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mfem/examples
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Tzanio 4a2449d87f Small adjustment in ceed-cuda skipped sample runs
2020-06-27 18:15:43 -07:00
..
ginkgo
Some updates in the examples
2020-03-09 18:05:47 -07:00
hiop
Some updates in the examples
2020-03-09 18:05:47 -07:00
petsc
The release number for mfem-4.1 is LLNL-CODE-806117
2020-03-02 17:18:46 -08:00
pumi
Adds notes about the location of pumi mesh/models
2020-05-15 01:37:21 -04:00
sundials
The release number for mfem-4.1 is LLNL-CODE-806117
2020-03-02 17:18:46 -08:00
CMakeLists.txt
Merge remote-tracking branch 'origin/master' into bc-example-dev
2020-04-21 11:41:29 -07:00
ex1.cpp
Modifying new mesh names in sample runs
2020-06-07 12:08:58 -07:00
ex1p.cpp
Small adjustment in ceed-cuda skipped sample runs
2020-06-27 18:15:43 -07:00
ex2.cpp
Melding with master
2018-10-12 15:58:30 -07:00
ex2p.cpp
Requested changes from PR review
2020-03-18 09:42:04 -04:00
ex3.cpp
minor
2020-02-29 09:07:23 -08:00
ex3p.cpp
Requested changes from PR review
2020-03-18 09:42:04 -04:00
ex4.cpp
Adding some comments.
2020-04-07 09:54:34 -07:00
ex4p.cpp
Adding some comments.
2020-04-07 09:54:34 -07:00
ex5.cpp
Fixed memory leaks.
2020-04-27 22:32:07 -07:00
ex5p.cpp
Added the option to specify the number of serial mesh refinements in ex5p, and changed the star.mesh -pa sample run to use a coarser mesh.
2020-04-28 11:50:34 -07:00
ex6.cpp
Small adjustment in ceed-cuda skipped sample runs
2020-06-27 18:15:43 -07:00
ex6p.cpp
Small adjustment in ceed-cuda skipped sample runs
2020-06-27 18:15:43 -07:00
ex7.cpp
In example 7/7p, when using a non-conforming mesh, ensure it remains
2020-04-10 20:03:56 -07:00
ex7p.cpp
In example 7/7p, when using a non-conforming mesh, ensure it remains
2020-04-10 20:03:56 -07:00
ex8.cpp
Adding sample runs with mixed or wedge meshes
2018-09-13 15:35:53 -07:00
ex8p.cpp
Requested changes from PR review
2020-03-18 09:42:04 -04:00
ex9.cpp
Revert "Use a string option to select the Assembly Level."
2020-04-27 16:56:31 -07:00
ex9p.cpp
Revert "Use a string option to select the Assembly Level."
2020-04-27 16:56:31 -07:00
ex10.cpp
Final adjustments
2020-01-29 15:00:00 -08:00
ex10p.cpp
Make ex10p.cpp look just like master.
2020-04-28 13:53:09 -07:00
ex11p.cpp
Modifying new mesh names in sample runs
2020-06-07 12:08:58 -07:00
ex12p.cpp
Updated examples
2020-03-29 12:37:43 -07:00
ex13p.cpp
Requested changes from PR review
2020-03-18 09:42:04 -04:00
ex14.cpp
Adding sample runs with mixed or wedge meshes
2018-09-13 15:35:53 -07:00
ex14p.cpp
Add final argument to Monitor, and demonstrate a custom monitor in ex14p
2020-04-16 09:51:17 -07:00
ex15.cpp
Both ex15 and ex15p now prefer nonconforming refinement. Updated descriptions too.
2020-01-07 14:10:59 +01:00
ex15p.cpp
Both ex15 and ex15p now prefer nonconforming refinement. Updated descriptions too.
2020-01-07 14:10:59 +01:00
ex16.cpp
Adding sample runs with mixed or wedge meshes
2018-09-13 15:35:53 -07:00
ex16p.cpp
Updated examples
2020-03-29 12:37:43 -07:00
ex17.cpp
In ex17.cpp, recover the original print level, 3, for
2019-11-15 17:05:03 -08:00
ex17p.cpp
Requested changes from PR review
2020-03-18 09:42:04 -04:00
ex18.cpp
Overwriting ex18 with new one from master
2018-05-25 12:12:43 -07:00
ex18.hpp
Removing uses of deprecated FaceElementTransformations features
2020-04-21 15:01:23 -07:00
ex18p.cpp
Requested changes from PR review
2020-03-18 09:42:04 -04:00
ex19.cpp
Move residual monitor demonstration from ex10 to ex19
2020-04-28 13:28:19 -07:00
ex19p.cpp
Fix style
2020-04-28 16:10:44 -07:00
ex20.cpp
Minor styling in Example 20.
2018-07-24 19:53:59 -07:00
ex20p.cpp
Requested changes from PR review
2020-03-18 09:42:04 -04:00
ex21.cpp
Renumbering ex21
2019-08-21 16:25:39 -07:00
ex21p.cpp
Renumbering ex21
2019-08-21 16:25:39 -07:00
ex22.cpp
final changes
2020-03-25 17:36:47 -07:00
ex22p.cpp
In ex22p.cpp, to support 64-bit 'HYPRE_Int', use 'int' for the
2020-03-09 21:43:37 -07:00
ex23.cpp
Merge branch 'master' into mfem-4.1-dev
2020-03-10 16:56:01 -07:00
ex24.cpp
Incorporating some of Yohann's suggestions.
2020-04-07 14:17:22 -07:00
ex24p.cpp
Incorporating some of Yohann's suggestions.
2020-04-07 14:17:22 -07:00
ex25.cpp
final changes
2020-03-25 17:36:47 -07:00
ex25p.cpp
Prevent sockets streams from mixing in ex25p
2020-03-25 18:18:08 -07:00
ex26.cpp
minor
2020-04-18 18:07:43 -07:00
ex26p.cpp
minor
2020-04-18 18:07:43 -07:00
ex27.cpp
Replacing error measure with a proper l2 error and average in one function
2020-04-15 13:23:27 -07:00
ex27p.cpp
Replacing error measure with a proper l2 error and average in one function
2020-04-15 13:23:27 -07:00
makefile
Merge remote-tracking branch 'origin/master' into bc-example-dev
2020-04-21 11:41:29 -07:00
README.html
Updated/added README.html files
2020-03-10 09:55:25 -07:00

README.html

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<h1 id="example-codes-and-miniapps">Example Codes and Miniapps</h1>
<p>This page provides a brief overview of MFEM's example codes and miniapps. For
detailed documentation of the MFEM sources, including the examples, see the
<a href="http://mfem.github.io/doxygen/html/index.html">online Doxygen documentation</a>,
or the <code>doc</code> directory in the distribution.</p>
<p>The goal of the example codes is to provide a step-by-step introduction to MFEM
in simple model settings. The miniapps are more complex, and are intended to be
more representative of the advanced usage of the library in physics/application
codes. We recommend that new users start with the example codes before moving to
the miniapps.</p>
<p>Clicking on any of the categories below displays examples and miniapps that contain the
described feature. <em>All examples support (arbitrarily) high-order meshes and
finite element spaces</em>.
The numerical results from the example codes can be visualized using the
GLVis visualization tool (based on MFEM). See the
<a href="http://glvis.org">GLVis website</a> for more details.</p>
<p>Users are encouraged to submit any example codes and miniapps that they have created and
would like to share. <br>
<em>Contact a member of the MFEM team to report
<a href="https://github.com/mfem/mfem/issues/new?labels=bug">bugs</a>
or post <a href="https://github.com/mfem/mfem/issues/new?labels=question">questions</a> or <a href="https://github.com/mfem/mfem/issues/new?labels=comment">comments</a></em>.</p>
<div class="row">
<p></p><div class="col-sm-6 col-md-2 small" markdown="1">
   <h5><strong>Application (PDE)</strong></h5>
   <label><input type="radio" id="all1" onchange="update(this.id);" checked="checked"> All</label><br>
   <label><input type="radio" id="laplace" onchange="update(this.id);"> Laplace</label><br>
   <label><input type="radio" id="elasticity" onchange="update(this.id);"> Elasticity</label><br>
   <label><input type="radio" id="maxwell" onchange="update(this.id);"> Electromagnetics</label><br>
   <label><input type="radio" id="graddiv" onchange="update(this.id);"> grad-div</label><br>
   <label><input type="radio" id="darcy" onchange="update(this.id);"> Darcy</label><br>
   <label><input type="radio" id="advection" onchange="update(this.id);"> Advection</label><br>
   <label><input type="radio" id="conduction" onchange="update(this.id);"> Conduction</label><br>
   <label><input type="radio" id="wave" onchange="update(this.id);"> Wave</label><br>
   <label><input type="radio" id="hydro" onchange="update(this.id);"> Hydrodynamics</label><br>
   <label><input type="radio" id="meshing" onchange="update(this.id);"> Meshing</label><br>
   <label><input type="radio" id="hpc" onchange="update(this.id);"> High-performance</label><br>
</div>
<div class="col-sm-6 col-md-3 small" markdown="1">
   <h5><strong>Finite Elements</strong></h5>
   <label><input type="radio" id="all2" onchange="update(this.id);" checked="checked"> All</label><br>
   <label><input type="radio" id="l2" onchange="update(this.id);"> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1"><span class="MJXp-msubsup" id="MJXp-Span-2"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-3" style="margin-right: 0.05em;">L</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-4" style="vertical-align: -0.4em;">2</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-1-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.637ex" height="2.362ex" viewBox="0 -761.8 1135.4 1017.1" role="img" focusable="false" style="vertical-align: -0.593ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="963" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-1">L_2</script> discontinuous elements</label><br>
   <label><input type="radio" id="h1" onchange="update(this.id);"> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-5"><span class="MJXp-msubsup" id="MJXp-Span-6"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-7" style="margin-right: 0.05em;">H</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-8" style="vertical-align: 0.5em;">1</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-2-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.158ex" height="2.509ex" viewBox="0 -951.8 1359.6 1080.4" role="img" focusable="false" style="vertical-align: -0.299ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="1280" y="513"></use></g></svg></span><script type="math/tex" id="MathJax-Element-2">H^1</script> nodal elements</label><br>
   <label><input type="radio" id="hcurl" onchange="update(this.id);"> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-9"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-10">H</span><span class="MJXp-mo" id="MJXp-Span-11" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-12">c</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-13">u</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-14">r</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-15">l</span><span class="MJXp-mo" id="MJXp-Span-16" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-3-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.951ex" height="2.656ex" viewBox="0 -825.1 3423.5 1143.7" role="img" focusable="false" style="vertical-align: -0.74ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="888" y="0"></use><use xlink:href="#MJMATHI-63" x="1278" y="0"></use><use xlink:href="#MJMATHI-75" x="1711" y="0"></use><use xlink:href="#MJMATHI-72" x="2284" y="0"></use><use xlink:href="#MJMATHI-6C" x="2735" y="0"></use><use xlink:href="#MJMAIN-29" x="3034" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-3">H(curl)</script> Nedelec elements</label><br>
   <label><input type="radio" id="hdiv" onchange="update(this.id);"> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-17"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-18">H</span><span class="MJXp-mo" id="MJXp-Span-19" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-20">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-21">i</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-22">v</span><span class="MJXp-mo" id="MJXp-Span-23" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-4-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.019ex" height="2.656ex" viewBox="0 -825.1 3022 1143.7" role="img" focusable="false" style="vertical-align: -0.74ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="888" y="0"></use><use xlink:href="#MJMATHI-64" x="1278" y="0"></use><use xlink:href="#MJMATHI-69" x="1801" y="0"></use><use xlink:href="#MJMATHI-76" x="2147" y="0"></use><use xlink:href="#MJMAIN-29" x="2632" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-4">H(div)</script> Raviart-Thomas elements</label><br>
   <label><input type="radio" id="h12" onchange="update(this.id);"> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-24"><span class="MJXp-msubsup" id="MJXp-Span-25"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-26" style="margin-right: 0.05em;">H</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-27" style="vertical-align: 0.5em;"><span class="MJXp-mo" id="MJXp-Span-28">−</span><span class="MJXp-mn" id="MJXp-Span-29">1</span><span class="MJXp-mrow" id="MJXp-Span-30"><span class="MJXp-mo" id="MJXp-Span-31">/</span></span><span class="MJXp-mn" id="MJXp-Span-32">2</span></span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-5-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.08ex" height="2.656ex" viewBox="0 -1015.1 2617.9 1143.7" role="img" focusable="false" style="vertical-align: -0.299ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><g transform="translate(905,362)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2F" x="1279" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="1779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-5">H^{-1/2}</script> interfacial elements</label><br>
</div>
<div class="clearfix hidden-md hidden-lg"></div>
<div class="col-sm-6 col-md-3 small" markdown="1">
   <h5><strong>Discretization</strong></h5>
   <label><input type="radio" id="all3" onchange="update(this.id);" checked="checked"> All</label><br>
   <label><input type="radio" id="galerkin" onchange="update(this.id);"> Galerkin FEM</label><br>
   <label><input type="radio" id="mixed" onchange="update(this.id);"> Mixed FEM</label><br>
   <label><input type="radio" id="dg" onchange="update(this.id);"> Discontinuous Galerkin (DG)</label><br>
   <label><input type="radio" id="dpg" onchange="update(this.id);"> Discont. Petrov-Galerkin (DPG)</label><br>
   <label><input type="radio" id="hybr" onchange="update(this.id);"> Hybridization</label><br>
   <label><input type="radio" id="staticcond" onchange="update(this.id);"> Static condensation</label><br>
   <label><input type="radio" id="nurbs" onchange="update(this.id);"> Isogeometric analysis (NURBS)</label><br>
   <label><input type="radio" id="amr" onchange="update(this.id);"> Adaptive mesh refinement (AMR)</label><br>
</div>
<div class="col-sm-6 col-md-4 small" markdown="1">
   <h5><strong>Solver</strong></h5>
   <label><input type="radio" id="all4" onchange="update(this.id);" checked="checked"> All</label><br>
