "python3": [20, 46, 121], "q": [7, 9, 10, 12, 22, 27, 30, 34, 53, 54, 55, 56, 58, 62, 78, 79, 81, 84, 85, 100, 101, 121], "q2": 86, "q_h": [7, 54, 86], "q_i": [34, 88], "qf": 121, "qquad": [0, 6, 8, 9, 12, 13, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 41, 43, 47, 49, 50, 51, 52, 53, 57, 62, 63, 64, 65, 66, 67, 73, 74, 76, 79, 80, 82, 83, 84, 88, 89, 92, 95, 97, 98, 100, 101, 102, 103, 104, 105, 106, 111, 112, 113, 114, 115], "qr": 115, "quad": [27, 30, 31, 32, 34, 40, 49, 53, 54, 55, 56, 64, 65, 69, 73, 76, 82, 88, 89, 90, 98, 100, 103, 105, 107, 109, 112], "quad_domin": [10, 80], "quadrat": [9, 50, 76, 81, 84, 105], "quadrilater": [9, 10, 62, 80], "qualiti": 51, "quantifi": 64, "quantiti": 51, "quasi": [8, 28, 31, 36, 54, 66, 83, 109], "queri": [10, 20, 76], "quickest": 20, "quickli": 66, "quotient": [51, 101, 103, 105, 112, 113], "r": [0, 1, 5, 6, 8, 9, 12, 13, 16, 18, 19, 21, 22, 24, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 44, 46, 47, 48, 49, 50, 51, 52, 58, 63, 68, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 88, 89, 92, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 109, 110, 112, 113, 115, 121], "r1": 34, "r2": 34, "r_": [34, 49], "r_0": 49, "r_e": 34, "r_h": 83, "r_i": [24, 51, 83, 105], "r_j": 88, "r_k": 49, "r_n": 49, "r_t": [8, 34], "radiu": [48, 51, 99, 102, 105], "rafetsed": 67, "rais": 121, "random": 64, "rang": [0, 10, 11, 14, 15, 18, 19, 20, 21, 22, 23, 31, 33, 35, 37, 38, 39, 40, 41, 42, 44, 47, 48, 49, 51, 55, 63, 64, 66, 70, 74, 76, 77, 79, 80, 81, 82, 83, 84, 88, 100, 103, 104, 105, 108, 109, 113, 114, 115, 118, 121], "rank": [14, 15, 16, 18, 19, 20, 21, 22, 24, 42, 84, 105, 106], "rare": 47, "rate": [8, 28, 48, 77, 79], "ratio": 8, "ration": 80, "raviart": [7, 9, 12, 61, 86, 88, 101], "rayleigh": [51, 103, 105], "raymond": 5, "rcll": 100, "re": [7, 34, 36, 48, 50, 51, 63, 82, 113], "reach": 40, "reaction": 28, "read": [25, 34, 72, 76, 84, 105, 113], "reader": [25, 81], "readi": [25, 83], "readonli": [10, 76], "readthedoc": 14, "real": [31, 112], "realiz": [4, 41], "rearrang": 61, "reason": [12, 56, 121], "recal": 24, "recast": 84, "receiv": [15, 42], "recent": [46, 55, 58, 62, 118, 119, 120, 121], "recommend": [72, 76], "reconstruct": [82, 86], "recov": [8, 33, 38, 53, 105], "rect": [12, 35, 37, 38, 39, 82, 89, 108, 118, 121], "rectangl": [0, 7, 12, 16, 18, 19, 33, 34, 35, 37, 38, 39, 55, 78, 81, 82, 85, 89, 97, 103, 108, 112, 118, 121], "rectangular": [24, 42, 53, 82, 105, 115], "recurr": [47, 79], "recurs": [33, 49, 63, 64, 65], "recv": 20, "redraw": [0, 4, 5, 16, 55, 70, 113, 114, 118, 119, 120, 121], "reduc": [0, 15, 30, 40, 45, 48, 49, 50, 64, 67, 73, 82, 101, 112, 114], "reduct": [15, 47, 48, 50, 79, 82], "ref": [12, 79, 100], "refer": [8, 9, 10, 12, 13, 36, 52, 62, 64, 88, 106, 112, 121], "refin": [10, 11, 14, 15, 18, 19, 22, 23, 35, 40, 63, 64, 65, 66, 75, 76, 77, 83, 84, 103, 108, 109, 118], "refinefromtre": 76, "refinehp": [8, 76], "refinementflag": 76, "reformul": [31, 49, 64, 105], "refract": 121, "reg": 78, "regex": 76, "regexpr": [10, 76], "region": [0, 10, 75, 76, 78, 108, 112, 121], "region_wis": 55, "regioncf": 76, "regular": [8, 9, 10, 33, 34, 39, 53, 62, 64, 76, 83, 84, 91, 98, 104, 105, 109], "reinforc": 91, "reinstal": 20, "reissner": [45, 62], "rel": [34, 112], "relat": [8, 13, 29, 47, 50, 64, 78, 101, 108], "relationship": 45, "relax": 50, "relev": 40, "reliabl": [33, 34, 35], "remain": 49, "remark": [12, 34, 82], "remov": [20, 33, 47], "renam": 56, "render": 46, "repeat": [27, 51], "rephras": [53, 54], "replac": [4, 12, 33, 36, 54, 56, 63, 65, 73, 75, 82, 101, 103, 106, 112, 113], "report": 26, "repositori": 80, "repres": [20, 24, 29, 34, 37, 49, 63, 65, 83, 98, 99, 105, 108, 109, 115], "represent": [24, 31, 45, 46, 47, 76, 103, 105, 106, 109, 115], "reproduc": 66, "reproven": 105, "requir": [3, 8, 12, 24, 35, 53, 61, 63, 65, 76, 82, 84, 86, 101, 104, 109, 121], "rescal": [12, 47, 109], "resf": [22, 81, 85], "resg": [22, 81, 85], "reshap": [0, 55], "residu": [14, 21, 33, 38, 39, 46, 47, 48, 50, 51, 63, 77, 83, 84, 91, 105, 110], "residuum": [34, 51, 73], "resolut": [10, 76], "reson": 46, "resourc": 121, "resp": 62, "respect": [27, 28, 31, 34, 35, 38, 41, 47, 48, 49, 53, 79, 84, 99, 100, 102, 105], "rest": [12, 36, 63, 66, 73, 83, 87, 104, 109], "restoresubvector": 23, "restrict": [1, 5, 13, 22, 30, 63, 66, 73, 76, 79, 85, 89, 98, 99, 100, 101, 104, 109], "result": [8, 15, 20, 24, 37, 51, 62, 65, 66, 69, 71, 76, 79, 81, 83, 97, 99, 100, 101, 106, 109, 110], "retriev": 15, "return": [0, 2, 3, 6, 7, 10, 15, 17, 18, 19, 24, 33, 42, 47, 49, 55, 58, 62, 63, 66, 67, 69, 70, 74, 76, 81, 103, 108, 109, 110, 112, 121], "rev": [65, 105], "revers": [7, 16, 18, 19, 81, 85, 105], "rewrit": [51, 53, 57, 61, 76, 100, 103], "rewritten": [49, 76], "rh": [79, 84, 91], "rho": [4, 10, 47, 48, 51, 76, 105], "rho_": [47, 48, 50], "rho_n": 47, "rhscurl": 17, "ri": [22, 85], "rich": [50, 53, 58], "richardson": [45, 46, 47, 49, 50, 52, 84], "riesz": [28, 31, 45, 46, 53], "right": [0, 1, 5, 6, 8, 10, 11, 12, 21, 23, 24, 28, 30, 31, 34, 35, 36, 37, 38, 41, 44, 47, 48, 49, 51, 52, 53, 55, 61, 62, 64, 65, 68, 69, 73, 76, 78, 79, 80, 81, 82, 83, 84, 89, 90, 91, 92, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 112, 113, 114, 121], "rightarrow": [2, 6, 8, 9, 12, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 50, 53, 54, 56, 58, 65, 66, 68, 69, 74, 75, 76, 78, 79, 83, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 109, 112, 113, 115], "rightdomain": 44, "rigid": [59, 62], "rigor": [33, 35], "ring": 46, "ring_resonator_import": [119, 120], "ritz": 49, "rk": 111, "rk2": [4, 111], "rk3": [111, 112], "rk4": [4, 111], "rll": 53, "robert": 118, "robin": [43, 51, 73, 104], "robust": [1, 2, 34, 46, 71, 110], "rodrigu": 79, "role": [57, 89, 103], "root": [15, 20, 28, 51], "rotat": [0, 8, 33, 55, 59, 61, 62, 67, 69, 70], "rotmat": [0, 55], "rotor": 0, "rough": 64, "roughli": 8, "roundoff": 49, "row": [12, 52, 56, 60, 61, 81, 82, 84, 111], "row_pardof": [14, 42], "rt": [7, 18, 19, 88, 91], "rt0": 88, "rt_0": 88, "rt_k": [86, 87, 88, 91], "rtol": [14, 23], "rule": [0, 8, 12, 13, 88, 100, 106, 109, 111, 114], "run": [20, 25, 26, 46, 48, 51], "rung": [4, 5, 45], "ruuth": 5, "s1": [47, 111], "s11": 62, "s2": 47, "s_0": 100, "s_1": 100, "s_i": 100, "s_j": [88, 100], "s_l": [64, 88], "s_m": 100, "saad": [47, 49], "sabin": [72, 80, 88], "saddl": [38, 56, 81, 82, 83, 91], "sai": [9, 28, 32, 38, 73, 98, 101], "same": [1, 2, 8, 9, 10, 12, 20, 21, 24, 28, 32, 34, 36, 38, 40, 41, 51, 55, 56, 63, 66, 68, 76, 79, 88, 89, 91, 97, 99, 101, 109, 110, 111, 112, 121], "sampl": 115, "satisfi": [27, 28, 30, 34, 36, 47, 51, 58, 61, 64, 65, 67, 69, 75, 76, 79, 81, 83, 84, 98, 109, 111, 112, 113], "scalar": [60, 82, 86, 112], "scale": [12, 27, 34, 36, 38, 42, 47, 64, 65, 77, 79, 81, 87, 97, 100, 108, 120, 121], "scaledchebi": 47, "scalematrix": 120, "scalingmat": 42, "scatter": [20, 44], "scene": [0, 4, 5, 16, 55, 76, 113, 114, 118, 119, 120, 121], "schmidt": [49, 115], "schoeberl": 25, "schuller": 