Update documentation

_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": {

_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",

_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."
    ]
   },
   {

_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",

timedependent/waves/wave-leapfrogDG.html

@@ -32,7 +32,7 @@
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+    <link rel="stylesheet" type="text/css" href="../../_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css" />
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