   <label><input type="radio" id="jacobi" onchange="update(this.id);"> Jacobi</label> <br>
   <label><input type="radio" id="gs" onchange="update(this.id);"> Gauss-Seidel</label> <br>
   <label><input type="radio" id="pcg" onchange="update(this.id);"> PCG</label> <br>
   <label><input type="radio" id="minres" onchange="update(this.id);"> MINRES</label> <br>
   <label><input type="radio" id="gmres" onchange="update(this.id);"> GMRES</label> <br>
   <label><input type="radio" id="amg" onchange="update(this.id);"> Algebraic Multigrid (BoomerAMG)</label> <br>
   <label><input type="radio" id="ams" onchange="update(this.id);"> Auxiliary-space Maxwell Solver (AMS)</label> <br>
   <label><input type="radio" id="ads" onchange="update(this.id);"> Auxiliary-space Divergence Solver (ADS)</label> <br>
   <label><input type="radio" id="superlu" onchange="update(this.id);"> SuperLU/STRUMPACK (parallel direct)</label><br>
   <label><input type="radio" id="umfpack" onchange="update(this.id);"> UMFPACK (serial direct)</label><br>
   <label><input type="radio" id="newton" onchange="update(this.id);"> Newton method (nonlinear solver)</label><br>
   <label><input type="radio" id="rk" onchange="update(this.id);"> Explicit Runge-Kutta (ODE integration)</label><br>
   <label><input type="radio" id="sdirk" onchange="update(this.id);"> Implicit Runge-Kutta (ODE integration)</label><br>
   <label><input type="radio" id="newmark" onchange="update(this.id);"> Newmark (ODE Integration)</label><br>
   <label><input type="radio" id="symplectic" onchange="update(this.id);"> Symplectic Algorithm (ODE Integration)</label><br>
   <label><input type="radio" id="lobpcg" onchange="update(this.id);"> LOBPCG, AME (eigensolvers)</label><br>
   <label><input type="radio" id="sundials" onchange="update(this.id);"> SUNDIALS solvers</label><br>
   <label><input type="radio" id="petsc" onchange="update(this.id);"> PETSc solvers</label><br>
   <label><input type="radio" id="hiop" onchange="update(this.id);"> HiOp solvers</label><br>
</div><p></p>
</div>
<hr>

<!-- ------------------------------------------------------------------------- -->

<div id="ex1" style="display: block;">
<h2 id="example-1-laplace-problem">Example 1: Laplace Problem</h2>
<p><img class="floatright" src="../doc/web/examples/ex1.png"></p>
<p>This example code demonstrates the use of MFEM to define a
simple isoparametric finite element discretization of the
Laplace problem <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-33"><span class="MJXp-mo" id="MJXp-Span-34" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi" id="MJXp-Span-35">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-36">u</span><span class="MJXp-mo" id="MJXp-Span-37" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-38">1</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.335ex" height="2.302ex" viewBox="0 -824.7 4019.1 991.3" role="img" focusable="false" style="vertical-align: -0.387ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMAIN-394" x="778" y="0"></use><use xlink:href="#MJMATHI-75" x="1612" y="0"></use><use xlink:href="#MJMAIN-3D" x="2462" y="0"></use><use xlink:href="#MJMAIN-31" x="3518" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">-\Delta u = 1</script> with homogeneous Dirichlet
boundary conditions. Specifically, we discretize with the
finite element space coming from the mesh (linear by default, quadratic
for quadratic curvilinear mesh, NURBS for NURBS mesh, etc.)</p>
<p>The example highlights the use of mesh refinement, finite
element grid functions, as well as linear and bilinear forms
corresponding to the left-hand side and right-hand side of the
discrete linear system. We also cover the explicit elimination
of essential boundary conditions, static condensation, and the optional
connection to the <a href="http://glvis.org">GLVis</a> tool for visualization.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex1.cpp">ex1.cpp</a>),
a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex1p.cpp">ex1p.cpp</a>),
and HPC versions: <a href="https://github.com/mfem/mfem/blob/master/miniapps/performance/ex1.cpp">performance/ex1.cpp</a>,
<a href="https://github.com/mfem/mfem/blob/master/miniapps/performance/ex1p.cpp">performance/ex1p.cpp</a>.
It also has a PETSc modification in <a href="https://github.com/mfem/mfem/blob/master/examples/petsc">examples/petsc</a>
, a PUMI modification in <a href="https://github.com/mfem/mfem/blob/master/examples/pumi">examples/pumi</a> and a Ginkgo modification
in <a href="https://github.com/mfem/mfem/tree/master/examples/ginkgo">examples/ginkgo</a>.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex2" style="display: block;">
<h2 id="example-2-linear-elasticity">Example 2: Linear Elasticity</h2>
<p><img class="floatright" src="../doc/web/examples/ex2.png"></p>
<p>This example code solves a simple linear elasticity problem
describing a multi-material cantilever beam.
Specifically, we approximate the weak form of
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-39"><span class="MJXp-mo" id="MJXp-Span-40" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mrow" id="MJXp-Span-41"><span class="MJXp-mi" id="MJXp-Span-42">d</span><span class="MJXp-mi" id="MJXp-Span-43">i</span><span class="MJXp-mi" id="MJXp-Span-44">v</span></span><span class="MJXp-mo" id="MJXp-Span-45" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-46"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-47">σ</span></span><span class="MJXp-mo" id="MJXp-Span-48" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-49"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-50">u</span></span><span class="MJXp-mo" id="MJXp-Span-51" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-52" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-53" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-54">0</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.669ex" height="2.684ex" viewBox="0 -824.7 6746.6 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-28" x="2142" y="0"></use><use xlink:href="#MJMATHI-3C3" x="2531" y="0"></use><use xlink:href="#MJMAIN-28" x="3104" y="0"></use><use xlink:href="#MJMAINB-75" x="3493" y="0"></use><use xlink:href="#MJMAIN-29" x="4133" y="0"></use><use xlink:href="#MJMAIN-29" x="4522" y="0"></use><use xlink:href="#MJMAIN-3D" x="5189" y="0"></use><use xlink:href="#MJMAIN-30" x="6246" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-7">-{\rm div}({\sigma}({\bf u})) = 0</script>
where
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-55"><span class="MJXp-mrow" id="MJXp-Span-56"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-57">σ</span></span><span class="MJXp-mo" id="MJXp-Span-58" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-59"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-60">u</span></span><span class="MJXp-mo" id="MJXp-Span-61" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-62" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-63">λ</span><span class="MJXp-mspace" id="MJXp-Span-64" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-65"><span class="MJXp-mi" id="MJXp-Span-66">d</span><span class="MJXp-mi" id="MJXp-Span-67">i</span><span class="MJXp-mi" id="MJXp-Span-68">v</span></span><span class="MJXp-mo" id="MJXp-Span-69" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-70"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-71">u</span></span><span class="MJXp-mo" id="MJXp-Span-72" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mspace" id="MJXp-Span-73" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-74">I</span><span class="MJXp-mo" id="MJXp-Span-75" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-76">μ</span><span class="MJXp-mspace" id="MJXp-Span-77" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mo" id="MJXp-Span-78" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi" id="MJXp-Span-79">∇</span><span class="MJXp-mrow" id="MJXp-Span-80"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-81">u</span></span><span class="MJXp-mo" id="MJXp-Span-82" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi" id="MJXp-Span-83">∇</span><span class="MJXp-msubsup" id="MJXp-Span-84"><span class="MJXp-mrow" id="MJXp-Span-85" style="margin-right: 0.05em;"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-86">u</span></span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-87" style="vertical-align: 0.5em;">T</span></span><span class="MJXp-mo" id="MJXp-Span-88" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="34.996ex" height="3.067ex" viewBox="0 -989.3 15067.6 1320.3" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="572" y="0"></use><use xlink:href="#MJMAINB-75" x="962" y="0"></use><use xlink:href="#MJMAIN-29" x="1601" y="0"></use><use xlink:href="#MJMAIN-3D" x="2268" y="0"></use><use xlink:href="#MJMATHI-3BB" x="3325" y="0"></use><g transform="translate(4075,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-28" x="5438" y="0"></use><use xlink:href="#MJMAINB-75" x="5828" y="0"></use><use xlink:href="#MJMAIN-29" x="6467" y="0"></use><use xlink:href="#MJMATHI-49" x="7023" y="0"></use><use xlink:href="#MJMAIN-2B" x="7750" y="0"></use><use xlink:href="#MJMATHI-3BC" x="8751" y="0"></use><use xlink:href="#MJMAIN-28" x="9521" y="0"></use><use xlink:href="#MJMAIN-2207" x="9911" y="0"></use><use xlink:href="#MJMAINB-75" x="10744" y="0"></use><use xlink:href="#MJMAIN-2B" x="11606" y="0"></use><use xlink:href="#MJMAIN-2207" x="12606" y="0"></use><g transform="translate(13440,0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="904" y="583"></use></g><use xlink:href="#MJMAIN-29" x="14678" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-8">{\sigma}({\bf u}) = \lambda\, {\rm div}({\bf u})\,I + \mu\,(\nabla{\bf u} + \nabla{\bf u}^T)</script>
is the stress tensor corresponding to displacement field <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-89"><span class="MJXp-mrow" id="MJXp-Span-90"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-91">u</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-9-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.485ex" height="1.538ex" viewBox="0 -550.5 639.5 662.2" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-9">{\bf u}</script>, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-92"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-93">λ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-10-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.355ex" height="2.048ex" viewBox="0 -769.9 583.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-10">\lambda</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-94"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-95">μ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-11-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.92ex" viewBox="0 -550.5 603.5 826.7" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-11">\mu</script>
are the material Lame constants. The boundary conditions are
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-96"><span class="MJXp-mrow" id="MJXp-Span-97"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-98">u</span></span><span class="MJXp-mo" id="MJXp-Span-99" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-100">0</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-12-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.746ex" height="2.048ex" viewBox="0 -769.9 2474.1 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use><use xlink:href="#MJMAIN-3D" x="917" y="0"></use><use xlink:href="#MJMAIN-30" x="1973" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-12">{\bf u}=0</script> on the fixed part of the boundary with attribute 1, and
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-101"><span class="MJXp-mrow" id="MJXp-Span-102"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-103">σ</span></span><span class="MJXp-mo" id="MJXp-Span-104" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-105"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-106">u</span></span><span class="MJXp-mo" id="MJXp-Span-107" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-108" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-109">n</span><span class="MJXp-mo" id="MJXp-Span-110" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-111">f</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-13-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.075ex" height="2.684ex" viewBox="0 -824.7 5199 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="572" y="0"></use><use xlink:href="#MJMAINB-75" x="962" y="0"></use><use xlink:href="#MJMAIN-29" x="1601" y="0"></use><use xlink:href="#MJMAIN-22C5" x="2213" y="0"></use><use xlink:href="#MJMATHI-6E" x="2713" y="0"></use><use xlink:href="#MJMAIN-3D" x="3592" y="0"></use><use xlink:href="#MJMATHI-66" x="4648" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-13">{\sigma}({\bf u})\cdot n = f</script> on the remainder with <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-112"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-113">f</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-14-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.279ex" height="2.43ex" viewBox="0 -769.9 550.5 1046.1" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-66" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-14">f</script> being
a constant pull down vector on boundary elements with attribute 2, and zero
otherwise. The geometry of the domain is assumed to be as follows:</p>
<p><img alt="" src="../doc/web/examples/ex2-domain.png"></p>
<p>The example demonstrates the use of high-order and NURBS vector
finite element spaces with the linear elasticity bilinear form,
meshes with curved elements, and the definition of piece-wise
constant and vector coefficient objects. Static condensation is
also illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex2.cpp">ex2.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex2p.cpp">ex2p.cpp</a>) version.