63, "schur": [37, 38, 42, 82], "schurdir": [38, 39], "schwab": 79, "schwarz": [12, 27, 36, 37, 45, 46, 63, 64, 65, 66, 83, 100, 102, 109], "sch\u00f6berl": [7, 34, 45, 46, 62, 71, 72, 83, 119, 120], "scienc": 46, "scientif": [23, 46, 111], "scipi": [1, 2, 46, 52, 76, 115], "scipymat": [1, 2], "scope": 15, "scott": 46, "script": [76, 98], "scroll": 37, "sd": 70, "search": [30, 34, 50, 55, 60, 61, 69, 76, 83, 103, 113], "secod": 6, "secon": 57, "second": [4, 5, 6, 10, 31, 33, 40, 50, 51, 55, 56, 58, 64, 67, 70, 76, 81, 83, 84, 86, 89, 97, 98, 103, 106, 109, 110, 112, 114, 115, 118], "secondari": 0, "secondord": 10, "section": [46, 50, 52, 75, 109], "sector": [8, 104], "see": [1, 5, 10, 12, 31, 37, 46, 53, 54, 58, 59, 60, 65, 73, 76, 79, 80, 81, 84, 88, 89, 100, 103, 115], "seem": 103, "seen": [6, 49, 53, 63, 64, 83, 91, 99], "segment": 9, "seidel": [63, 64, 105, 110], "select": 76, "self": [1, 2, 10, 42, 48, 51, 63, 66, 76, 109, 121], "semi": [33, 36, 83, 99, 100, 101, 103, 106, 109, 112, 113], "send": [20, 40, 46], "sens": [9, 34, 55, 62, 64, 76, 78, 83, 100, 121], "sensit": 47, "sent": 15, "separ": [2, 41, 84], "septemb": 72, "sequenc": [27, 31, 32, 33, 63, 66, 75, 97, 99, 100, 101, 103], "sequenti": [24, 63], "seri": [41, 65, 100, 102, 104], "serial": 20, "server": 46, "set": [0, 3, 4, 5, 7, 9, 10, 12, 13, 14, 15, 17, 18, 19, 22, 24, 27, 28, 30, 33, 34, 35, 37, 38, 39, 41, 48, 51, 53, 54, 55, 56, 57, 58, 62, 63, 64, 66, 70, 73, 75, 76, 79, 80, 81, 83, 85, 87, 88, 91, 98, 99, 101, 104, 105, 106, 108, 109, 113, 114, 115, 116, 118, 121], "set_minmax": 121, "set_titl": 47, "set_ylim": 47, "set_yscal": 47, "setcouplingtyp": [10, 18, 19, 76], "setdefinedon": [10, 76], "setdeform": [0, 55, 76], "setelementord": 76, "setislocalmat": 23, "setlgmap": 23, "setmateri": 44, "setminu": [100, 107, 108, 109], "setnormtyp": [14, 23], "setnumthread": [4, 5], "setoper": [14, 23], "setord": [10, 58, 76, 81], "setparallelstatu": 42, "setpml": 76, "setrandom": [48, 51, 64], "setrefinementflag": [33, 35, 76], "setsiz": 23, "settoler": [14, 23], "settyp": [14, 23], "setup": [21, 77, 78, 97, 108, 109], "setupkorn": 103, "setuppoincareproblem": 103, "setupproblem": 103, "setuptraceproblem": 103, "sever": [5, 12, 40, 58, 64, 108], "shape": [1, 7, 8, 9, 10, 16, 18, 19, 22, 24, 33, 34, 35, 42, 46, 63, 76, 77, 78, 80, 81, 85, 88, 89, 103, 108, 109, 118, 121], "share": [15, 24, 110], "sharp": [100, 103, 106], "shat": 82, "shear": [69, 70], "shift": [0, 25, 47, 112, 114], "ship": [14, 20], "short": [33, 63, 76, 88, 100, 108, 112, 115], "should": [10, 33, 46, 76, 100, 108, 121], "show": [10, 12, 29, 30, 31, 33, 34, 36, 41, 46, 47, 53, 64, 65, 74, 79, 81, 84, 97, 102, 106, 108, 109, 112], "shown": [34, 40, 53, 60, 64, 65, 79, 83, 90, 105], "shutdown": [14, 15, 20, 21, 22, 23, 24], "siam": [24, 65, 105], "side": [0, 8, 10, 11, 12, 21, 23, 24, 30, 31, 33, 34, 35, 36, 37, 41, 53, 55, 64, 65, 73, 76, 78, 79, 82, 89, 91, 98, 100, 101, 102, 103, 104, 105, 106, 112, 121], "sigma": [18, 19, 37, 47, 48, 53, 57, 60, 61, 62, 64, 67, 68, 69, 70, 71, 74, 76, 81, 83, 84, 86, 87, 88, 89, 90, 91, 109, 112, 119, 120], "sigma1": 47, "sigma_": [34, 62, 67, 90, 91], "sigma_h": [62, 67, 68, 71, 86, 87, 91], "sigma_n": [34, 57, 62, 89, 90], "sigman": 70, "sign": [12, 55, 81], "signatur": [10, 76], "silenc": [38, 39], "sim": 73, "simeq": [1, 8, 12, 34, 36, 65, 100, 101, 103, 104], "similar": [12, 21, 27, 33, 36, 40, 50, 52, 61, 63, 64, 65, 74, 78, 79, 83, 88, 91, 100, 102, 108, 113, 114, 115], "similarli": [2, 24, 76, 81], "simpl": [0, 3, 6, 20, 24, 31, 48, 51, 73, 83, 100, 105, 109], "simpler": 12, "simplest": [33, 58], "simpli": [3, 27, 50, 79, 83], "simplic": 79, "simplici": [76, 79], "simplif": 12, "simplifi": [30, 47, 51, 109], "simul": [44, 46], "simultan": 8, "sin": [0, 6, 41, 55, 66, 73, 80, 100, 101, 102, 104, 119], "sinc": [1, 2, 3, 4, 7, 8, 10, 12, 15, 27, 28, 31, 32, 33, 34, 36, 37, 39, 40, 48, 49, 50, 51, 60, 61, 64, 65, 76, 79, 83, 84, 86, 88, 97, 98, 100, 101, 104, 105, 106, 108, 109, 111, 112, 114, 115, 121], "sine": [100, 108], "singl": [1, 24, 45, 51, 65, 111], "singular": [33, 35, 39, 40, 58, 111], "sinum": 118, "sinv": 84, "sinwel": 62, "sisc": 118, "sit": [8, 9, 110], "site": 121, "situat": [51, 52], "six": 62, "size": [1, 2, 8, 12, 15, 20, 23, 47, 48, 63, 65, 75, 76, 77, 79, 83, 84, 97, 103, 105, 106, 108, 109, 112, 114], "skel_dof2proc": 42, "skel_pardof": 42, "skeleton": [0, 2, 6, 38, 121], "sketch": [27, 101], "skew": 61, "skew2vec": [18, 19], "skip": [2, 12, 30, 31, 58, 65, 83, 98], "sleep": [0, 55, 113, 114, 115, 118], "slider": [74, 76], "slightli": [62, 112], "small": [8, 33, 35, 40, 45, 46, 49, 56, 64, 69, 78, 79, 83, 84, 86, 103, 108, 109, 115, 118, 121], "smaller": [20, 37, 64, 88, 108, 109, 118], "smallest": [0, 24, 27, 34, 47, 97, 103, 106, 108], "smooth": [6, 8, 9, 13, 33, 41, 60, 63, 64, 90, 98, 99, 100, 104, 106], "smoothback": 63, "smoother": 64, "smoothli": 26, "smuggl": 1, "so": [1, 2, 3, 6, 7, 27, 29, 33, 51, 52, 53, 55, 66, 73, 76, 77, 79, 82, 83, 91, 105, 106, 108, 109, 111], "sobolev": [8, 64, 79, 90, 101, 103, 104], "softwar": 20, "sol": [22, 66, 81, 85, 91], "sol_lam": 55, "sol_sigma": 86, "sol_u": [55, 86], "solid": [45, 46, 77, 103], "solsigma": 70, "solut": [0, 1, 2, 6, 8, 19, 21, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 46, 47, 48, 49, 50, 53, 54, 58, 64, 65, 73, 75, 76, 79, 82, 83, 84, 86, 91, 100, 104, 105, 108, 109, 110, 111, 112, 113, 115, 121], "solv": [4, 5, 8, 10, 14, 18, 23, 28, 30, 34, 35, 37, 38, 40, 45, 46, 52, 53, 56, 63, 64, 70, 73, 77, 80, 82, 84, 85, 86, 100, 104, 105, 109, 111, 112, 114, 115], "solvabl": [28, 30, 31, 53], "solveestimatemark": [33, 35], "solveiniti": 16, "solvem": [10, 76], "solver": [0, 1, 5, 14, 18, 23, 24, 40, 49, 83, 84, 91, 92, 96, 97, 103, 108, 109, 121], "solverparamet": 18, "solvestok": 58, "some": [0, 3, 6, 8, 10, 12, 28, 29, 30, 31, 32, 34, 35, 39, 41, 46, 48, 49, 54, 63, 64, 73, 75, 76, 81, 82, 86, 89, 99, 100, 101, 104, 105, 109, 110, 111, 112, 113, 121], "someth": [20, 102], "sometim": 8, "soon": 51, "sophist": [1, 2], "sort": 33, "sourc": [6, 20, 33, 34, 35, 36, 43, 44, 46, 55, 57, 76, 89, 113], "sp": [1, 2, 121], "space": [0, 2, 3, 6, 7, 8, 9, 10, 12, 14, 27, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 47, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 69, 73, 75, 78, 79, 80, 82, 83, 84, 86, 88, 89, 91, 98, 103, 104, 106, 109, 110, 114, 115, 121], "span": [9, 31, 49, 76, 82, 100, 104, 106, 108, 115], "spars": [1, 2, 23, 47, 52, 76, 77, 108], "sparsecholeski": [0, 1, 2, 5, 38, 39, 44, 62, 77, 85, 103, 109, 121], "sparsematrixd": 42, "sparseschur": 42, "sparsiti": [10, 76], "spatial": [63, 113, 114], "spd": [47, 48, 50, 51, 52, 64, 84], "special": [15, 83, 99], "specialcf": [0, 1, 2, 3, 4, 5, 6, 7, 17, 18, 19, 22, 34, 57, 62, 67, 69, 70, 85, 91, 116, 121], "specif": [10, 65, 76], "specifi": [6, 34, 73, 76, 104, 111], "spectral": [48, 49, 51, 64, 81, 105, 106, 108], "spectrum": [37, 47, 81, 84, 109], "spend": 103, "sphere": 121, "spiteri": 5, "splinegeometri": [44, 62], "split": [20, 