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31 changes: 15 additions & 16 deletions DG/Nitsche.html
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
Expand Down Expand Up @@ -559,7 +559,7 @@ <h1><span class="section-number">35. </span>Splitting Methods for the time-depen
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;ngsolve.comp.LinearForm at 0x10e081970&gt;
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;ngsolve.comp.LinearForm at 0x116ba17b0&gt;
</pre></div>
</div>
</div>
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</div>
</div>
<div class="cell_output docutils container">
<script type="application/vnd.jupyter.widget-view+json">{"version_major": 2, "version_minor": 0, "model_id": "284774df2d88470db00555d9860e9b12"}</script><div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>factor parallel atormic
</pre></div>
</div>
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<script type="application/vnd.jupyter.widget-view+json">{"version_major": 2, "version_minor": 0, "model_id": "8b99ba606d724af4aa0fdcb2487299f5"}</script><div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">KeyboardInterrupt</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="n">line</span> <span class="mi">13</span>
<span class="g g-Whitespace"> </span><span class="mi">11</span> <span class="n">cnt</span> <span class="o">=</span> <span class="mi">0</span>
<span class="g g-Whitespace"> </span><span class="mi">12</span> <span class="n">SetNumThreads</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="ne">---&gt; </span><span class="mi">13</span> <span class="k">while</span> <span class="n">t</span> <span class="o">&lt;</span> <span class="n">tend</span><span class="p">:</span>
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="n">line</span> <span class="mi">15</span>
<span class="g g-Whitespace"> </span><span class="mi">13</span> <span class="k">while</span> <span class="n">t</span> <span class="o">&lt;</span> <span class="n">tend</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">14</span> <span class="n">t</span> <span class="o">+=</span> <span class="n">tau</span>
<span class="g g-Whitespace"> </span><span class="mi">15</span> <span class="n">aconv</span><span class="o">.</span><span class="n">Apply</span><span class="p">(</span><span class="n">gfu</span><span class="o">.</span><span class="n">vec</span><span class="p">,</span> <span class="n">convu</span><span class="p">)</span>
<span class="ne">---&gt; </span><span class="mi">15</span> <span class="n">aconv</span><span class="o">.</span><span class="n">Apply</span><span class="p">(</span><span class="n">gfu</span><span class="o">.</span><span class="n">vec</span><span class="p">,</span> <span class="n">convu</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">16</span> <span class="n">r</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">vec</span> <span class="o">-</span> <span class="n">convu</span> <span class="o">-</span> <span class="n">adiff</span><span class="o">.</span><span class="n">mat</span> <span class="o">*</span> <span class="n">gfu</span><span class="o">.</span><span class="n">vec</span>
<span class="g g-Whitespace"> </span><span class="mi">17</span> <span class="n">w</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">inv</span> <span class="o">*</span> <span class="n">r</span>

<span class="ne">KeyboardInterrupt</span>:
</pre></div>
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2 changes: 1 addition & 1 deletion FEM/erroranalysis.html
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
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2 changes: 1 addition & 1 deletion FEM/finiteelements.html
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
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12 changes: 6 additions & 6 deletions FEM/myFiniteElement.html
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18 changes: 9 additions & 9 deletions FEM/nonconforming.html
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
Expand Down Expand Up @@ -534,8 +534,8 @@ <h1><span class="section-number">19. </span>Non-conforming Finite Element Method
&amp; = &amp; A(u-v_h, w_h) + [ A(v_h, w_h) - A_h(v_h, w_h) ] + [ f_h(w_h) - f(w_h)]
\end{eqnarray*}\]</div>
<p>Divide by <span class="math notranslate nohighlight">\(\| u_h - v_h \| = \| w_h \|\)</span>, and use the continuity of <span class="math notranslate nohighlight">\(A(.,.)\)</span>:</p>
<div class="amsmath math notranslate nohighlight" id="equation-7c2b63b6-ffbe-4190-91d7-4cd56447dc3f">
<span class="eqno">(19.1)<a class="headerlink" href="#equation-7c2b63b6-ffbe-4190-91d7-4cd56447dc3f" title="Permalink to this equation">#</a></span>\[\begin{equation}
<div class="amsmath math notranslate nohighlight" id="equation-5b381b0a-1e18-49b0-ba1f-d09e3b5d158d">
<span class="eqno">(19.1)<a class="headerlink" href="#equation-5b381b0a-1e18-49b0-ba1f-d09e3b5d158d" title="Permalink to this equation">#</a></span>\[\begin{equation}
\label{equ_strang1a}
\| u_h - v_h \| \preceq \| u - v_h \| + \frac{|A(v_h, w_h) - A_h(v_h, w_h)|}{\| w_h \|} + \frac{ | f(w_h) - f_h(w_h) | } { \| w_h \| }
\end{equation}\]</div>
Expand Down Expand Up @@ -576,16 +576,16 @@ <h1><span class="section-number">19. </span>Non-conforming Finite Element Method
\right.