It also has a PETSc modification in <a href="https://github.com/mfem/mfem/blob/master/examples/petsc">examples/petsc</a>
and a PUMI modification in <a href="https://github.com/mfem/mfem/blob/master/examples/pumi">examples/pumi</a>.
We recommend viewing Example 1 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex3" style="display: block;">
<h2 id="example-3-definite-maxwell-problem">Example 3: Definite Maxwell Problem</h2>
<p><img class="floatright" src="../doc/web/examples/ex3.png"></p>
<p>This example code solves a simple 3D electromagnetic diffusion
problem corresponding to the second order definite Maxwell
equation <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-114"><span class="MJXp-mi" id="MJXp-Span-115">∇</span><span class="MJXp-mo" id="MJXp-Span-116" style="margin-left: 0.267em; margin-right: 0.267em;">×</span><span class="MJXp-mi" id="MJXp-Span-117">∇</span><span class="MJXp-mo" id="MJXp-Span-118" style="margin-left: 0.267em; margin-right: 0.267em;">×</span><span class="MJXp-mspace" id="MJXp-Span-119" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-120">E</span><span class="MJXp-mo" id="MJXp-Span-121" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-122">E</span><span class="MJXp-mo" id="MJXp-Span-123" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-124">f</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="20.708ex" height="2.43ex" viewBox="0 -769.9 8916.1 1046.1" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="1055" y="0"></use><use xlink:href="#MJMAIN-2207" x="2056" y="0"></use><use xlink:href="#MJMAIN-D7" x="3112" y="0"></use><use xlink:href="#MJMATHI-45" x="4279" y="0"></use><use xlink:href="#MJMAIN-2B" x="5266" y="0"></use><use xlink:href="#MJMATHI-45" x="6267" y="0"></use><use xlink:href="#MJMAIN-3D" x="7309" y="0"></use><use xlink:href="#MJMATHI-66" x="8365" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-15">\nabla\times\nabla\times\, E + E = f</script>
with boundary condition <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-125"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-126">E</span><span class="MJXp-mo" id="MJXp-Span-127" style="margin-left: 0.267em; margin-right: 0.267em;">×</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-128">n</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-16-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.011ex" height="2.048ex" viewBox="0 -769.9 2587.9 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="986" y="0"></use><use xlink:href="#MJMATHI-6E" x="1987" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-16"> E \times n </script> = "given tangential field".
Here, we use a given exact solution <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-129"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-130">E</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-17-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.048ex" viewBox="0 -769.9 764.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-17">E</script> and compute the corresponding r.h.s.
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-131"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-132">f</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-18-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.279ex" height="2.43ex" viewBox="0 -769.9 550.5 1046.1" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-66" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-18">f</script>. We discretize with Nedelec finite elements in 2D or 3D.</p>
<p>The example demonstrates the use of <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-133"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-134">H</span><span class="MJXp-mo" id="MJXp-Span-135" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-136">c</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-137">u</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-138">r</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-139">l</span><span class="MJXp-mo" id="MJXp-Span-140" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-19-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.951ex" height="2.684ex" viewBox="0 -824.7 3423.5 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="888" y="0"></use><use xlink:href="#MJMATHI-63" x="1278" y="0"></use><use xlink:href="#MJMATHI-75" x="1711" y="0"></use><use xlink:href="#MJMATHI-72" x="2284" y="0"></use><use xlink:href="#MJMATHI-6C" x="2735" y="0"></use><use xlink:href="#MJMAIN-29" x="3034" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-19">H(curl)</script> finite element
spaces with the curl-curl and the (vector finite element) mass
bilinear form, as well as the computation of discretization
error when the exact solution is known. Static condensation is
also illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex3.cpp">ex3.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex3p.cpp">ex3p.cpp</a>) version.
It also has a PETSc modification in <a href="https://github.com/mfem/mfem/blob/master/examples/petsc">examples/petsc</a>.
We recommend viewing examples 1-2 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex4" style="display: block;">
<h2 id="example-4-grad-div-problem">Example 4: Grad-div Problem</h2>
<p><img class="floatright" src="../doc/web/examples/ex4.png"></p>
<p>This example code solves a simple 2D/3D <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-141"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-142">H</span><span class="MJXp-mo" id="MJXp-Span-143" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-144">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-145">i</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-146">v</span><span class="MJXp-mo" id="MJXp-Span-147" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-20-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.019ex" height="2.684ex" viewBox="0 -824.7 3022 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="888" y="0"></use><use xlink:href="#MJMATHI-64" x="1278" y="0"></use><use xlink:href="#MJMATHI-69" x="1801" y="0"></use><use xlink:href="#MJMATHI-76" x="2147" y="0"></use><use xlink:href="#MJMAIN-29" x="2632" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-20">H(div)</script>
diffusion problem corresponding to the second order definite equation
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-148"><span class="MJXp-mo" id="MJXp-Span-149" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mrow" id="MJXp-Span-150"><span class="MJXp-mi" id="MJXp-Span-151">g</span><span class="MJXp-mi" id="MJXp-Span-152">r</span><span class="MJXp-mi" id="MJXp-Span-153">a</span><span class="MJXp-mi" id="MJXp-Span-154">d</span></span><span class="MJXp-mo" id="MJXp-Span-155" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-156">α</span><span class="MJXp-mspace" id="MJXp-Span-157" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-158"><span class="MJXp-mi" id="MJXp-Span-159">d</span><span class="MJXp-mi" id="MJXp-Span-160">i</span><span class="MJXp-mi" id="MJXp-Span-161">v</span></span><span class="MJXp-mo" id="MJXp-Span-162" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-163">F</span><span class="MJXp-mo" id="MJXp-Span-164" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-165" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-166" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-167">β</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-168">F</span><span class="MJXp-mo" id="MJXp-Span-169" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-170">f</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="27.028ex" height="2.684ex" viewBox="0 -824.7 11637.2 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#MJMAIN-67" x="0" y="0"></use><use xlink:href="#MJMAIN-72" x="500" y="0"></use><use xlink:href="#MJMAIN-61" x="893" y="0"></use><use xlink:href="#MJMAIN-64" x="1393" y="0"></use></g><use xlink:href="#MJMAIN-28" x="2728" y="0"></use><use xlink:href="#MJMATHI-3B1" x="3118" y="0"></use><g transform="translate(3925,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-28" x="5288" y="0"></use><use xlink:href="#MJMATHI-46" x="5678" y="0"></use><use xlink:href="#MJMAIN-29" x="6427" y="0"></use><use xlink:href="#MJMAIN-29" x="6817" y="0"></use><use xlink:href="#MJMAIN-2B" x="7428" y="0"></use><use xlink:href="#MJMATHI-3B2" x="8429" y="0"></use><use xlink:href="#MJMATHI-46" x="9003" y="0"></use><use xlink:href="#MJMAIN-3D" x="10030" y="0"></use><use xlink:href="#MJMATHI-66" x="11086" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-21">-{\rm grad}(\alpha\,{\rm div}(F)) + \beta F = f</script>
with boundary condition <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-171"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-172">F</span><span class="MJXp-mo" id="MJXp-Span-173" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-174">n</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-22-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.815ex" height="2.048ex" viewBox="0 -769.9 2072.9 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-46" x="0" y="0"></use><use xlink:href="#MJMAIN-22C5" x="971" y="0"></use><use xlink:href="#MJMATHI-6E" x="1472" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-22">F \cdot n</script> = "given normal field".
Here we use a given exact solution <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-175"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-176">F</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-23-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.741ex" height="2.048ex" viewBox="0 -769.9 749.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-46" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-23">F</script> and compute the corresponding
right hand side <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-177"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-178">f</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-24-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.279ex" height="2.43ex" viewBox="0 -769.9 550.5 1046.1" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-66" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-24">f</script>.  We discretize with the Raviart-Thomas finite elements.</p>
<p>The example demonstrates the use of <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-179"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-180">H</span><span class="MJXp-mo" id="MJXp-Span-181" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-182">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-183">i</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-184">v</span><span class="MJXp-mo" id="MJXp-Span-185" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-25-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.019ex" height="2.684ex" viewBox="0 -824.7 3022 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="888" y="0"></use><use xlink:href="#MJMATHI-64" x="1278" y="0"></use><use xlink:href="#MJMATHI-69" x="1801" y="0"></use><use xlink:href="#MJMATHI-76" x="2147" y="0"></use><use xlink:href="#MJMAIN-29" x="2632" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-25">H(div)</script>
finite element spaces with the grad-div and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-186"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-187">H</span><span class="MJXp-mo" id="MJXp-Span-188" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-189">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-190">i</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-191">v</span><span class="MJXp-mo" id="MJXp-Span-192" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-26-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.019ex" height="2.684ex" viewBox="0 -824.7 3022 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="888" y="0"></use><use xlink:href="#MJMATHI-64" x="1278" y="0"></use><use xlink:href="#MJMATHI-69" x="1801" y="0"></use><use xlink:href="#MJMATHI-76" x="2147" y="0"></use><use xlink:href="#MJMAIN-29" x="2632" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-26">H(div)</script>
vector finite element mass bilinear form, as well as the computation of discretization
error when the exact solution is known.