31, 33, 37, 38, 39, 40, 45, 46, 51, 53, 63, 64, 73, 83, 98, 100, 105, 106, 108, 109], "splitelements_alfeld": 76, "springer": 63, "spy": [1, 2], "sqrt": [0, 1, 30, 31, 33, 34, 35, 41, 47, 49, 50, 51, 52, 64, 65, 79, 80, 81, 84, 88, 92, 97, 100, 103, 111, 112, 119], "sqrt2": 100, "squar": [0, 8, 28, 41, 51, 55, 65, 76, 101, 103, 104, 106, 108], "src": [20, 119], "ss": 111, "stab_e": 112, "stab_i": 112, "stab_improvede": 112, "stab_rk3": 112, "stab_trapez": 112, "stabil": [1, 2, 49, 60, 61, 86, 109, 111, 115], "stabl": [6, 30, 45, 46, 54, 56, 60, 66, 79, 83, 109, 111, 112], "stackrel": [68, 86], "stage": 111, "stand": [37, 39], "standard": [10, 20, 21, 23, 34, 57, 60, 77, 80, 88, 91, 108], "star": 46, "start": [0, 1, 2, 14, 15, 20, 21, 22, 23, 24, 26, 28, 30, 33, 34, 40, 46, 48, 55, 64, 65, 76, 80, 81, 86, 100, 109, 111, 121], "start_and_connect": [14, 15, 16, 18, 19, 20, 21, 22, 23, 24], "state": [28, 31, 60, 98, 99, 100, 101, 103, 112], "statement": [27, 104], "static": [10, 76], "stationari": [4, 45, 46, 103, 105], "stdflux": 89, "stdout": [14, 15, 20, 21, 22, 23, 24], "stdtemp": 89, "steel_": 76, "steeper": 66, "stefano": 14, "stenberg": 61, "stencil": 118, "step": [4, 5, 8, 12, 13, 21, 34, 35, 36, 40, 45, 46, 47, 48, 50, 51, 63, 64, 70, 79, 81, 82, 84, 105, 106, 108, 109, 111, 115], "steven": 5, "stiff": [112, 113, 118], "still": [39, 46, 60, 63, 82, 100], "stoke": [5, 45, 46, 56, 84, 101], "stokesa": [18, 19], "stop": [33, 49, 121], "storag": 24, "store": [24, 52, 82, 108, 109, 121], "str": [10, 20, 37, 38, 39, 47, 76, 108], "straight": [12, 91], "strain": 74, "strang": 56, "strategi": [8, 48], "stream": 20, "strength": 6, "strengthen": 105, "stress": [60, 61, 70, 74], "stretch": 62, "strictli": [51, 111], "string": [10, 76], "strip": 121, "strong": [0, 33, 55, 64, 76, 88, 98, 100], "stronger": [64, 100], "structur": [10, 34, 45, 46, 56], "student": 46, "studi": [8, 40, 58, 75, 100], "sturm": 79, "sub": [2, 8, 12, 23, 24, 27, 30, 32, 34, 37, 38, 39, 40, 42, 49, 53, 54, 63, 65, 73, 76, 77, 79, 82, 83, 84, 88, 98, 103, 104, 106, 108, 109, 111, 112, 113], "subdivis": 9, "subdomain": 108, "submatrix": 91, "subplot": 47, "subset": [6, 8, 9, 12, 15, 27, 29, 31, 34, 36, 37, 40, 41, 47, 49, 54, 63, 64, 66, 67, 68, 76, 79, 81, 83, 84, 88, 92, 95, 97, 98, 100, 101, 102, 104, 105, 106, 109, 111, 112, 113], "subspac": [27, 28, 29, 31, 45, 46, 65, 105], "substep": 118, "substitut": [13, 47, 51, 52, 58, 65, 79], "substructur": 37, "subtrac": 81, "subtract": 12, "succ": 53, "succeq": [6, 8, 36, 53], "successfulli": 26, "sudo": 20, "suffici": [0, 1, 2, 8, 31], "suggest": [8, 46, 100], "suit": 64, "sum": [13, 15, 22, 24, 33, 34, 35, 36, 41, 49, 52, 55, 64, 65, 66, 74, 83, 85, 98, 100, 102, 105, 106, 108, 109, 111, 112, 121], "sum_": [2, 6, 8, 9, 12, 13, 24, 33, 34, 35, 36, 38, 40, 41, 42, 47, 49, 51, 62, 65, 66, 76, 79, 87, 90, 91, 99, 100, 101, 102, 104, 105, 106, 111, 113, 121], "sum_i": [76, 100, 105, 106, 109, 111], "sum_j": [48, 105], "sum_k": [65, 100], "sum_l": [65, 100], "sum_t": [1, 2, 3, 12, 24, 34, 36, 54, 62, 67, 86, 87, 91, 106, 112], "sum_v": [34, 62], "sumlocdof": 15, "summand": 106, "summat": [12, 15], "summatrix": [120, 121], "summer": 46, "sumup": 15, "sundai": 26, "sung": 3, "sup": [6, 27, 45, 46, 53, 54, 86], "sup_": [12, 27, 28, 29, 30, 31, 34, 36, 37, 41, 47, 48, 53, 54, 75, 83, 84, 86, 87, 90, 97, 103, 105, 112], "sup_u": [6, 37, 84], "sup_v": [29, 34, 53], "super": [33, 42, 63, 76, 109, 121], "supp": 98, "supplement": 72, "support": [3, 10, 76, 90, 98, 99, 100, 108, 109], "suppos": 33, "supremum": [30, 75], "supset": 37, "surfac": [6, 35, 70, 76], "surfacel2": 55, "surject": [30, 31, 53, 100], "swap": [30, 89, 103], "switch": 3, "sym": [56, 60, 62, 67, 74, 82, 103], "symbol": [10, 76], "symbolicbfi": [62, 67, 69, 70], "symbolicenergi": 70, "symboliclfi": [62, 67, 69], "symmetr": [0, 1, 2, 5, 8, 27, 28, 30, 31, 37, 45, 46, 48, 51, 52, 53, 60, 61, 62, 63, 64, 67, 69, 70, 74, 77, 81, 84, 91, 101, 103, 112, 121], "symmetri": 45, "symplect": 121, "syng": 67, "system": [1, 5, 27, 31, 34, 37, 38, 41, 42, 46, 47, 49, 50, 51, 52, 54, 56, 57, 58, 61, 67, 76, 77, 81, 82, 84, 89, 91, 111, 112, 113, 114], "t": [0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 16, 22, 24, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 46, 47, 49, 50, 51, 52, 54, 56, 57, 61, 62, 63, 64, 65, 66, 67, 69, 70, 73, 74, 76, 79, 81, 82, 83, 84, 85, 87, 88, 89, 91, 96, 97, 100, 103, 105, 106, 108, 109, 111, 112, 113, 114, 115, 118, 119, 120, 121], "t_": [4, 13, 34, 36, 47, 111, 112], "t_0": [34, 47, 112], "t_1": [9, 34, 36, 47, 112], "t_2": [34, 36], "t_envelop": 119, "t_i": [9, 34], "t_j": [9, 111, 112, 113], "t_l": 65, "t_m": 9, "t_n": [4, 34, 47, 112], "tabl": [10, 76], "table_i": [10, 76], "tag": 20, "take": [6, 8, 14, 25, 29, 31, 32, 33, 35, 36, 40, 41, 46, 53, 54, 62, 65, 76, 78, 84, 87, 89, 99, 100, 104, 106, 108, 112], "taken": [40, 47, 103], "talk": [40, 100], "tall": 105, "tang": [7, 17, 18, 19, 62, 67, 69, 70], "tangenti": [7, 33, 61, 68, 70], "tangentialfacetfespac": [7, 17, 18, 19, 116], "target": 35, "tartar": [45, 101, 103, 104], "taskmanag": [4, 16, 34, 44, 70, 74, 77, 83, 85, 118, 119, 120, 121], "tau": [4, 5, 18, 19, 28, 34, 47, 53, 57, 60, 61, 62, 67, 69, 70, 79, 83, 86, 88, 89, 91, 105, 111, 112, 113, 114, 115, 118, 119, 120], "tau_": [47, 62, 67, 91], "tau_1": [47, 91], "tau_2": [47, 91], "tau_h": [62, 67, 86, 91], "tau_i": 47, "tau_j": 112, "tau_k": 47, "tau_n": [47, 57, 88, 89, 91], "taun": 70, "tauopt": 47, "taylor": [58, 111, 112], "td": 67, "tdnn": [45, 69, 70], "teach": [25, 46], "team": [15, 20, 42], "tear": 38, "technic": [83, 88, 109], "techniqu": [6, 35, 40, 45, 46, 64, 65, 79, 81, 100, 101, 105, 112], "technologi": 20, "telescop": 65, "tell": [29, 76], "temperatur": [73, 76, 78, 113], "tend": [4, 5, 16, 113, 114, 115, 118, 119, 120, 121], "tensor": [60, 62, 74, 76, 79], "term": [0, 1, 2, 4, 5, 6, 7, 8, 12, 13, 30, 33, 34, 36, 40, 46, 47, 48, 49, 55, 56, 58, 62, 64, 65, 69, 73, 76, 79, 81, 83, 84, 86, 87, 89, 90, 91, 97, 100, 101, 104, 108, 121], "termin": 33, "test": [1, 2, 6, 8, 10, 13, 42, 55, 57, 58, 64, 73, 76, 89, 100, 113, 115], "testfunct": [0, 1, 3, 5, 7, 10, 12, 18, 19, 34, 44, 55, 57, 58, 62, 67, 69, 70, 76, 86, 89, 90, 91, 121], "testspac": [38, 39, 116, 121], "tet": 62, "tetrahedr": [10, 110, 118], "tetrahedra": 9, "text": [6, 30, 34, 43, 47, 48, 49, 50, 51, 53, 55, 57, 58, 60, 61, 62, 63, 65, 67, 68, 70, 73, 74, 76, 79, 80, 82, 83, 84, 88, 89, 90, 100, 102, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115], "textbook": [72, 101], "tfrac": [8, 30, 32, 40, 47, 49, 50, 53, 56, 61, 74, 79, 84, 92, 96, 97, 100, 103, 105, 108, 109, 112, 114, 121], "th": [20, 34, 51, 67, 79, 105, 109], "than": [1, 10, 15, 33, 34, 37, 55, 65, 66, 76, 88, 98, 99, 100, 108], "thank": [64, 81, 86, 108, 109, 121], "thei": [1, 7, 12, 24, 30, 63, 64, 73, 79, 83, 108, 109, 111, 121], "them": [3, 10, 21, 24, 46], "theo_subdomainh1": 100, "theorem": [8, 12, 28, 30, 31, 32, 36, 37, 45, 46, 53, 54, 64, 65, 66, 76, 79, 81, 83, 