\)</span>$</p>
<p>To apply the first lemma of Strang, we have to verify the uniform coercivity</p>
<div class="amsmath math notranslate nohighlight" id="equation-a20394f9-182f-4b6b-b2a2-78dc34a23e22">
<span class="eqno">(19.2)<a class="headerlink" href="#equation-a20394f9-182f-4b6b-b2a2-78dc34a23e22" title="Permalink to this equation">#</a></span>\[\begin{equation}
<div class="amsmath math notranslate nohighlight" id="equation-00d7ef8b-716d-40f7-b8ad-9bc5d5f4a044">
<span class="eqno">(19.2)<a class="headerlink" href="#equation-00d7ef8b-716d-40f7-b8ad-9bc5d5f4a044" title="Permalink to this equation">#</a></span>\[\begin{equation}
\label{equ_uniformell}
\sum_T \frac{|T|}{3} \sum_{\alpha = 1}^3 |v_h(x_{T,\alpha})|^2 \geq
\alpha_1 \sum_T \int_T | v_h |^2 \, dx \qquad \forall \, v_h \in V_h,
\end{equation}\]</div>
<p>which is done by transformation to the reference element.
The consistency error can be estimated by</p>
<div class="amsmath math notranslate nohighlight" id="equation-102dcc08-6f8a-426d-bff0-b6364245256b">
<span class="eqno">(19.3)<a class="headerlink" href="#equation-102dcc08-6f8a-426d-bff0-b6364245256b" title="Permalink to this equation">#</a></span>\[\begin{equation}
<div class="amsmath math notranslate nohighlight" id="equation-47966c2c-fc39-42f0-85b8-fc82657dc7e3">
<span class="eqno">(19.3)<a class="headerlink" href="#equation-47966c2c-fc39-42f0-85b8-fc82657dc7e3" title="Permalink to this equation">#</a></span>\[\begin{equation}
\label{equ_consist}
| \int_T u_h v_h \, dx - \frac{|T|}{3} \sum_{\alpha=1}^3 u_h(x_\alpha) v_h(x_\alpha) |
\preceq h_T^2 \, \| \nabla u_h \|_{L_2(T)} \, \| \nabla v_h \|_{L_2(T)}
Expand Down Expand Up @@ -632,8 +632,8 @@ <h1><span class="section-number">19. </span>Non-conforming Finite Element Method
<p>The error can now be measured only in the discrete norm <span class="math notranslate nohighlight">\(\| u - u_h \|_{V_h}\)</span>.
\begin{lemma}
Under the above assumptions there holds</p>
<div class="amsmath math notranslate nohighlight" id="equation-7fdbcad3-c47e-42e2-afc9-17e45e993ec0">
<span class="eqno">(19.4)<a class="headerlink" href="#equation-7fdbcad3-c47e-42e2-afc9-17e45e993ec0" title="Permalink to this equation">#</a></span>\[\begin{equation}
<div class="amsmath math notranslate nohighlight" id="equation-55de7545-28e0-4d67-932c-dda941cb2974">
<span class="eqno">(19.4)<a class="headerlink" href="#equation-55de7545-28e0-4d67-932c-dda941cb2974" title="Permalink to this equation">#</a></span>\[\begin{equation}
\label{equ_strang2}
\| u - u_h \|_h \preceq \inf_{v_h \in V_h} \| u - v_h \|_h +
\sup_{w_h \in V_h} \frac{| A_h(u,w_h) - f_h(w_h) |}{\| w_h \|_h}
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
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4 changes: 2 additions & 2 deletions MPIparallel/PETSc_interface.html
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
Expand Down Expand Up @@ -489,7 +489,7 @@ <h1><span class="section-number">78. </span>NGSolve - PETSc interface<a class="h
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Starting 4 engines with &lt;class &#39;ipyparallel.cluster.launcher.MPIEngineSetLauncher&#39;&gt;
</pre></div>
</div>
<script type="application/vnd.jupyter.widget-view+json">{"version_major": 2, "version_minor": 0, "model_id": "d6bc7702d2b547e5bc809b8bd8ed07f6"}</script><div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0, 1, 2, 3]
<script type="application/vnd.jupyter.widget-view+json">{"version_major": 2, "version_minor": 0, "model_id": "34b99b7a62b14927bbb7879f739ec464"}</script><div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0, 1, 2, 3]
</pre></div>
</div>
</div>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Sub-space Correction Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/asm.html">51. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">52. Same Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">52. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/minimaldd.html">53. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/overlapping.html">54. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/exercise.html">55. Exercise: Robust preconditioners</a></li>
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