Bilinear form hybridization and static condensation are also illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex4.cpp">ex4.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex4p.cpp">ex4p.cpp</a>) version.
It also has a PETSc modification in <a href="https://github.com/mfem/mfem/blob/master/examples/petsc">examples/petsc</a>.
We recommend viewing examples 1-3 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex5" style="display: block;">
<h2 id="example-5-darcy-problem">Example 5: Darcy Problem</h2>
<p><img class="floatright" src="../doc/web/examples/ex5.png"></p>
<p>This example code solves a simple 2D/3D mixed Darcy problem
corresponding to the saddle point system
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-193"><span class="MJXp-mtable" id="MJXp-Span-194"><span><span class="MJXp-mtr" id="MJXp-Span-195" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-196" style="text-align: right;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-197">k</span><span class="MJXp-mspace" id="MJXp-Span-198" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-199"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-200">u</span></span><span class="MJXp-mo" id="MJXp-Span-201" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mrow" id="MJXp-Span-202"><span class="MJXp-mi" id="MJXp-Span-203">g</span><span class="MJXp-mi" id="MJXp-Span-204">r</span><span class="MJXp-mi" id="MJXp-Span-205">a</span><span class="MJXp-mi" id="MJXp-Span-206">d</span></span><span class="MJXp-mspace" id="MJXp-Span-207" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-208">p</span></span><span class="MJXp-mtd" id="MJXp-Span-209" style="padding-left: 1em; text-align: center;"><span class="MJXp-mo" id="MJXp-Span-210" style="margin-left: 0.333em; margin-right: 0.333em;">=</span></span><span class="MJXp-mtd" id="MJXp-Span-211" style="padding-left: 1em; text-align: left;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-212">f</span></span></span><span class="MJXp-mtr" id="MJXp-Span-213" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-214" style="padding-top: 0.4em; text-align: right;"><span class="MJXp-mo" id="MJXp-Span-215" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mrow" id="MJXp-Span-216"><span class="MJXp-mi" id="MJXp-Span-217">d</span><span class="MJXp-mi" id="MJXp-Span-218">i</span><span class="MJXp-mi" id="MJXp-Span-219">v</span></span><span class="MJXp-mspace" id="MJXp-Span-220" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-221"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-222">u</span></span></span><span class="MJXp-mtd" id="MJXp-Span-223" style="padding-left: 1em; padding-top: 0.4em; text-align: center;"><span class="MJXp-mo" id="MJXp-Span-224" style="margin-left: 0.333em; margin-right: 0.333em;">=</span></span><span class="MJXp-mtd" id="MJXp-Span-225" style="padding-left: 1em; padding-top: 0.4em; text-align: left;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-226">g</span></span></span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-27-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="20.485ex" height="5.997ex" viewBox="0 -1537.7 8820.1 2581.8" role="img" focusable="false" style="vertical-align: -2.425ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-14,0)"><g transform="translate(0,656)"><use xlink:href="#MJMATHI-6B" x="0" y="0"></use><use xlink:href="#MJMAINB-75" x="688" y="0"></use><use xlink:href="#MJMAIN-2B" x="1549" y="0"></use><g transform="translate(2550,0)"><use xlink:href="#MJMAIN-67" x="0" y="0"></use><use xlink:href="#MJMAIN-72" x="500" y="0"></use><use xlink:href="#MJMAIN-61" x="893" y="0"></use><use xlink:href="#MJMAIN-64" x="1393" y="0"></use></g><use xlink:href="#MJMATHI-70" x="4667" y="0"></use></g><g transform="translate(2222,-751)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAINB-75" x="2308" y="0"></use></g></g><g transform="translate(6157,0)"><use xlink:href="#MJMAIN-3D" x="0" y="656"></use><use xlink:href="#MJMAIN-3D" x="0" y="-751"></use></g><g transform="translate(7935,0)"><use xlink:href="#MJMATHI-66" x="0" y="656"></use><use xlink:href="#MJMATHI-67" x="0" y="-751"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-27"> \begin{array}{rcl}
   k\,{\bf u} + {\rm grad}\,p &=& f \\
   -{\rm div}\,{\bf u} &=& g
\end{array} </script>
with natural boundary condition <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-227"><span class="MJXp-mo" id="MJXp-Span-228" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-229">p</span><span class="MJXp-mo" id="MJXp-Span-230" style="margin-left: 0.333em; margin-right: 0.333em;">=</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-28-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.431ex" height="2.175ex" viewBox="0 -660.2 2338.3 936.4" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMATHI-70" x="778" y="0"></use><use xlink:href="#MJMAIN-3D" x="1559" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-28">-p = </script> "given pressure".
Here we use a given exact solution <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-231"><span class="MJXp-mo" id="MJXp-Span-232" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-233"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-234">u</span></span><span class="MJXp-mo" id="MJXp-Span-235" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-236">p</span><span class="MJXp-mo" id="MJXp-Span-237" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-29-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.498ex" height="2.684ex" viewBox="0 -824.7 2367.2 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28" x="0" y="0"></use><use xlink:href="#MJMAINB-75" x="389" y="0"></use><use xlink:href="#MJMAIN-2C" x="1029" y="0"></use><use xlink:href="#MJMATHI-70" x="1474" y="0"></use><use xlink:href="#MJMAIN-29" x="1977" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-29">({\bf u},p)</script> and compute the
corresponding right hand side <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-238"><span class="MJXp-mo" id="MJXp-Span-239" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-240">f</span><span class="MJXp-mo" id="MJXp-Span-241" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-242">g</span><span class="MJXp-mo" id="MJXp-Span-243" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-30-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.238ex" height="2.684ex" viewBox="0 -824.7 2255.2 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28" x="0" y="0"></use><use xlink:href="#MJMATHI-66" x="389" y="0"></use><use xlink:href="#MJMAIN-2C" x="940" y="0"></use><use xlink:href="#MJMATHI-67" x="1385" y="0"></use><use xlink:href="#MJMAIN-29" x="1865" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-30">(f, g)</script>. We discretize with Raviart-Thomas
finite elements (velocity <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-244"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-245">u</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-31-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.485ex" height="1.538ex" viewBox="0 -550.5 639.5 662.2" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-31">\bf u</script>) and piecewise discontinuous
polynomials (pressure <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-246"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-247">p</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-32-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.259ex" height="1.92ex" viewBox="-38.5 -550.5 542 826.7" role="img" focusable="false" style="vertical-align: -0.642ex; margin-left: -0.089ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-70" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-32">p</script>).</p>
<p>The example demonstrates the use of the BlockMatrix and BlockOperator
classes, as well as the collective saving of several grid functions in
<a href="http://visit.llnl.gov">VisIt</a> and <a href="https://www.paraview.org">ParaView</a>
formats.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex5.cpp">ex5.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex5p.cpp">ex5p.cpp</a>) version.
It also has a PETSc modification in <a href="https://github.com/mfem/mfem/blob/master/examples/petsc">examples/petsc</a>.
We recommend viewing examples 1-4 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex6" style="display: block;">
<h2 id="example-6-laplace-problem-with-amr">Example 6: Laplace Problem with AMR</h2>
<p><img class="floatright" src="../doc/web/examples/ex6.png"></p>
<p>This is a version of Example 1 with a simple adaptive mesh
refinement loop. The problem being solved is again the Laplace
equation <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-248"><span class="MJXp-mo" id="MJXp-Span-249" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi" id="MJXp-Span-250">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-251">u</span><span class="MJXp-mo" id="MJXp-Span-252" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-253">1</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-33-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.335ex" height="2.302ex" viewBox="0 -824.7 4019.1 991.3" role="img" focusable="false" style="vertical-align: -0.387ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMAIN-394" x="778" y="0"></use><use xlink:href="#MJMATHI-75" x="1612" y="0"></use><use xlink:href="#MJMAIN-3D" x="2462" y="0"></use><use xlink:href="#MJMAIN-31" x="3518" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-33">-\Delta u = 1</script> with homogeneous Dirichlet boundary
conditions. The problem is solved on a sequence of meshes which
are locally refined in a conforming (triangles, tetrahedrons)
or non-conforming (quadrilaterals, hexahedra) manner according
to a simple ZZ error estimator.</p>
<p>The example demonstrates MFEM's capability to work with both
conforming and nonconforming refinements, in 2D and 3D, on
linear, curved and surface meshes. Interpolation of functions
from coarse to fine meshes, as well as persistent <a href="http://glvis.org">GLVis</a>
visualization are also illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex6.cpp">ex6.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex6p.cpp">ex6p.cpp</a>) version.
It also has a PETSc modification in <a href="https://github.com/mfem/mfem/blob/master/examples/petsc">examples/petsc</a>
and a PUMI modification in <a href="https://github.com/mfem/mfem/blob/master/examples/pumi">examples/pumi</a>.
We recommend viewing Example 1 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex7" style="display: block;">
<h2 id="example-7-surface-meshes">Example 7: Surface Meshes</h2>
<p><img class="floatright" src="../doc/web/examples/ex7.png"></p>
<p>This example code demonstrates the use of MFEM to define a
triangulation of a unit sphere and a simple isoparametric
finite element discretization of the Laplace problem with mass
term, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-254"><span class="MJXp-mo" id="MJXp-Span-255" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi" id="MJXp-Span-256">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-257">u</span><span class="MJXp-mo" id="MJXp-Span-258" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-259">u</span><span class="MJXp-mo" id="MJXp-Span-260" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-261">f</span><span class="MJXp-mo" id="MJXp-Span-262" style="margin-left: 0em; margin-right: 0.222em;">.</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-34-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.268ex" height="2.557ex" viewBox="0 -824.7 6143 1101" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMAIN-394" x="778" y="0"></use><use xlink:href="#MJMATHI-75" x="1612" y="0"></use><use xlink:href="#MJMAIN-2B" x="2406" y="0"></use><use xlink:href="#MJMATHI-75" x="3407" y="0"></use><use xlink:href="#MJMAIN-3D" x="4257" y="0"></use><use xlink:href="#MJMATHI-66" x="5313" y="0"></use><use xlink:href="#MJMAIN-2E" x="5864" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-34">-\Delta u + u = f.</script></p>
<p>The example highlights mesh generation, the use of mesh
refinement, high-order meshes and finite elements, as well as
surface-based linear and bilinear forms corresponding to the
left-hand side and right-hand side of the discrete linear
system. Simple local mesh refinement is also demonstrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex7.cpp">ex7.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex7p.cpp">ex7p.cpp</a>) version.