84, 88, 90, 99, 101, 102, 103, 105, 108, 109], "theoret": 46, "theori": [25, 31, 51, 63, 64, 65, 72, 79, 101, 108, 109], "therefor": 32, "thermal": 73, "thesi": [1, 7, 59, 62, 72], "theta": 47, "thi": [0, 1, 2, 5, 6, 7, 8, 10, 11, 12, 13, 20, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 60, 61, 62, 63, 64, 65, 66, 67, 69, 72, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 97, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115, 121], "thick": [69, 70], "thin": [71, 121], "think": [40, 52, 53, 63], "third": [2, 6, 53, 86, 91, 98, 99, 111], "thm": 101, "thoma": [7, 9, 61, 86, 88], "thread": 11, "three": [47, 62, 73, 79, 82, 88], "threorem": 45, "through": [25, 32, 35, 88], "thu": [1, 2, 3, 4, 6, 12, 13, 27, 28, 29, 31, 32, 34, 40, 46, 47, 48, 50, 51, 53, 54, 55, 56, 57, 58, 62, 64, 65, 66, 69, 73, 76, 79, 80, 83, 84, 86, 87, 88, 90, 91, 92, 98, 99, 100, 101, 103, 104, 105, 106, 110, 112, 115, 118, 121], "tightli": 67, "tild": [9, 30, 33, 37, 41, 42, 50, 51, 66, 73, 79, 84, 97, 100, 104, 108], "till": 8, "time": [0, 1, 4, 6, 8, 9, 12, 13, 15, 24, 25, 27, 28, 30, 31, 35, 37, 40, 49, 52, 53, 54, 55, 58, 61, 63, 64, 68, 75, 76, 79, 81, 82, 84, 97, 98, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 115, 119, 120, 121], "timer2list": 121, "timestep": 16, "tk": 62, "tn": 2, "tnt": [0, 2, 4, 6, 10, 11, 12, 14, 17, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 47, 48, 49, 50, 51, 55, 56, 63, 64, 66, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 121], "todo": 20, "togeth": [0, 7, 8, 10, 11, 24, 27, 34, 49, 63, 76, 81, 83, 103, 106], "tol": [47, 49, 74, 81, 85], "told": 47, "toler": 33, "too": [53, 58, 121], "tool": [46, 63, 65, 83, 102, 105, 108, 109], "toolbar_vis": 47, "toolkit": 23, "top": [10, 34, 44, 56, 62, 73, 76, 80, 82], "topologi": 110, "tordman": 118, "total": [9, 55, 57, 119, 120], "touch": 46, "toward": [8, 76], "tpeak": 119, "tr": [8, 12, 38, 41, 74, 90, 96, 97, 100, 101, 103, 104], "trace": [3, 6, 12, 34, 45, 46, 53, 55, 62, 70, 74, 76, 79, 96, 108, 121], "traceback": [58, 119, 120, 121], "tracenorm": 97, "traceop": [116, 121], "traceoper": [10, 76, 121], "tracespac": [10, 76], "traction": 74, "tran": 70, "transfer": [14, 83, 100, 121], "transform": [8, 9, 10, 12, 13, 45, 46, 51, 76, 82, 85, 97, 100, 106], "transfrom": 81, "transistor": 40, "transmiss": 73, "transport": [5, 45, 46, 111], "transpos": [13, 63, 84, 109], "trapezoid": [111, 114], "travel": 121, "treat": [2, 3, 5, 55], "treatment": [5, 74], "trial": [6, 10, 76, 78], "trialfunct": [0, 1, 3, 5, 7, 10, 12, 18, 19, 34, 44, 55, 57, 58, 62, 67, 69, 70, 76, 86, 89, 91, 121], "trialspac": [38, 39, 116, 121], "triangl": [8, 9, 10, 33, 34, 36, 54, 66, 83, 86, 88, 106, 118], "triangul": [8, 9, 76, 110], "triangular": [10, 51, 52, 53, 67, 80, 88, 110, 111, 118], "trick": [1, 8], "tricki": [1, 2, 55, 63], "trig": [58, 81, 118], "trigonometr": [0, 47], "tripl": 9, "trivial": [27, 53, 66, 83, 87, 99, 100, 101, 106], "trottenberg": [63, 64], "true": [0, 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 28, 29, 33, 34, 35, 36, 42, 43, 44, 47, 48, 50, 51, 53, 54, 55, 56, 62, 63, 66, 67, 69, 70, 73, 74, 76, 81, 82, 83, 84, 85, 88, 91, 92, 100, 101, 103, 104, 108, 109, 110, 113, 114, 115, 116, 118, 121], "try": [25, 30, 33, 46, 58, 75, 83], "tt": 70, "tu": [25, 45, 46, 56], "tu_1": 27, "tucker": [53, 84], "tupl": [10, 20, 76], "tutori": [11, 20, 24, 25, 75], "tuwel": 46, "tuwien": [10, 11], "tv": 27, "twice": [64, 82, 104], "two": [0, 2, 8, 9, 21, 24, 30, 32, 34, 36, 38, 47, 48, 50, 52, 63, 73, 75, 76, 80, 82, 88, 91, 97, 100, 101, 105, 106, 108, 109, 114], "typ": 121, "type": [10, 14, 21, 23, 24, 34, 65, 67, 76, 78, 84, 109, 121], "type1": 62, "typic": [1, 2, 9, 40, 52, 53, 76, 88, 105, 113, 114], "typo": 55, "u": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 115, 118, 121], "u0": [73, 113, 114, 115, 118], "u1": [22, 70, 85], "u2": [70, 86], "u3": 70, "u_": [6, 24, 31, 34, 36, 42, 53, 64, 65, 76, 92, 95, 97, 104, 106, 115], "u_0": [4, 32, 53, 65, 66, 73, 82, 100, 106, 113, 114, 115], "u_1": [2, 27, 29, 30, 31, 32, 53, 55, 61, 76, 82, 91, 107, 113, 115], "u_2": [2, 27, 29, 30, 31, 55, 61, 65, 91, 107], "u_d": [0, 2, 8, 41, 53, 55, 57, 73, 74, 83, 84, 89, 104, 113], "u_f": [73, 83, 106, 108, 109], "u_g": 100, "u_h": [6, 8, 12, 13, 28, 29, 31, 33, 34, 35, 36, 54, 56, 62, 65, 76, 83, 86, 87, 91, 103, 105, 108, 109, 112, 113], "u_i": [13, 24, 31, 38, 40, 42, 64, 76, 82, 83, 100, 105, 106, 108, 109, 113, 115], "u_j": [38, 40, 100, 105], "u_k": [41, 65, 98, 100, 101], "u_l": [29, 64, 65, 66, 78, 100, 101], "u_m": [27, 100, 115], "u_n": [12, 27, 31, 76, 97, 100, 113], "u_p": 79, "u_r": [31, 78], "u_t": [6, 62], "u_x": 82, "ud": [2, 73], "uext": 66, "uh": 70, "uhat": [0, 1, 5, 7, 18, 19, 91], "ui": [37, 38, 39], "uin": [7, 16, 18, 19, 22, 81, 85], "ulrich": 63, "umberto": 14, "umfpackinvers": 58, "un": 115, "unbound": [98, 99, 119, 120], "unconstrain": 82, "under": [8, 12, 47, 79, 84, 99], "underbrac": [1, 28, 29, 32, 47, 48, 51, 56, 79, 86, 101], "underli": 65, "underlin": [8, 13, 63, 65, 84, 105, 109], "understand": [84, 90], "understood": [24, 34, 69, 76, 78, 87, 102], "undocu": 55, "unew": 118, "unfortun": [35, 39], "uniform": [8, 12, 15, 33, 48, 65, 77], "uniformli": [12, 100, 103], "union": 76, "uniqu": [24, 27, 28, 29, 30, 31, 32, 50, 53, 76, 79, 83, 88, 90, 104, 105], "unisolv": 34, "unit": [8, 11, 75, 76, 79, 102, 104, 105, 108], "unit_cub": [46, 55, 116], "unit_squar": [0, 1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 15, 17, 21, 23, 24, 42, 43, 47, 48, 49, 50, 51, 55, 56, 57, 58, 63, 64, 66, 67, 69, 73, 76, 80, 84, 86, 88, 91, 103, 106, 108, 109, 110, 113, 114, 115, 118, 121], "uniti": 109, "unknown": [33, 40, 58, 59, 77, 79, 112, 113, 114], "unless": [83, 98], "unlik": 37, "unnecessari": 33, "unset": 76, "unsetdeform": [0, 55, 76], "unsetpml": 76, "uold": 118, "up": [1, 6, 8, 12, 13, 14, 15, 24, 28, 30, 31, 49, 51, 58, 63, 66, 79, 83, 97, 98, 99, 101, 109, 111, 112, 113, 121], "updat": [0, 10, 11, 21, 47, 49, 50, 51, 55, 58, 63, 76, 81, 83, 114], "updatedoft": [10, 76], "upgrad": [20, 103], "upost": 86, "upper": [33, 34, 35, 51, 81, 89, 103, 109], "upwind": [4, 6], "uri": 5, "us": [0, 1, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 21, 24, 25, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 67, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 90, 91, 97, 98, 100, 101, 102, 103, 105, 106, 108, 109, 111, 112, 113, 118, 121], "usag": 77, "user": 46, "usethread": 15, "usual": [1, 6, 9, 13, 33, 35, 40, 55, 68, 81, 84, 86, 108, 111], "ut": 115, "uup": [4, 5, 6], "uvec": 70, "uvert": [37, 39], "uxi": 82, "uzawa": 84, "v": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 115, 118, 121], "v0": [114, 118], "v1": [3, 22, 23, 70, 85], "v1loc": 23, "v2": [3, 23, 70, 86], "v2loc": 23, "v3": 70, "v4": 20, "v_": [8, 9, 12, 13, 31, 34, 42, 53, 63, 64, 67, 79, 83, 100, 103], "v_0": [8, 53, 63, 65, 66, 67, 82, 83, 100, 101, 104, 114], "v_1": [2, 28, 30, 31, 55, 63, 66, 84, 100, 101, 107], "v_2": [2, 28, 30, 31, 55, 84, 107], "v_d": [8, 