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JSchoeberl committed Mar 13, 2024
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6 changes: 3 additions & 3 deletions DG/splitting.html
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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
Expand Down Expand Up @@ -562,7 +562,7 @@ <h1><span class="section-number">38. </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 0x10c3c9c70&gt;
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;ngsolve.comp.LinearForm at 0x11ada72b0&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": "ba9046f118044b23a0bcffd9fe208ff8"}</script><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": "05f633c270554354b0804043dec12fd6"}</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">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>
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2 changes: 1 addition & 1 deletion FEM/erroranalysis.html
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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
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2 changes: 1 addition & 1 deletion FEM/finiteelements.html
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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
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18 changes: 9 additions & 9 deletions FEM/nonconforming.html
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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
Expand Down Expand Up @@ -553,8 +553,8 @@ <h2><span class="section-number">20.1. </span>The First Lemma of Strang<a class=

\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-11938fb5-577b-47e8-948a-86722fc3850d">
<span class="eqno">(20.1)<a class="headerlink" href="#equation-11938fb5-577b-47e8-948a-86722fc3850d" title="Permalink to this equation">#</a></span>\[\begin{equation}
<div class="amsmath math notranslate nohighlight" id="equation-0281bdfe-2ee2-484c-a7c2-c00264d06f1c">
<span class="eqno">(20.1)<a class="headerlink" href="#equation-0281bdfe-2ee2-484c-a7c2-c00264d06f1c" 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 @@ -599,16 +599,16 @@ <h2><span class="section-number">20.1. </span>The First Lemma of Strang<a class=
\right.
\end{split}\]</div>
<p>To apply the first lemma of Strang, we have to verify the uniform coercivity</p>
<div class="amsmath math notranslate nohighlight" id="equation-087c8978-72c0-4557-9da8-a5d4272143a9">
<span class="eqno">(20.2)<a class="headerlink" href="#equation-087c8978-72c0-4557-9da8-a5d4272143a9" title="Permalink to this equation">#</a></span>\[\begin{equation}
<div class="amsmath math notranslate nohighlight" id="equation-f20a8984-8fba-4b60-8011-6c4059497919">
<span class="eqno">(20.2)<a class="headerlink" href="#equation-f20a8984-8fba-4b60-8011-6c4059497919" 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-212407d6-e1e9-4e10-bff3-6acbd2d691d3">
<span class="eqno">(20.3)<a class="headerlink" href="#equation-212407d6-e1e9-4e10-bff3-6acbd2d691d3" title="Permalink to this equation">#</a></span>\[\begin{equation}
<div class="amsmath math notranslate nohighlight" id="equation-04940b44-70dc-4741-a7bb-e155e82d37fa">
<span class="eqno">(20.3)<a class="headerlink" href="#equation-04940b44-70dc-4741-a7bb-e155e82d37fa" 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 @@ -662,8 +662,8 @@ <h2><span class="section-number">20.2. </span>The Second Lemma of Strang<a class
<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-d798fd69-5e09-495c-b305-b709ee7a9f11">
<span class="eqno">(20.4)<a class="headerlink" href="#equation-d798fd69-5e09-495c-b305-b709ee7a9f11" title="Permalink to this equation">#</a></span>\[\begin{equation}
<div class="amsmath math notranslate nohighlight" id="equation-f4f7b6ea-b838-4610-8f7d-7921c1592694">
<span class="eqno">(20.4)<a class="headerlink" href="#equation-f4f7b6ea-b838-4610-8f7d-7921c1592694" 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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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
Expand Down Expand Up @@ -492,7 +492,7 @@ <h1><span class="section-number">81. </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": "dfccf567a05d462d9b226f0f7c46fc89"}</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": "5143a3e9659f45cb9742ba10c1639de1"}</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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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
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<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/asm.html">53. Additive Schwarz Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/examples.html">54. Some Examples of ASM preconditioners</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/pversion.html">55. Schwarz preconditioners for high order finite elements</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/pversion.html">55. high order fem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/minimaldd.html">56. Domain Decomposition with minimal overlap</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/overlapping.html">57. Overlapping Domain Decomposition Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../subspacecorrection/exercise.html">58. Exercise: Robust preconditioners</a></li>
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