We recommend viewing Example 1 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex8" style="display: block;">
<h2 id="example-8-dpg-for-the-laplace-problem">Example 8: DPG for the Laplace Problem</h2>
<p><img class="floatright" src="../doc/web/examples/ex8.png"></p>
<p>This example code demonstrates the use of the Discontinuous
Petrov-Galerkin (DPG) method in its primal 2x2 block form as a
simple finite element discretization of the Laplace problem
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-263"><span class="MJXp-mo" id="MJXp-Span-264" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi" id="MJXp-Span-265">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-266">u</span><span class="MJXp-mo" id="MJXp-Span-267" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-268">f</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-35-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.451ex" height="2.557ex" viewBox="0 -824.7 4069.1 1101" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMAIN-394" x="778" y="0"></use><use xlink:href="#MJMATHI-75" x="1612" y="0"></use><use xlink:href="#MJMAIN-3D" x="2462" y="0"></use><use xlink:href="#MJMATHI-66" x="3518" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-35">-\Delta u = f</script> with homogeneous Dirichlet boundary conditions. We
use high-order continuous trial space, a high-order interfacial
(trace) space, and a high-order discontinuous test space
defining a local dual (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-269"><span class="MJXp-msubsup" id="MJXp-Span-270"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-271" style="margin-right: 0.05em;">H</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-272" style="vertical-align: 0.5em;"><span class="MJXp-mo" id="MJXp-Span-273">−</span><span class="MJXp-mn" id="MJXp-Span-274">1</span></span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-36-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.436ex" height="2.43ex" viewBox="0 -934.4 1910 1046.1" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><g transform="translate(905,362)"><use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-36">H^{-1}</script>) norm.
We use the primal form of DPG, see
<a href="http://dx.doi.org/10.1016/j.camwa.2013.06.029">"A primal DPG method without a first-order reformulation"</a>,
Demkowicz and Gopalakrishnan, CAM 2013.</p>
<p>The example highlights the use of interfacial (trace) finite
elements and spaces, trace face integrators and the definition
of block operators and preconditioners.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex8.cpp">ex8.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex8p.cpp">ex8p.cpp</a>) version.
We recommend viewing examples 1-5 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex9" style="display: block;">
<h2 id="example-9-dg-advection">Example 9: DG Advection</h2>
<p><img class="floatright" src="../doc/web/examples/ex9.png"></p>
<p>This example code solves the time-dependent advection equation
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-275"><span class="MJXp-mfrac" id="MJXp-Span-276" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi" id="MJXp-Span-277">∂</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-278">u</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi" id="MJXp-Span-279">∂</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-280">t</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-281" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-282">v</span><span class="MJXp-mo" id="MJXp-Span-283" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mi" id="MJXp-Span-284">∇</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-285">u</span><span class="MJXp-mo" id="MJXp-Span-286" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-287">0</span><span class="MJXp-mo" id="MJXp-Span-288" style="margin-left: 0em; margin-right: 0.222em;">,</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-37-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.304ex" height="5.36ex" viewBox="0 -1482.9 7450.4 2307.6" role="img" focusable="false" style="vertical-align: -1.915ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="1260" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMATHI-75" x="567" y="0"></use></g><g transform="translate(165,-694)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="567" y="0"></use></g></g><use xlink:href="#MJMAIN-2B" x="1722" y="0"></use><use xlink:href="#MJMATHI-76" x="2722" y="0"></use><use xlink:href="#MJMAIN-22C5" x="3430" y="0"></use><use xlink:href="#MJMAIN-2207" x="3931" y="0"></use><use xlink:href="#MJMATHI-75" x="4764" y="0"></use><use xlink:href="#MJMAIN-3D" x="5615" y="0"></use><use xlink:href="#MJMAIN-30" x="6671" y="0"></use><use xlink:href="#MJMAIN-2C" x="7171" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-37">\frac{\partial u}{\partial t} + v \cdot \nabla u = 0,</script> where <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-289"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-290">v</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-38-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.128ex" height="1.538ex" viewBox="0 -550.5 485.5 662.2" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-38">v</script> is a given fluid
velocity, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-291"><span class="MJXp-msubsup" id="MJXp-Span-292"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-293" style="margin-right: 0.05em;">u</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-294" style="vertical-align: -0.4em;">0</span></span><span class="MJXp-mo" id="MJXp-Span-295" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-296">x</span><span class="MJXp-mo" id="MJXp-Span-297" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-298" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-299">u</span><span class="MJXp-mo" id="MJXp-Span-300" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mn" id="MJXp-Span-301">0</span><span class="MJXp-mo" id="MJXp-Span-302" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-303">x</span><span class="MJXp-mo" id="MJXp-Span-304" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-39-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.286ex" height="2.684ex" viewBox="0 -824.7 6581.6 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-75" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="809" y="-213"></use><use xlink:href="#MJMAIN-28" x="1026" y="0"></use><use xlink:href="#MJMATHI-78" x="1415" y="0"></use><use xlink:href="#MJMAIN-29" x="1988" y="0"></use><use xlink:href="#MJMAIN-3D" x="2655" y="0"></use><use xlink:href="#MJMATHI-75" x="3711" y="0"></use><use xlink:href="#MJMAIN-28" x="4284" y="0"></use><use xlink:href="#MJMAIN-30" x="4673" y="0"></use><use xlink:href="#MJMAIN-2C" x="5174" y="0"></use><use xlink:href="#MJMATHI-78" x="5619" y="0"></use><use xlink:href="#MJMAIN-29" x="6192" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-39">u_0(x)=u(0,x)</script> is a given initial condition.</p>
<p>The example demonstrates the use of Discontinuous Galerkin (DG) bilinear forms
in MFEM (face integrators), the use of explicit and implicit (with block ILU
preconditioning) ODE time integrators, the definition of periodic boundary
conditions through periodic meshes, as well as the use of
<a href="http://glvis.org">GLVis</a> for persistent visualization of a time-evolving
solution. The saving of time-dependent data files for external visualization
with <a href="http://visit.llnl.gov">VisIt</a> and <a href="https://www.paraview.org">ParaView</a> is also illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex9.cpp">ex9.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex9p.cpp">ex9p.cpp</a>) version.
It also has a SUNDIALS modification in <a href="https://github.com/mfem/mfem/blob/master/examples/sundials">examples/sundials</a>
, a PETSc modification in <a href="https://github.com/mfem/mfem/blob/master/examples/petsc">examples/petsc</a>,
and a HiOp modification in <a href="https://github.com/mfem/mfem/blob/master/examples/hiop">examples/hiop</a>.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex10" style="display: block;">
<h2 id="example-10-nonlinear-elasticity">Example 10: Nonlinear Elasticity</h2>
<p><img class="floatright" src="../doc/web/examples/ex10.png"></p>
<p>This example solves a time dependent nonlinear elasticity problem of the form
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-305"><span class="MJXp-mfrac" id="MJXp-Span-306" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-307">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-308">v</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-309">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-310">t</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-311" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-312">H</span><span class="MJXp-mo" id="MJXp-Span-313" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-314">x</span><span class="MJXp-mo" id="MJXp-Span-315" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-316" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-317">S</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-318">v</span><span class="MJXp-mspace" id="MJXp-Span-319" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mo" id="MJXp-Span-320" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mspace" id="MJXp-Span-321" style="width: 2em; height: 0em;"></span><span class="MJXp-mfrac" id="MJXp-Span-322" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-323">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-324">x</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-325">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-326">t</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-327" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-328">v</span><span class="MJXp-mspace" id="MJXp-Span-329" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mo" id="MJXp-Span-330" style="margin-left: 0em; margin-right: 0.222em;">,</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-40-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.656ex" height="5.232ex" viewBox="0 -1482.9 13629.6 2252.8" role="img" focusable="false" style="vertical-align: -1.788ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="1129" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-76" x="523" y="0"></use></g><g transform="translate(122,-686)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="523" y="0"></use></g></g><use xlink:href="#MJMAIN-3D" x="1646" y="0"></use><use xlink:href="#MJMATHI-48" x="2703" y="0"></use><use xlink:href="#MJMAIN-28" x="3591" y="0"></use><use xlink:href="#MJMATHI-78" x="3981" y="0"></use><use xlink:href="#MJMAIN-29" x="4553" y="0"></use><use xlink:href="#MJMAIN-2B" x="5165" y="0"></use><use xlink:href="#MJMATHI-53" x="6166" y="0"></use><use xlink:href="#MJMATHI-76" x="6811" y="0"></use><use xlink:href="#MJMAIN-2C" x="7463" y="0"></use><g transform="translate(9742,0)"><g transform="translate(286,0)"><rect stroke="none" width="1216" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-78" x="523" y="0"></use></g><g transform="translate(165,-686)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="523" y="0"></use></g></g></g><use xlink:href="#MJMAIN-3D" x="11642" y="0"></use><use xlink:href="#MJMATHI-76" x="12698" y="0"></use><use xlink:href="#MJMAIN-2C" x="13351" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-40"> \frac{dv}{dt} = H(x) + S v\,,\qquad \frac{dx}{dt} = v\,, </script>
where <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-331"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-332">H</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-41-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.064ex" height="2.048ex" viewBox="0 -769.9 888.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-41">H</script> is a hyperelastic model and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-333"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-334">S</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-42-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.499ex" height="2.048ex" viewBox="0 -769.9 645.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-53" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-42">S</script> is a viscosity operator of
Laplacian type. The geometry of the domain is assumed to be as follows:</p>
<p><img alt="" src="../doc/web/examples/ex10-domain.png"></p>
<p>The example demonstrates the use of nonlinear operators, as well as their
implicit time integration using a Newton method for solving an associated
reduced backward-Euler type nonlinear equation. Each Newton step requires the
inversion of a Jacobian matrix, which is done through a (preconditioned) inner
solver.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex10.cpp">ex10.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex10p.cpp">ex10p.cpp</a>) version.
It also has a SUNDIALS modification in <a href="https://github.com/mfem/mfem/blob/master/examples/sundials">examples/sundials</a>
and a PETSc modification in <a href="https://github.com/mfem/mfem/blob/master/examples/petsc">examples/petsc</a>.
We recommend viewing examples 2 and 9 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex11" style="display: block;">
<h2 id="example-11-laplace-eigenproblem">Example 11: Laplace Eigenproblem</h2>
<p><img class="floatright" src="../doc/web/examples/ex11.png"></p>
<p>This example code demonstrates the use of MFEM to solve the eigenvalue problem
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-335"><span class="MJXp-mo" id="MJXp-Span-336" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi" id="MJXp-Span-337">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-338">u</span><span class="MJXp-mo" id="MJXp-Span-339" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-340">λ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-341">u</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-43-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.857ex" height="2.302ex" viewBox="0 -824.7 4674.6 991.3" role="img" focusable="false" style="vertical-align: -0.387ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMAIN-394" x="778" y="0"></use><use xlink:href="#MJMATHI-75" x="1612" y="0"></use><use xlink:href="#MJMAIN-3D" x="2462" y="0"></use><use xlink:href="#MJMATHI-3BB" x="3518" y="0"></use><use xlink:href="#MJMATHI-75" x="4102" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-43">-\Delta u = \lambda u</script> with homogeneous Dirichlet boundary conditions.</p>
<p>We compute a number of the lowest eigenmodes by discretizing the Laplacian and
Mass operators using a finite element space of the specified order, or an
isoparametric/isogeometric space if order &lt; 1 (quadratic for quadratic
curvilinear mesh, NURBS for NURBS mesh, etc.)</p>
<p>The example highlights the use of the LOBPCG eigenvalue solver together with the
BoomerAMG preconditioner in HYPRE, as well as optionally the SuperLU or
STRUMPACK parallel direct solvers. Reusing a single <a href="http://glvis.org">GLVis</a>
visualization window for multiple eigenfunctions is also illustrated.</p>
<p><em>The example has only a parallel
(<a href="https://github.com/mfem/mfem/blob/master/examples/ex11p.cpp">ex11p.cpp</a>) version.