101, 104], "v_g": 31, "v_h": [6, 7, 8, 12, 13, 24, 28, 29, 31, 34, 35, 36, 54, 62, 65, 68, 76, 83, 84, 86, 87, 91, 103, 105, 106, 109, 112, 113], "v_i": [31, 38, 40, 42, 83, 104, 105, 108, 109], "v_j": [40, 99, 105, 108], "v_k": 65, "v_l": [63, 64, 65, 66, 121], "v_m": [32, 115], "v_n": [3, 27, 32, 104, 121], "v_r": [31, 121], "v_t": [8, 9, 24, 34, 62, 88], "val": [23, 42, 76], "valid": 98, "valu": [0, 1, 2, 3, 12, 15, 24, 27, 31, 33, 34, 35, 36, 41, 42, 47, 48, 50, 55, 58, 61, 62, 64, 66, 73, 76, 78, 79, 88, 89, 90, 97, 100, 101, 103, 105, 106, 108, 109, 112, 113, 121], "van": 118, "vanish": [12, 34, 36, 49, 79, 80, 86, 98, 100], "varepsilon": [5, 27, 32, 47, 48, 49, 56, 60, 62, 69, 71, 74, 83, 90, 99, 103, 106, 107], "varepsilon_": 70, "vari": [78, 112], "variabl": [1, 3, 10, 15, 20, 37, 38, 39, 40, 46, 51, 53, 54, 55, 56, 57, 61, 64, 76, 89, 91, 112, 113, 121], "variant": [84, 105, 121], "variat": [8, 12, 30, 34, 45, 46, 53, 60, 67, 73, 75, 76, 78, 82, 83, 84, 86, 87, 91, 101, 104, 105, 114], "varphi": [9, 13, 32, 34, 62, 64, 80, 90, 98, 99, 100], "varphi_": [8, 9, 13, 64, 66, 79, 90, 106], "varphi_i": [9, 12, 13, 36, 80, 99, 105, 106, 108], "varphi_j": [12, 13, 105], "varphi_l": 64, "varphi_t": [9, 13, 62], "vb": [76, 78], "vdot": [105, 111], "vec": [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 33, 34, 35, 37, 38, 39, 42, 43, 44, 47, 48, 49, 50, 51, 55, 56, 57, 58, 62, 63, 64, 66, 67, 69, 70, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 106, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "vecmap": 14, "vector": [3, 4, 6, 10, 11, 12, 14, 21, 23, 27, 29, 33, 34, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 60, 62, 63, 64, 66, 69, 70, 73, 75, 76, 78, 82, 83, 84, 88, 96, 100, 101, 105, 106, 108, 109, 115, 121], "vectorh1": [0, 17, 19, 22, 33, 55, 56, 58, 74, 81, 82, 85, 103], "vectori": 72, "vectorl2": [18, 19, 116, 121], "vectormap": 14, "vectorvalu": 12, "vegt": 118, "vel": 7, "veloc": [7, 12, 16, 58, 59, 84, 114], "venv": 46, "veri": [0, 6, 21, 24, 33, 46, 49, 51, 55, 64, 65, 66, 74, 76, 82, 83, 89, 98, 103, 108, 112, 115], "verif": 75, "verifi": [12, 28, 30, 41, 48, 50, 54, 64, 83, 89, 99, 100, 103, 106, 111, 112], "versa": [104, 121], "version": [1, 2, 3, 8, 20, 46, 50, 72, 79, 81, 103, 105, 110, 121], "vert": 10, "vertex": [9, 12, 33, 34, 36, 37, 39, 62, 76, 80, 83, 104, 109, 110], "vertic": [8, 12, 33, 36, 37, 39, 54, 59, 67, 69, 76, 79, 104, 109, 110], "vf": [7, 34], "vhat": [0, 1, 3, 5, 7, 18, 19, 91], "vi": [37, 38, 39], "via": [1, 3, 20, 23, 28, 30, 40, 46, 47, 51, 64, 65, 79, 99, 105, 106, 108, 109, 115, 121], "vice": [104, 121], "view": 14, "violat": [12, 36], "virtual": [46, 55], "viscos": 7, "visibl": [10, 76], "visual": [15, 17, 25, 109], "vnum": 76, "vol": [10, 11, 15, 22, 33, 34, 35, 38, 39, 76, 78, 96, 108, 109], "vol_or_bnd": [10, 76], "voltag": 82, "volum": [0, 6, 8, 55, 75, 76, 78, 108], "vorb": [10, 76, 78], "vt": 7, "vvert": [37, 39], "w": [0, 3, 4, 5, 6, 8, 9, 12, 18, 19, 21, 27, 28, 30, 31, 32, 33, 34, 35, 41, 42, 47, 49, 51, 53, 63, 64, 67, 68, 69, 70, 73, 75, 79, 81, 82, 88, 98, 100, 101, 104, 105, 109, 113, 118, 119, 120, 121], "w_": [49, 83, 99], "w_0": 53, "w_2": 99, "w_e": 100, "w_h": [12, 28, 31, 35, 67, 68, 69, 83], "w_i": [51, 83, 105, 109], "w_k": [49, 66], "w_l": [65, 66], "w_n": [3, 49], "w_p": 99, "wa": [1, 28, 36, 37, 65, 100, 103, 105], "wai": [1, 3, 12, 13, 25, 31, 33, 35, 75, 82, 83, 106, 110], "wall": [7, 12, 16, 18, 19, 22, 81, 85], "want": [5, 20, 25, 34, 35, 40, 49, 54, 55, 66, 76, 82, 83, 84, 101, 103, 113, 121], "warbuton": 121, "warn": [38, 39, 55], "watch": 77, "wave": [43, 46, 112, 118], "wavi": 6, "wb_fulledg": 76, "wb_withedg": [76, 77], "we": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 69, 73, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 97, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 111, 112, 113, 114, 115, 118, 121], "weak": [8, 12, 29, 31, 43, 46, 56, 57, 58, 73, 78, 83, 88, 90, 98, 99, 100, 106, 113], "weaker": [8, 31, 55], "weakli": 55, "webglscen": 121, "webgui": [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 33, 34, 35, 37, 38, 39, 42, 43, 46, 47, 48, 49, 55, 56, 57, 58, 63, 64, 66, 73, 74, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 92, 96, 97, 103, 108, 109, 110, 113, 114, 115, 116, 118, 119, 120, 121], "webgui_jupyter_widget": [46, 121], "weight": [8, 13, 34, 35, 79, 104, 111], "well": [12, 27, 36, 40, 41, 51, 61, 62, 64, 65, 67, 76, 79, 81, 83, 84, 90, 97, 100, 104, 108, 109, 112, 115, 118, 121], "were": [32, 62, 91, 99], "wf": 34, "what": [1, 2, 3, 8, 23, 28, 30, 34, 36, 50, 51, 75, 76, 77, 86, 87, 88, 89, 91, 105], "wheel": 20, "when": [0, 25, 40, 47, 48, 63, 88, 98], "whenev": 48, "where": [0, 1, 2, 3, 6, 8, 9, 12, 20, 27, 33, 34, 35, 36, 37, 39, 42, 51, 54, 56, 61, 62, 63, 64, 65, 66, 68, 69, 73, 76, 78, 79, 80, 82, 83, 84, 100, 102, 103, 105, 106, 108, 109, 112, 113, 115, 118, 121], "whether": 6, "which": [0, 1, 5, 8, 10, 12, 15, 20, 23, 27, 28, 29, 30, 32, 34, 35, 37, 38, 40, 46, 47, 49, 50, 54, 55, 58, 60, 63, 64, 65, 66, 76, 79, 81, 82, 83, 84, 88, 97, 98, 99, 100, 102, 103, 105, 106, 108, 109, 110, 112, 113], "while": [4, 5, 16, 33, 35, 48, 50, 51, 58, 63, 64, 119, 120, 121], "who": 105, "whole": [53, 55, 73, 90, 100, 104, 105], "whose": 100, "why": [30, 55, 100], "widehat": [1, 3, 5, 8, 9, 12, 13, 36, 81, 88, 91, 99, 106], "wider": 9, "widetild": [9, 33, 37, 42, 47, 73, 86, 99, 100, 104, 109], "widget": [47, 121], "width": [23, 40, 66, 109], "wien": [25, 45, 46], "wiki": [52, 111], "wikipedia": [52, 111], "wind": [4, 6], "window": 20, "winther": [60, 83], "wip": 76, "wirebasket": 76, "wirebasket_dof": [18, 19], "wise": [6, 15, 24, 33, 34, 37, 56, 58, 60, 70, 76, 86, 88, 106, 108, 121], "witch": 37, "within": [1, 10, 15, 20, 25, 29, 46, 65, 80, 101, 108, 111, 121], "without": [0, 21, 25, 30, 34, 39, 46, 50, 54, 65, 84], "wolfgang": 63, "work": [6, 15, 24, 25, 26, 40, 46, 47, 62, 70, 100, 108, 118], "worker": 15, "world": [15, 20], "worri": 73, "wors": 33, "worth": 103, "would": [5, 6, 35, 54], "wr": [49, 81], "wrap": 14, "write": [11, 13, 28, 47, 51, 54, 56, 64, 76, 79, 82, 105, 106, 109, 112], "written": [28, 86, 111, 115], "wrn": [49, 81], "wrt": 54, "www": [20, 23], "x": [0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 28, 30, 34, 36, 37, 38, 39, 41, 42, 43, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 62, 63, 64, 66, 67, 69, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 85, 86, 88, 89, 91, 92, 96, 97, 98, 99, 100, 101, 102, 103, 104, 108, 109, 110, 112, 113, 114, 115, 116, 118, 121], "x2": 86, "x_": [8, 12, 49], "x_0": 49, "x_1": [13, 61, 98], "x_2": 61, "x_i": [8, 12, 33, 36, 51], "x_j": [12, 51], "x_k": [13, 49], "x_m": 13, "x_n": [49, 98], "xaux": [17, 19], "xc": 66, "xi": [64, 79, 100], "xlabel": [103, 115], "xlam": 34, "xscale": 33, "xsigma": 34, "xu": [31, 65, 105], "xw": 34, "xwf": 34, "xx": 46, "xy": 75, "y": [0, 2, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 22, 30, 34, 37, 38, 39, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 55, 63, 66, 70, 74, 75, 76, 78, 79, 80, 81, 84, 85, 88, 89, 92, 96, 97, 100, 101, 102, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118, 121], "y_": [111, 112], "y_0": [112, 115], "y_j": [111, 112], "y_k": 112, "yet": [0, 49, 66], "ylabel": 115, "you": [10, 20, 25, 26, 46, 51, 73, 75, 77, 103, 108], "young": [30, 81], "your": [20, 25, 26, 46, 75, 77, 102], "yousef": 47, "yscale": [33, 48, 49, 50, 51, 80, 115], "z": [22, 29, 30, 41, 47, 48, 49, 53, 64, 74, 76, 83, 85, 100, 101, 104, 105, 111, 112, 115, 116, 118, 121], "z_": 101, "z_0": [41, 100], "z_i": 105, "z_k": [41, 100], "z_l": 100, "za": 111, "zaglmayr": [72, 80, 88], "zampini": 14, "zerbinati": 14, "zero": [1, 2, 12, 53, 58, 66, 73, 81, 84, 100, 106, 108, 112, 121], "zhao": 83, "zikatanov": 31, "zip": [33, 38, 39, 42, 80, 103, 115], "znew": 118, "zold": 118, "zulehn": 67, "zz": 33}, "titles": ["40. Nitsche\u2019s Method for boundary and interface conditions", "42. Hybrid DG for elliptic equations", "41. DG - Methods for elliptic problems", "44. Fourth Order Equation", "39. Instationary Transport Equation", "43. Splitting Methods for the time-dependent convection diffusion equation", "38. Stationary Transport Equation", "45. H(div)-conforming Stokes", "24. Finite element error analysis", "20. Finite Element Method", "21. Implementation of Finite Elements", "23. Implement our own system assembling", "25. Non-conforming Finite Element Methods", "22. Finite element system assembling", "88. NGSolve - PETSc interface", "84. Distributed Meshes and Spaces", "<no title>", "<no title>", "<no title>", "<no title>", "83. Introduction to MPI with mpi4py", "86. Iteration methods in parallel", "89. Solving Stokes in parallel", "87. Using PETSc", "85. Consistent and Distributed Vectors", "This repository contains an interactive introduction to the Finite Element Method", "<no title>", "7. Basic properties", "10. Coercive variational problems and their approximation", "9. Riesz representation theorem and symmetric variational problems", "12. Exercises", "11. Inf-sup stable variational problems", "8. Projection onto subspaces", "26. A posteriori error estimates", "29. Equilibrated Residual Error Estimates", "28. Goal driven error estimates", "27. The residual error estimator", "82. BDDC - Preconditioner", "80. FETI methods", "81. FETI-DP", "77. Introduction to Non-overlapping Domain Decomposition", "78. Traces spaces", "Implement a parallel BDDC preconditioner", "Helmholtz Equation", "Grating", "An Interactive Introduction to the Finite Element Method", "An Interactive Introduction to the Finite Element Method", "61. The Chebyshev Method", "58. The Richardson Iteration", "62. Conjugate Gradients", "59. The Gradient Method", "60. Preconditioning", "57. Basic Iterative Methods", "35. Abstract Theory", "36. Abstract theory for mixed finite element methods", "33. Boundary Conditions", "37. Parameter Dependent Problems", "34. Mixed Methods for second order equations", "32. Stokes Equation", "Non-linear dynamics", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\nHellinger Reissner mixed formulation", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\n\\(\\DeclareMathOperator{\\opcurl}{curl}\\)\n\\(\\DeclareMathOperator{\\eps}{\\varepsilon}\\)\nReduced symmetry methods", "Tangential displacement normal normal stress continuous finite elements", "69. Multigrid and Multilevel Methods", "71. Analysis of the Multigrid Iteration", "70. Analysis of the multi-level preconditioner", "72. Multi-level Extension", "The Hellan Herrmann Johnson Method for Kirchhoff plates", "Relationship between HHJ and TDNNS", "Reissner Mindlin Plates", "Nonlinear Shells", "3D-TDNNS Elasticity and Reissner Mindlin Plate elements", "Preamble", "2. Boundary Conditions", "5. 3D Solid Mechanics", "6. Exercises", "1. Solving the Poisson Equation", "4. Iterative Solvers", "3. Variable Coefficients", "30. hp - Finite Elements", "31. Implementation of High Order Finite Elements", "74. The Bramble-Pasciak Transformation", "75. A Small Number of Constraints", "76. Parameter Dependent Problems", "73. Structure of Saddle-point Problems", "<no title>", "49. Finite Element Error Analysis", "50. Error Analysis in \\(L_2 \\times H^1\\)", "48. Finite Elements in \\(H(\\operatorname{div})\\)", "46. Application of the abstract theory", "47. The function space \\(H(\\operatorname{div})\\)", "51. Hybridization Techniques", "Friedrichs\u2019 Inequality", "<no title>", "<no title>", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "13. Generalized derivatives", "14. Sobolev spaces", "15. Trace theorems and their applications", "16. Equivalent norms on \\(H^1\\) and on sub-spaces", "19. Exercises", "18. Experiments with norms", "17. The weak formulation of the Poisson equation", "63. Additive Schwarz Methods", "64. Some Examples of ASM preconditioners", "68. Exercise: Robust preconditioners", "66. Domain Decomposition with minimal overlap", "67. Overlapping Domain Decomposition Methods", "65. Schwarz preconditioners for high order finite elements", "Runge Kutta Methods", "Single-step methods", "52. Heat Equation", "53. Wave Equation", "Exponential Integrators for Parabolic Equations", "<no title>", "<no title>", "54. Mass-lumping and Local time-stepping", "56. Nano-optics: A ring-resonator", "Nano-optics: A ring-resonator", "55. Discontinuous Galerkin for the Wave Equation"], "titleterms": {"": [0, 38, 41, 103], "0": 3, "1": [12, 68, 76, 87, 100, 101, 106], "1d": [79, 102], "2": [100, 109], "2d": 30, "3d": [71, 74], "A": [33, 45, 46, 82, 107, 119, 120], "One": 30, "The": [12, 15, 20, 33, 36, 45, 46, 47, 48, 50, 51, 62, 64, 67, 76, 79, 81, 90, 96, 97, 100, 104, 105, 118], "With": 108, "abstract": [45, 46, 53, 54, 89], "ad": 108, "addit": [45, 105], "algebra": 75, "algorithm": [33, 64, 66], "alpha": 48, "an": [25, 45, 46], "analysi": [8, 45, 64, 65, 83, 86, 87, 108, 109], "appendix": 45, "applic": [89, 100], "approxim": [8, 28, 31, 64], "around": 8, "asm": 106, "assembl": [11, 13], "b": 107, "base": [79, 108, 112], "basi": [76, 80], "basic": [27, 52], "bddc": [37, 42], "between": 68, "bilienarform": 75, "bilinear": [1, 30, 76], "bisect": 33, "block": [30, 84, 105, 110], "bottom": 41, "bound": 105, "boundari": [0, 2, 8, 41, 53, 55, 56, 57, 66, 73, 83], "brambl": [81, 102], "build": 30, "butcher": 111, "c": 3, "calcul": 11, "can": 11, "case": 79, "chebyshev": 47, "class": 15, "classif": 112, "cl\u00e9ment": 36, "coars": [83, 108, 109], "coeffici": [33, 78], "coefficientfunct": 75, "coerciv": [1, 28, 30], "commut": 68, "comparison": 109, "complement": 84, "complex": 30, "comput": [13, 34, 66, 75], "conda": 20, "condit": [0, 2, 8, 30, 53, 55, 56, 57, 73, 83, 112], "conform": [7, 12], "conjug": 49, "consist": 24, "constrain": [53, 84], "constraint": 82, "contain": 25, "continu": [1, 3, 62], "convect": 5, "converg": [64, 111], "correct": [45, 46, 83], "curl": 61, "cycl": 64, "data": 66, "dd": 109, "declaremathoper": [60, 61], "decomposit": [40, 45, 46, 66, 108, 109], "definit": 100, "delta": 34, "depend": [5, 45, 46, 56, 83], "deriv": [30, 98], "detail": 