We recommend viewing Example 1 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex12" style="display: block;">
<h2 id="example-12-linear-elasticity-eigenproblem">Example 12: Linear Elasticity Eigenproblem</h2>
<p><img class="floatright" src="../doc/web/examples/ex12.png"></p>
<p>This example code solves the linear elasticity eigenvalue
problem for a multi-material cantilever beam.
Specifically, we compute a number of the lowest eigenmodes by approximating the weak form of
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-342"><span class="MJXp-mo" id="MJXp-Span-343" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mrow" id="MJXp-Span-344"><span class="MJXp-mi" id="MJXp-Span-345">d</span><span class="MJXp-mi" id="MJXp-Span-346">i</span><span class="MJXp-mi" id="MJXp-Span-347">v</span></span><span class="MJXp-mo" id="MJXp-Span-348" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-349"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-350">σ</span></span><span class="MJXp-mo" id="MJXp-Span-351" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-352"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-353">u</span></span><span class="MJXp-mo" id="MJXp-Span-354" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-355" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-356" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-357">λ</span><span class="MJXp-mrow" id="MJXp-Span-358"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-359">u</span></span><span class="MJXp-mspace" id="MJXp-Span-360" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mo" id="MJXp-Span-361" style="margin-left: 0em; margin-right: 0.222em;">,</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-44-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.381ex" height="2.684ex" viewBox="0 -824.7 7914.2 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-28" x="2142" y="0"></use><use xlink:href="#MJMATHI-3C3" x="2531" y="0"></use><use xlink:href="#MJMAIN-28" x="3104" y="0"></use><use xlink:href="#MJMAINB-75" x="3493" y="0"></use><use xlink:href="#MJMAIN-29" x="4133" y="0"></use><use xlink:href="#MJMAIN-29" x="4522" y="0"></use><use xlink:href="#MJMAIN-3D" x="5189" y="0"></use><use xlink:href="#MJMATHI-3BB" x="6246" y="0"></use><use xlink:href="#MJMAINB-75" x="6829" y="0"></use><use xlink:href="#MJMAIN-2C" x="7635" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-44">-{\rm div}({\sigma}({\bf u})) = \lambda {\bf u} \,,</script>
where
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-362"><span class="MJXp-mrow" id="MJXp-Span-363"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-364">σ</span></span><span class="MJXp-mo" id="MJXp-Span-365" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-366"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-367">u</span></span><span class="MJXp-mo" id="MJXp-Span-368" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-369" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-370">λ</span><span class="MJXp-mspace" id="MJXp-Span-371" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-372"><span class="MJXp-mi" id="MJXp-Span-373">d</span><span class="MJXp-mi" id="MJXp-Span-374">i</span><span class="MJXp-mi" id="MJXp-Span-375">v</span></span><span class="MJXp-mo" id="MJXp-Span-376" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-377"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-378">u</span></span><span class="MJXp-mo" id="MJXp-Span-379" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mspace" id="MJXp-Span-380" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-381">I</span><span class="MJXp-mo" id="MJXp-Span-382" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-383">μ</span><span class="MJXp-mspace" id="MJXp-Span-384" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mo" id="MJXp-Span-385" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi" id="MJXp-Span-386">∇</span><span class="MJXp-mrow" id="MJXp-Span-387"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-388">u</span></span><span class="MJXp-mo" id="MJXp-Span-389" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi" id="MJXp-Span-390">∇</span><span class="MJXp-msubsup" id="MJXp-Span-391"><span class="MJXp-mrow" id="MJXp-Span-392" style="margin-right: 0.05em;"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-393">u</span></span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-394" style="vertical-align: 0.5em;">T</span></span><span class="MJXp-mo" id="MJXp-Span-395" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-45-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="34.996ex" height="3.067ex" viewBox="0 -989.3 15067.6 1320.3" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="572" y="0"></use><use xlink:href="#MJMAINB-75" x="962" y="0"></use><use xlink:href="#MJMAIN-29" x="1601" y="0"></use><use xlink:href="#MJMAIN-3D" x="2268" y="0"></use><use xlink:href="#MJMATHI-3BB" x="3325" y="0"></use><g transform="translate(4075,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-28" x="5438" y="0"></use><use xlink:href="#MJMAINB-75" x="5828" y="0"></use><use xlink:href="#MJMAIN-29" x="6467" y="0"></use><use xlink:href="#MJMATHI-49" x="7023" y="0"></use><use xlink:href="#MJMAIN-2B" x="7750" y="0"></use><use xlink:href="#MJMATHI-3BC" x="8751" y="0"></use><use xlink:href="#MJMAIN-28" x="9521" y="0"></use><use xlink:href="#MJMAIN-2207" x="9911" y="0"></use><use xlink:href="#MJMAINB-75" x="10744" y="0"></use><use xlink:href="#MJMAIN-2B" x="11606" y="0"></use><use xlink:href="#MJMAIN-2207" x="12606" y="0"></use><g transform="translate(13440,0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="904" y="583"></use></g><use xlink:href="#MJMAIN-29" x="14678" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-45">{\sigma}({\bf u}) = \lambda\, {\rm div}({\bf u})\,I + \mu\,(\nabla{\bf u} + \nabla{\bf u}^T)</script>
is the stress tensor corresponding to displacement field <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-396"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-397">u</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-46-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.485ex" height="1.538ex" viewBox="0 -550.5 639.5 662.2" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-46">\bf u</script>, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-398"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-399">λ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-47-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.355ex" height="2.048ex" viewBox="0 -769.9 583.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-47">\lambda</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-400"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-401">μ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-48-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.92ex" viewBox="0 -550.5 603.5 826.7" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-48">\mu</script>
are the material Lame constants. The boundary conditions are
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-402"><span class="MJXp-mrow" id="MJXp-Span-403"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-404">u</span></span><span class="MJXp-mo" id="MJXp-Span-405" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-406">0</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-49-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.746ex" height="2.048ex" viewBox="0 -769.9 2474.1 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use><use xlink:href="#MJMAIN-3D" x="917" y="0"></use><use xlink:href="#MJMAIN-30" x="1973" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-49">{\bf u}=0</script> on the fixed part of the boundary with attribute 1, and
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-407"><span class="MJXp-mrow" id="MJXp-Span-408"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-409">σ</span></span><span class="MJXp-mo" id="MJXp-Span-410" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-411"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-412">u</span></span><span class="MJXp-mo" id="MJXp-Span-413" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-414" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-415">n</span><span class="MJXp-mo" id="MJXp-Span-416" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-417">f</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-50-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.075ex" height="2.684ex" viewBox="0 -824.7 5199 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="572" y="0"></use><use xlink:href="#MJMAINB-75" x="962" y="0"></use><use xlink:href="#MJMAIN-29" x="1601" y="0"></use><use xlink:href="#MJMAIN-22C5" x="2213" y="0"></use><use xlink:href="#MJMATHI-6E" x="2713" y="0"></use><use xlink:href="#MJMAIN-3D" x="3592" y="0"></use><use xlink:href="#MJMATHI-66" x="4648" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-50">{\sigma}({\bf u})\cdot n = f</script> on the remainder.
The geometry of the domain is assumed to be as follows:</p>
<p><img alt="" src="../doc/web/examples/ex12-domain.png"></p>
<p>The example highlights the use of the LOBPCG eigenvalue solver together with the
BoomerAMG preconditioner in HYPRE.
Reusing a single <a href="http://glvis.org">GLVis</a> visualization window for multiple
eigenfunctions is also illustrated.</p>
<p><em>The example has only a parallel
(<a href="https://github.com/mfem/mfem/blob/master/examples/ex12p.cpp">ex12p.cpp</a>) version.
We recommend viewing examples 2 and 11 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex13" style="display: block;">
<h2 id="example-13-maxwell-eigenproblem">Example 13: Maxwell Eigenproblem</h2>
<p><img class="floatright" src="../doc/web/examples/ex13.png"></p>
<p>This example code solves the Maxwell (electromagnetic)
eigenvalue problem
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-418"><span class="MJXp-mi" id="MJXp-Span-419">∇</span><span class="MJXp-mo" id="MJXp-Span-420" style="margin-left: 0.267em; margin-right: 0.267em;">×</span><span class="MJXp-mi" id="MJXp-Span-421">∇</span><span class="MJXp-mo" id="MJXp-Span-422" style="margin-left: 0.267em; margin-right: 0.267em;">×</span><span class="MJXp-mspace" id="MJXp-Span-423" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-424">E</span><span class="MJXp-mo" id="MJXp-Span-425" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-426">λ</span><span class="MJXp-mspace" id="MJXp-Span-427" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-428">E</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-51-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.332ex" height="2.048ex" viewBox="0 -769.9 7892.8 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="1055" y="0"></use><use xlink:href="#MJMAIN-2207" x="2056" y="0"></use><use xlink:href="#MJMAIN-D7" x="3112" y="0"></use><use xlink:href="#MJMATHI-45" x="4279" y="0"></use><use xlink:href="#MJMAIN-3D" x="5321" y="0"></use><use xlink:href="#MJMATHI-3BB" x="6378" y="0"></use><use xlink:href="#MJMATHI-45" x="7128" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-51">\nabla\times\nabla\times\, E = \lambda\, E </script>
with  homogeneous Dirichlet boundary conditions <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-429"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-430">E</span><span class="MJXp-mo" id="MJXp-Span-431" style="margin-left: 0.267em; margin-right: 0.267em;">×</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-432">n</span><span class="MJXp-mo" id="MJXp-Span-433" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-434">0</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-52-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.272ex" height="2.048ex" viewBox="0 -769.9 4422.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use><use xlink:href="#MJMAIN-D7" x="986" y="0"></use><use xlink:href="#MJMATHI-6E" x="1987" y="0"></use><use xlink:href="#MJMAIN-3D" x="2865" y="0"></use><use xlink:href="#MJMAIN-30" x="3922" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-52">E \times n = 0</script>.</p>
<p>We compute a number of the lowest nonzero eigenmodes by
discretizing the curl curl operator using a Nedelec finite element space of
the specified order in 2D or 3D.</p>
<p>The example highlights the use of the AME subspace eigenvalue
solver from HYPRE, which uses LOBPCG and AMS internally.
Reusing a single <a href="http://glvis.org">GLVis</a> visualization window for multiple
eigenfunctions is also illustrated.</p>
<p><em>The example has only a parallel
(<a href="https://github.com/mfem/mfem/blob/master/examples/ex13p.cpp">ex13p.cpp</a>) version.
We recommend viewing examples 3 and 11 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex14" style="display: block;">
<h2 id="example-14-dg-diffusion">Example 14: DG Diffusion</h2>
<p><img class="floatright" src="../doc/web/examples/ex14.png"></p>
<p>This example code demonstrates the use of MFEM to define a
discontinuous Galerkin (DG) finite element discretization of
the Laplace problem  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-435"><span class="MJXp-mo" id="MJXp-Span-436" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi" id="MJXp-Span-437">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-438">u</span><span class="MJXp-mo" id="MJXp-Span-439" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-440">1</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-53-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.335ex" height="2.302ex" viewBox="0 -824.7 4019.1 991.3" role="img" focusable="false" style="vertical-align: -0.387ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><use xlink:href="#MJMAIN-394" x="778" y="0"></use><use xlink:href="#MJMATHI-75" x="1612" y="0"></use><use xlink:href="#MJMAIN-3D" x="2462" y="0"></use><use xlink:href="#MJMAIN-31" x="3518" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-53">-\Delta u = 1</script> with homogeneous Dirichlet
boundary conditions. Finite element spaces of any order,
including zero on regular grids, are supported. The example highlights the use
of discontinuous spaces and DG-specific face integrators.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex14.cpp">ex14.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex14p.cpp">ex14p.cpp</a>) version.