118, "dg": [1, 2], "diagon": [106, 111], "diagram": 68, "differenti": [45, 121], "diffus": 5, "dirichlet": [2, 8, 53, 56, 83], "discontinu": [6, 45, 46, 121], "discret": [1, 54, 121], "displac": 62, "distribut": [15, 24], "div": [7, 33, 60, 61, 88, 90], "diverg": 62, "domain": [40, 41, 45, 46, 90, 100, 102, 108, 109], "dp": 39, "driven": 35, "dual": [60, 75, 89], "dynam": 59, "edg": [33, 41, 100], "ee": 112, "effici": [36, 66, 121], "eigenvalu": 121, "elast": [45, 71, 74], "element": [8, 9, 10, 11, 12, 13, 15, 25, 45, 46, 54, 58, 62, 71, 76, 79, 80, 86, 88, 110], "ellipt": [1, 2], "enough": 30, "ep": 61, "eqival": 102, "equat": [1, 3, 4, 5, 6, 12, 43, 45, 46, 53, 57, 58, 67, 76, 83, 104, 112, 113, 114, 115, 121], "equilibr": 34, "equival": 101, "error": [8, 33, 34, 35, 36, 46, 54, 62, 79, 86, 87], "essenti": 73, "estim": [8, 33, 34, 35, 36, 46, 54, 62, 65, 79, 103], "euler": [112, 113], "evalu": 102, "exampl": [30, 35, 54, 56, 84, 106, 111], "exercis": [11, 30, 75, 102, 107, 111, 112, 121], "expand": 49, "experi": [48, 75, 103, 109], "explicit": [111, 112], "exponenti": 115, "extend": 66, "extens": [66, 100], "feti": [38, 39], "finest": 63, "finit": [8, 9, 10, 12, 13, 15, 25, 45, 46, 54, 58, 62, 76, 79, 80, 86, 88, 110], "first": [12, 30], "flux": [33, 83], "form": [1, 30, 76], "formul": [58, 60, 62, 74, 76, 83, 89, 104, 113], "fortin": 54, "fourth": 3, "fraction": 102, "framework": 34, "free": 65, "friedrich": [92, 95, 102, 103], "from": [30, 63], "function": [11, 62, 76, 80, 90, 102, 112], "galerkin": [6, 45, 46, 113, 121], "gauss": 51, "gener": [34, 98, 100, 105], "geometr": [8, 75], "geometri": 118, "get": 75, "goal": 35, "grade": 8, "gradient": [49, 50, 51], "graph": 108, "grate": 44, "green": 33, "grid": [83, 108, 109], "gridfunct": 75, "h": [7, 33, 41, 68, 76, 87, 88, 90, 100, 101, 106], "happen": 20, "hdg": 1, "heat": 113, "hellan": 67, "helling": 60, "helmholtz": [43, 45], "herrmann": 67, "hhj": 68, "high": [8, 46, 80, 110], "hilbert": [102, 105], "hp": 79, "hybrid": [0, 1, 3, 91], "hyperbol": 112, "i": 30, "ie": 112, "implement": [10, 11, 42, 80, 121], "implicit": [111, 112, 113], "improv": 112, "incompress": 56, "inequ": [45, 92, 95, 96, 97, 102, 103], "inf": [30, 31], "inner": 24, "inspect": 11, "instal": [20, 46], "instationari": 4, "integr": [11, 100, 112, 115], "interact": [25, 45, 46], "interfac": [0, 14, 55], "interior": 3, "interpol": [41, 79, 100, 102], "introduct": [20, 25, 40, 45, 46], "invers": 103, "ipyparallel": 20, "iter": [21, 45, 46, 47, 48, 51, 52, 64, 77, 84], "jacobi": [51, 105], "johnson": 67, "kirchhoff": 67, "korn": 103, "krylov": 49, "kutta": 111, "l_2": [8, 79, 87, 106], "lagrang": 40, "laplac": 121, "lbb": 54, "legendr": 79, "lemma": [12, 102, 105], "level": [63, 65, 66, 83, 109], "librari": 20, "lift": 34, "linear": [59, 74, 75, 76, 112], "linearform": 75, "literatur": 46, "local": [86, 118], "lump": 118, "mark": 33, "mass": 118, "materi": 56, "mathbb": 105, "matric": [13, 24, 63], "matrix": [11, 24], "maxwel": 83, "mean": [30, 102], "mechan": 74, "mesh": [8, 15, 33, 108], "method": [0, 2, 3, 5, 6, 9, 12, 21, 25, 38, 45, 46, 47, 50, 51, 52, 53, 54, 57, 60, 61, 63, 67, 76, 83, 105, 108, 109, 111, 112, 113, 114, 118], "mid": 112, "mindlin": [69, 71], "minim": [30, 49, 53, 84, 108, 109], "mix": [45, 46, 53, 54, 57, 60, 89], "ml": 65, "model": 75, "mpi": 20, "mpi4pi": 20, "multi": [65, 66], "multigrid": [45, 46, 63, 64], "multilevel": 63, "multipl": 24, "n": 105, "nano": [119, 120], "natur": [41, 73], "nearli": [56, 65], "netgen": 75, "newmark": 114, "ngsolv": [14, 33, 34, 46, 76], "nitsch": 0, "nn": 62, "non": [12, 40, 45, 46, 59], "nonlinear": 70, "norm": [8, 41, 75, 101, 103, 106], "normal": [62, 90], "number": 82, "numer": [45, 97], "one": 100, "onto": [30, 32], "opcurl": [61, 68], "opdiv": [60, 61], "opencascad": 75, "oper": [24, 36, 54, 100, 121], "operatornam": [33, 88, 90], "optic": [119, 120], "optim": [48, 64, 65], "order": [3, 8, 30, 45, 46, 53, 57, 80, 110], "ordinari": 45, "orthogon": 79, "our": 11, "over": 100, "overlap": [40, 45, 46, 105, 108, 109, 110], "own": 11, "p": 12, "packag": 20, "parabol": [45, 112, 115], "parallel": [21, 22, 42, 45, 46], "paralleldof": 15, "paramet": [40, 48, 56, 83], "part": 100, "partit": 108, "pasciak": 81, "penalti": [3, 56, 83], "petsc": [14, 20, 23], "piec": 62, "piola": 88, "pip": 20, "plate": [45, 67, 69, 71], "poincar\u00e9": [96, 102, 103], "point": [45, 46, 84, 102, 112], "poisson": [76, 104], "polynomi": [47, 79], "post": 86, "posteriori": [33, 46], "practic": 45, "preambl": 72, "precondit": [51, 63, 84], "precondition": [14, 37, 38, 42, 51, 63, 65, 82, 105, 106, 107, 109, 110], "primal": [60, 89], "problem": [2, 28, 29, 30, 31, 45, 46, 49, 53, 56, 83, 84], "process": 86, "product": 24, "project": [32, 63, 79, 82], "proof": 97, "properti": [27, 64], "prove": 54, "r": 105, "real": 30, "recoveri": 33, "red": 33, "reduc": 61, "refin": [8, 33], "regular": 65, "reissner": [60, 69, 71], "relationship": 68, "relax": 48, "reliabl": 36, "repeat": 30, "repositori": 25, "represent": 29, "residu": [34, 36], "reson": [119, 120], "richardson": [21, 48, 51], "riesz": 29, "ring": [119, 120], "rk2": 112, "robust": [83, 107], "rule": 112, "rung": 111, "saddl": [45, 46, 84], "scale": 102, "schur": 84, "schwarz": [105, 110], "second": [12, 30, 45, 46, 53, 57], "segment": 80, "seidel": 51, "shell": [45, 70], "shift": 104, "sigma": 34, "simpl": [111, 112], "singl": 112, "singular": 8, "small": [82, 106], "smooth": 62, "smoother": 83, "smoth": 64, "sobolev": [45, 46, 76, 99, 100, 102], "solid": 74, "solv": [12, 22, 49, 76, 121], "solver": [45, 46, 77], "some": [45, 106], "space": [15, 41, 45, 46, 49, 58, 76, 90, 99, 100, 101, 102, 105, 108, 113], "split": 5, "squar": [100, 102], "stabil": [6, 112], "stabl": 31, "start": 75, "stationari": 6, "step": [20, 102, 112, 113, 114, 118], "stoke": [7, 12, 22, 53, 54, 58, 83], "strang": 12, "stress": 62, "structur": 84, "sub": [41, 45, 46, 90, 100, 101, 105], "subspac": 32, "sup": [30, 31], "symmetr": 29, "symmetri": 61, "system": [11, 13, 30, 53], "tableau": 111, "tangenti": 62, "tdnn": [62, 68, 71], "techniqu": 91, "term": 106, "test": 121, "theorem": [29, 100, 104], "theori": [45, 46, 53, 54, 89], "thi": 25, "three": 20, "time": [5, 45, 46, 87, 113, 114, 118], "trace": [41, 90, 97, 100, 103], "transform": [81, 88], "transport": [4, 6], "trapezoid": 112, "triangl": [12, 79, 80], "two": 83, "upper": 105, "us": [11, 20, 23], "v": 64, "valu": [30, 102], "varepsilon": 61, "variabl": [33, 78], "variat": [28, 29, 31, 58, 62, 74, 113], "vector": [13, 24], "verif": 97, "verlet": 118, "version": 102, "vertex": 8, "visual": 76, "wave": [45, 114, 121], "we": 11, "weak": [76, 104], "wise": 62, "within": 53, "without": 20, "work": 75, "zhu": 33, "zienkiewicz": 33}}) \ No newline at end of file diff --git a/secondorder/erroranalysis.html b/secondorder/erroranalysis.html index 4ea34b78..5015b43e 100644 --- a/secondorder/erroranalysis.html +++ b/secondorder/erroranalysis.html @@ -32,7 +32,7 @@ - + @@ -59,7 +59,7 @@ - + @@ -534,38 +534,21 @@