We recommend viewing examples 1 and 9 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex15" style="display: block;">
<h2 id="example-15-dynamic-amr">Example 15: Dynamic AMR</h2>
<p><img class="floatright" src="../doc/web/examples/ex15.png"></p>
<p>Building on <a href="#ex6">Example 6</a>, this example demonstrates dynamic adaptive mesh refinement.
The mesh is adapted to a time-dependent solution by refinement
as well as by derefinement. For simplicity, the solution is
prescribed and no time integration is done. However, the error
estimation and refinement/derefinement decisions are realistic.</p>
<p>At each outer iteration the right hand side function is changed
to mimic a time dependent problem.  Within each inner iteration
the problem is solved on a sequence of meshes which are locally
refined according to a simple ZZ error estimator.  At the end
of the inner iteration the error estimates are also used to
identify any elements which may be over-refined and a single
derefinement step is performed.  After each refinement or
derefinement step a rebalance operation is performed to keep
the mesh evenly distributed among the available processors.</p>
<p>The example demonstrates MFEM's capability to refine, derefine
and load balance nonconforming meshes, in 2D and 3D, and on
linear, curved and surface meshes. Interpolation of functions
between coarse and fine meshes, persistent <a href="http://glvis.org">GLVis</a> visualization,
and saving of time-dependent fields for external visualization
with <a href="http://visit.llnl.gov">VisIt</a> are also illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex15.cpp">ex15.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex15p.cpp">ex15p.cpp</a>) version.
We recommend viewing examples 1, 6 and 9 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex16" style="display: block;">
<h2 id="example-16-time-dependent-heat-conduction">Example 16: Time Dependent Heat Conduction</h2>
<p><img class="floatright" src="../doc/web/examples/ex16.png"></p>
<p>This example code solves a simple 2D/3D time dependent nonlinear heat conduction problem
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-441"><span class="MJXp-mfrac" id="MJXp-Span-442" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-443">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-444">u</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-445">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-446">t</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-447" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi" id="MJXp-Span-448">∇</span><span class="MJXp-mo" id="MJXp-Span-449" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mrow" id="MJXp-Span-450"><span class="MJXp-mo" id="MJXp-Span-451" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-452">κ</span><span class="MJXp-mo" id="MJXp-Span-453" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-454">α</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-455">u</span><span class="MJXp-mo" id="MJXp-Span-456" style="margin-left: 0em; margin-right: 0em;">)</span></span><span class="MJXp-mi" id="MJXp-Span-457">∇</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-458">u</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-54-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.554ex" height="5.232ex" viewBox="0 -1482.9 9710.6 2252.8" role="img" focusable="false" style="vertical-align: -1.788ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="1216" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-75" x="523" y="0"></use></g><g transform="translate(165,-686)"><use xlink:href="#MJMATHI-64" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="523" y="0"></use></g></g><use xlink:href="#MJMAIN-3D" x="1733" y="0"></use><use xlink:href="#MJMAIN-2207" x="2790" y="0"></use><use xlink:href="#MJMAIN-22C5" x="3845" y="0"></use><g transform="translate(4346,0)"><use xlink:href="#MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#MJMATHI-3BA" x="0" y="0"></use><use xlink:href="#MJMAIN-2B" x="798" y="0"></use><use xlink:href="#MJMATHI-3B1" x="1799" y="0"></use><use xlink:href="#MJMATHI-75" x="2439" y="0"></use></g><use xlink:href="#MJMAIN-29" x="3401" y="0"></use></g><use xlink:href="#MJMAIN-2207" x="8304" y="0"></use><use xlink:href="#MJMATHI-75" x="9138" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-54">\frac{du}{dt} = \nabla \cdot \left( \kappa + \alpha u \right) \nabla u</script>
with a natural insulating boundary condition <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-459"><span class="MJXp-mfrac" id="MJXp-Span-460" style="vertical-align: 0.25em;"><span class="MJXp-box MJXp-script"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-461">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-462">u</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box MJXp-script"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-463">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-464">n</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-465" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-466">0</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-55-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.943ex" height="3.449ex" viewBox="0 -989.3 2989.3 1484.9" role="img" focusable="false" style="vertical-align: -1.151ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="914" height="60" x="0" y="220"></rect><g transform="translate(69,421)"><use transform="scale(0.707)" xlink:href="#MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-75" x="523" y="0"></use></g><g transform="translate(60,-405)"><use transform="scale(0.707)" xlink:href="#MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6E" x="523" y="0"></use></g></g><use xlink:href="#MJMAIN-3D" x="1432" y="0"></use><use xlink:href="#MJMAIN-30" x="2488" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-55">\frac{du}{dn} = 0</script>.
We linearize the problem by using the temperature field <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-467"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-468">u</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-56-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.538ex" viewBox="0 -550.5 572.5 662.2" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-75" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-56">u</script> from the previous time
step to compute the conductivity coefficient.</p>
<p>This example demonstrates both implicit and explicit time integration as well as a single
Picard step method for linearization. The saving of time dependent data files for external
visualization with <a href="http://visit.llnl.gov">VisIt</a> is also illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex16.cpp">ex16.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex16p.cpp">ex16p.cpp</a>) version.
We recommend viewing examples 2, 9, and 10 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex17" style="display: block;">
<h2 id="example-17-dg-linear-elasticity">Example 17: DG Linear Elasticity</h2>
<p><img class="floatright" src="../doc/web/examples/ex17.png"></p>
<p>This example code solves a simple linear elasticity problem
describing a multi-material cantilever beam using symmetric or
non-symmetric discontinuous Galerkin (DG) formulation.</p>
<p>Specifically, we approximate the weak form of
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-469"><span class="MJXp-mo" id="MJXp-Span-470" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mrow" id="MJXp-Span-471"><span class="MJXp-mi" id="MJXp-Span-472">d</span><span class="MJXp-mi" id="MJXp-Span-473">i</span><span class="MJXp-mi" id="MJXp-Span-474">v</span></span><span class="MJXp-mo" id="MJXp-Span-475" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-476"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-477">σ</span></span><span class="MJXp-mo" id="MJXp-Span-478" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-479"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-480">u</span></span><span class="MJXp-mo" id="MJXp-Span-481" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-482" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-483" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-484">0</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-57-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.669ex" height="2.684ex" viewBox="0 -824.7 6746.6 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-28" x="2142" y="0"></use><use xlink:href="#MJMATHI-3C3" x="2531" y="0"></use><use xlink:href="#MJMAIN-28" x="3104" y="0"></use><use xlink:href="#MJMAINB-75" x="3493" y="0"></use><use xlink:href="#MJMAIN-29" x="4133" y="0"></use><use xlink:href="#MJMAIN-29" x="4522" y="0"></use><use xlink:href="#MJMAIN-3D" x="5189" y="0"></use><use xlink:href="#MJMAIN-30" x="6246" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-57">-{\rm div}({\sigma}({\bf u})) = 0</script>
where
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-485"><span class="MJXp-mrow" id="MJXp-Span-486"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-487">σ</span></span><span class="MJXp-mo" id="MJXp-Span-488" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-489"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-490">u</span></span><span class="MJXp-mo" id="MJXp-Span-491" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-492" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-493">λ</span><span class="MJXp-mspace" id="MJXp-Span-494" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-495"><span class="MJXp-mi" id="MJXp-Span-496">d</span><span class="MJXp-mi" id="MJXp-Span-497">i</span><span class="MJXp-mi" id="MJXp-Span-498">v</span></span><span class="MJXp-mo" id="MJXp-Span-499" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-500"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-501">u</span></span><span class="MJXp-mo" id="MJXp-Span-502" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mspace" id="MJXp-Span-503" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-504">I</span><span class="MJXp-mo" id="MJXp-Span-505" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-506">μ</span><span class="MJXp-mspace" id="MJXp-Span-507" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mo" id="MJXp-Span-508" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi" id="MJXp-Span-509">∇</span><span class="MJXp-mrow" id="MJXp-Span-510"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-511">u</span></span><span class="MJXp-mo" id="MJXp-Span-512" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi" id="MJXp-Span-513">∇</span><span class="MJXp-msubsup" id="MJXp-Span-514"><span class="MJXp-mrow" id="MJXp-Span-515" style="margin-right: 0.05em;"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-516">u</span></span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-517" style="vertical-align: 0.5em;">T</span></span><span class="MJXp-mo" id="MJXp-Span-518" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-58-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="34.996ex" height="3.067ex" viewBox="0 -989.3 15067.6 1320.3" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="572" y="0"></use><use xlink:href="#MJMAINB-75" x="962" y="0"></use><use xlink:href="#MJMAIN-29" x="1601" y="0"></use><use xlink:href="#MJMAIN-3D" x="2268" y="0"></use><use xlink:href="#MJMATHI-3BB" x="3325" y="0"></use><g transform="translate(4075,0)"><use xlink:href="#MJMAIN-64" x="0" y="0"></use><use xlink:href="#MJMAIN-69" x="556" y="0"></use><use xlink:href="#MJMAIN-76" x="835" y="0"></use></g><use xlink:href="#MJMAIN-28" x="5438" y="0"></use><use xlink:href="#MJMAINB-75" x="5828" y="0"></use><use xlink:href="#MJMAIN-29" x="6467" y="0"></use><use xlink:href="#MJMATHI-49" x="7023" y="0"></use><use xlink:href="#MJMAIN-2B" x="7750" y="0"></use><use xlink:href="#MJMATHI-3BC" x="8751" y="0"></use><use xlink:href="#MJMAIN-28" x="9521" y="0"></use><use xlink:href="#MJMAIN-2207" x="9911" y="0"></use><use xlink:href="#MJMAINB-75" x="10744" y="0"></use><use xlink:href="#MJMAIN-2B" x="11606" y="0"></use><use xlink:href="#MJMAIN-2207" x="12606" y="0"></use><g transform="translate(13440,0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-54" x="904" y="583"></use></g><use xlink:href="#MJMAIN-29" x="14678" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-58">{\sigma}({\bf u}) = \lambda\, {\rm div}({\bf u})\,I + \mu\,(\nabla{\bf u} + \nabla{\bf u}^T)</script>
is the stress tensor corresponding to displacement field <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-519"><span class="MJXp-mrow" id="MJXp-Span-520"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-521">u</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-59-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.485ex" height="1.538ex" viewBox="0 -550.5 639.5 662.2" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-59">{\bf u}</script>, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-522"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-523">λ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-60-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.355ex" height="2.048ex" viewBox="0 -769.9 583.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-60">\lambda</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-524"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-525">μ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-61-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.92ex" viewBox="0 -550.5 603.5 826.7" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-61">\mu</script>
are the material Lame constants. The boundary conditions are
Dirichlet, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-526"><span class="MJXp-mrow" id="MJXp-Span-527"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-528">u</span></span><span class="MJXp-mo MJXp-bold" id="MJXp-Span-529" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-530"><span class="MJXp-msubsup" id="MJXp-Span-531"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-532" style="margin-right: 0.05em;">u</span><span class="MJXp-mi MJXp-bold MJXp-script" id="MJXp-Span-533" style="vertical-align: -0.4em;">D</span></span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-62-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.02ex" height="1.793ex" viewBox="0 -550.5 3453.1 771.9" role="img" focusable="false" style="vertical-align: -0.514ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use><use xlink:href="#MJMAINB-3D" x="917" y="0"></use><g transform="translate(2089,0)"><use xlink:href="#MJMAINB-75" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAINB-44" x="904" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-62">\bf{u}=\bf{u_D}</script>, on the fixed part of the boundary, namely
boundary attributes 1 and 2; on the rest of the boundary we use
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-534"><span class="MJXp-mrow" id="MJXp-Span-535"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-536">σ</span></span><span class="MJXp-mo" id="MJXp-Span-537" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-538"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-539">u</span></span><span class="MJXp-mo" id="MJXp-Span-540" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-541" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-542">n</span><span class="MJXp-mo" id="MJXp-Span-543" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-544"><span class="MJXp-mn MJXp-bold" id="MJXp-Span-545">0</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-63-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.133ex" height="2.684ex" viewBox="0 -824.7 5224 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use><use xlink:href="#MJMAIN-28" x="572" y="0"></use><use xlink:href="#MJMAINB-75" x="962" y="0"></use><use xlink:href="#MJMAIN-29" x="1601" y="0"></use><use xlink:href="#MJMAIN-22C5" x="2213" y="0"></use><use xlink:href="#MJMATHI-6E" x="2713" y="0"></use><use xlink:href="#MJMAIN-3D" x="3592" y="0"></use><use xlink:href="#MJMAINB-30" x="4648" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-63">{\sigma}({\bf u})\cdot n = {\bf 0}</script>. The geometry of the domain is assumed to be
as follows:</p>
<p><img alt="" src="../doc/web/examples/ex17-domain.png"></p>
<p>The example demonstrates the use of high-order DG vector finite
element spaces with the linear DG elasticity bilinear form,
meshes with curved elements, and the definition of piece-wise
constant and function vector-coefficient objects. The use of
non-homogeneous Dirichlet b.c. imposed weakly, is also
illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex17.cpp">ex17.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex17p.cpp">ex17p.cpp</a>) version.