49. Finite Element Error Analysis \[ \int (\sigma - I_h \sigma) \tau_h + -\int \underbrace{(I - P_h) \opdiv \sigma}_{\in V_h^\ast} \;\underbrace{ q_h}_{\in V_h} + \int \underbrace{ \opdiv \tau_h}_{\in V_h} \, \underbrace{ (u-P_h u) }_{\in V_h^\ast} +\int \underbrace{(I - P_h) \opdiv \sigma}_{\in V_h^\bot} \;\underbrace{ q_h}_{\in V_h} + \int \underbrace{ \opdiv \tau_h}_{\in V_h} \, \underbrace{ (u-P_h u) }_{\in V_h^\bot} \]

Thanks to orthogonality, the second and third term vanish !

-

Thus, we get the error estimate -$\( +

Thus, we get the error estimate

+
+\[ \| \sigma_h - I_h \sigma \|_{H(\opdiv)} + \| u_h - P_h u \|_{L_2} \preceq \sup_{\tau_h} \frac{ \int (\sigma - I_h \sigma) \tau_h} { \| \tau_h \|_{H(\opdiv)}} \leq \| \sigma - I_h \sigma \|_{L_2} -\)$

+\]

Using the triangle inequality for \(\sigma\), we get

\[ \| \sigma_h - \sigma \|_{L_2} + \| u_h - P_h u \|_{L_2} \preceq \| \sigma - I_h \sigma \|_{L_2} \]

The flux error is as good as we can interpolate into the flux space. Since the finite element space for \(u_h\) is of lower order, the error \(u - u_h\) is in general of lower order. But, the filtered error \(\| u_h - P_h u_h \|\) has the better order.

-
-

49.1. Local post-processing#

-

Since \(\nabla u = \lambda^{-1} \sigma\), we can reconstruct a better approximation \(\widetilde u\) by small, element-wise problems:

-
-\[ -\widetilde u = \operatorname{arg}\min_{v_h \in P^{k+1} \atop \int_T v_h = \int_T u_h} \| \lambda \nabla v_h -  \sigma \|_{L_2, \lambda^{-1}}^2 -\]
-

This optimization problems can be written as a mixed variational problem:

-

Find: \(\widetilde u \in P^{k+1,dc}\) and \(p_h \in P^0\):

-
-\[\begin{split} -\begin{array}{ccccll} -\sum_T \int_T \lambda \nabla \widetilde u \nabla \widetilde v -& + & \int_{\Omega} \widetilde v_h p_h & = & \sum_T \int_T \sigma_h \nabla \widetilde v_h & \forall \, \widetilde v_h \\ -\int_{\Omega} \widetilde u_h q_h & & & = & \int_{\Omega} u_h q_h & -\forall q_h -\end{array} -\end{split}\]
from ngsolve import *
@@ -605,7 +588,7 @@ 
49.1. Local post-processing
The flux
-
+

The scalar part of the solution:

@@ -615,9 +598,27 @@

49.1. Local post-processing -

+ -

Now do the postprocessing. This requires to solve decoupled problems on every element, what is cheap.

+
+

49.1. Local post-processing#

+

Since \(\nabla u = \lambda^{-1} \sigma\), we can reconstruct a better approximation \(\widetilde u\) by small, element-wise problems:

+
+\[ +\widetilde u = \operatorname{arg}\min_{v_h \in P^{k+1} \atop \int_T v_h = \int_T u_h} \| \lambda \nabla v_h -  \sigma \|_{L_2, \lambda^{-1}}^2 +\]
+

This optimization problems can be written as a mixed variational problem:

+

Find: \(\widetilde u \in P^{k+1,dc}\) and \(p_h \in P^0\):

+
+\[\begin{split} +\begin{array}{ccccll} +\sum_T \int_T \lambda \nabla \widetilde u \nabla \widetilde v +& + & \int_{\Omega} \widetilde v_h p_h & = & \sum_T \int_T \sigma_h \nabla \widetilde v_h & \forall \, \widetilde v_h \\ +\int_{\Omega} \widetilde u_h q_h & & & = & \int_{\Omega} u_h q_h & +\forall q_h +\end{array} +\end{split}\]
+

This requires to solve decoupled problems on every element, what is cheap.

V2 = L2(mesh, order=order+1)
@@ -641,7 +642,7 @@ 
49.1. Local post-processing
-
+
diff --git a/secondorder/finiteelements.html b/secondorder/finiteelements.html index b12ed59d..802ef0d6 100644 --- a/secondorder/finiteelements.html +++ b/secondorder/finiteelements.html @@ -59,7 +59,7 @@ - + @@ -514,11 +514,12 @@

48. Finite Elements in \(BDM_k\) elements are defined as -$\( +

The \(BDM_k\) elements are defined as

+
+\[ V_T = [P^k]^2 -\)$ -the degrees of freedom are

+\]
+

the degrees of freedom are