We recommend viewing examples 2 and 14 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex18" style="display: block;">
<h2 id="example-18-dg-euler-equations">Example 18: DG Euler Equations</h2>
<p><img class="floatright" src="../doc/web/examples/ex18.png"></p>
<p>This example code solves the compressible Euler system of equations, a model
nonlinear hyperbolic PDE, with a discontinuous Galerkin (DG) formulation. The
primary purpose is to show how a transient system of nonlinear equations can be
formulated in MFEM. The equations are solved in conservative form</p>
<p><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-546"><span class="MJXp-mfrac" id="MJXp-Span-547" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi" id="MJXp-Span-548">∂</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-549">u</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi" id="MJXp-Span-550">∂</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-551">t</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-552" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi" id="MJXp-Span-553">∇</span><span class="MJXp-mo" id="MJXp-Span-554" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mrow" id="MJXp-Span-555"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-556">F</span></span><span class="MJXp-mo" id="MJXp-Span-557" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-558">u</span><span class="MJXp-mo" id="MJXp-Span-559" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-560" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-561">0</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-64-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.022ex" height="5.36ex" viewBox="0 -1482.9 8189.9 2307.6" role="img" focusable="false" style="vertical-align: -1.915ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="1260" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMATHI-75" x="567" y="0"></use></g><g transform="translate(165,-694)"><use xlink:href="#MJMAIN-2202" x="0" y="0"></use><use xlink:href="#MJMATHI-74" x="567" y="0"></use></g></g><use xlink:href="#MJMAIN-2B" x="1722" y="0"></use><use xlink:href="#MJMAIN-2207" x="2722" y="0"></use><use xlink:href="#MJMAIN-22C5" x="3778" y="0"></use><use xlink:href="#MJMAINB-46" x="4279" y="0"></use><use xlink:href="#MJMAIN-28" x="5003" y="0"></use><use xlink:href="#MJMATHI-75" x="5393" y="0"></use><use xlink:href="#MJMAIN-29" x="5965" y="0"></use><use xlink:href="#MJMAIN-3D" x="6633" y="0"></use><use xlink:href="#MJMAIN-30" x="7689" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-64">\frac{\partial u}{\partial t} + \nabla \cdot {\bf F}(u) = 0</script></p>
<p>with a state vector <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-562"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-563">u</span><span class="MJXp-mo" id="MJXp-Span-564" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mo" id="MJXp-Span-565" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-566">ρ</span><span class="MJXp-mo" id="MJXp-Span-567" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-568">ρ</span><span class="MJXp-msubsup" id="MJXp-Span-569"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-570" style="margin-right: 0.05em;">v</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-571" style="vertical-align: -0.4em;">0</span></span><span class="MJXp-mo" id="MJXp-Span-572" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-573">ρ</span><span class="MJXp-msubsup" id="MJXp-Span-574"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-575" style="margin-right: 0.05em;">v</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-576" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-577" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-578">ρ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-579">E</span><span class="MJXp-mo" id="MJXp-Span-580" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-65-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.771ex" height="2.684ex" viewBox="0 -824.7 8512.4 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-75" x="0" y="0"></use><use xlink:href="#MJMAIN-3D" x="850" y="0"></use><use xlink:href="#MJMAIN-5B" x="1906" y="0"></use><use xlink:href="#MJMATHI-3C1" x="2185" y="0"></use><use xlink:href="#MJMAIN-2C" x="2702" y="0"></use><use xlink:href="#MJMATHI-3C1" x="3147" y="0"></use><g transform="translate(3665,0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="686" y="-213"></use></g><use xlink:href="#MJMAIN-2C" x="4604" y="0"></use><use xlink:href="#MJMATHI-3C1" x="5049" y="0"></use><g transform="translate(5567,0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="686" y="-213"></use></g><use xlink:href="#MJMAIN-2C" x="6506" y="0"></use><use xlink:href="#MJMATHI-3C1" x="6951" y="0"></use><use xlink:href="#MJMATHI-45" x="7469" y="0"></use><use xlink:href="#MJMAIN-5D" x="8233" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-65">u = [ \rho, \rho v_0, \rho v_1, \rho E ]</script>, where <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-581"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-582">ρ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-66-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.202ex" height="1.92ex" viewBox="0 -550.5 517.5 826.7" role="img" focusable="false" style="vertical-align: -0.642ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C1" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-66">\rho</script> is
the density, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-583"><span class="MJXp-msubsup" id="MJXp-Span-584"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-585" style="margin-right: 0.05em;">v</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-586" style="vertical-align: -0.4em;">i</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-67-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.927ex" height="1.793ex" viewBox="0 -550.5 829.8 771.9" role="img" focusable="false" style="vertical-align: -0.514ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="686" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-67">v_i</script> is the velocity in the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-587"><span class="MJXp-msubsup" id="MJXp-Span-588"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-589" style="margin-right: 0.05em;">i</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-590" style="vertical-align: 0.5em;"><span class="MJXp-mi" id="MJXp-Span-591">t</span><span class="MJXp-mi" id="MJXp-Span-592">h</span></span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-68-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.588ex" height="2.43ex" viewBox="0 -934.4 1114.4 1046.1" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-69" x="0" y="0"></use><g transform="translate(345,362)"><use transform="scale(0.707)" xlink:href="#MJMAIN-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-68" x="389" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-68">i^{\rm th}</script> direction, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-593"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-594">E</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-69-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.776ex" height="2.048ex" viewBox="0 -769.9 764.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-45" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-69">E</script> is the
total specific energy, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-595"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-596">H</span><span class="MJXp-mo" id="MJXp-Span-597" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-598">E</span><span class="MJXp-mo" id="MJXp-Span-599" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-600">p</span><span class="MJXp-mrow" id="MJXp-Span-601"><span class="MJXp-mo" id="MJXp-Span-602" style="margin-left: 0.111em; margin-right: 0.111em;">/</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-603">ρ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-70-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.312ex" height="2.684ex" viewBox="0 -824.7 5731.5 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-48" x="0" y="0"></use><use xlink:href="#MJMAIN-3D" x="1166" y="0"></use><use xlink:href="#MJMATHI-45" x="2222" y="0"></use><use xlink:href="#MJMAIN-2B" x="3209" y="0"></use><use xlink:href="#MJMATHI-70" x="4210" y="0"></use><use xlink:href="#MJMAIN-2F" x="4713" y="0"></use><use xlink:href="#MJMATHI-3C1" x="5214" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-70">H = E + p / \rho</script> is the total specific enthalpy.
The pressure, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-604"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-605">p</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-71-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.259ex" height="1.92ex" viewBox="-38.5 -550.5 542 826.7" role="img" focusable="false" style="vertical-align: -0.642ex; margin-left: -0.089ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-70" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-71">p</script> is computed through a simple equation of state (EOS) call.
The conservative hydrodynamic flux <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-606"><span class="MJXp-mrow" id="MJXp-Span-607"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-608">F</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-72-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.683ex" height="2.048ex" viewBox="0 -769.9 724.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-46" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-72">{\bf F}</script> in each direction <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-609"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-610">i</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-73-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="2.048ex" viewBox="0 -769.9 345.5 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-73">i</script> is</p>
<p><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-611"><span class="MJXp-mrow" id="MJXp-Span-612"><span class="MJXp-msubsup" id="MJXp-Span-613"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-614" style="margin-right: 0.05em;">F</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-615" style="vertical-align: -0.4em;"><span class="MJXp-mi undefined" id="MJXp-Span-616">i</span></span></span></span><span class="MJXp-mo" id="MJXp-Span-617" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mo" id="MJXp-Span-618" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-619">ρ</span><span class="MJXp-msubsup" id="MJXp-Span-620"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-621" style="margin-right: 0.05em;">v</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-622" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-623" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-624">ρ</span><span class="MJXp-msubsup" id="MJXp-Span-625"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-626" style="margin-right: 0.05em;">v</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-627" style="vertical-align: -0.4em;">0</span></span><span class="MJXp-msubsup" id="MJXp-Span-628"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-629" style="margin-right: 0.05em;">v</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-630" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-631" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-632">p</span><span class="MJXp-msubsup" id="MJXp-Span-633"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-634" style="margin-right: 0.05em;">δ</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-635" style="vertical-align: -0.4em;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-636">i</span><span class="MJXp-mo" id="MJXp-Span-637">,</span><span class="MJXp-mn" id="MJXp-Span-638">0</span></span></span><span class="MJXp-mo" id="MJXp-Span-639" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-640">ρ</span><span class="MJXp-msubsup" id="MJXp-Span-641"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-642" style="margin-right: 0.05em;">v</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-643" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-msubsup" id="MJXp-Span-644"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-645" style="margin-right: 0.05em;">v</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-646" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-647" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-648">p</span><span class="MJXp-msubsup" id="MJXp-Span-649"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-650" style="margin-right: 0.05em;">δ</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-651" style="vertical-align: -0.4em;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-652">i</span><span class="MJXp-mo" id="MJXp-Span-653">,</span><span class="MJXp-mn" id="MJXp-Span-654">1</span></span></span><span class="MJXp-mo" id="MJXp-Span-655" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-656">ρ</span><span class="MJXp-msubsup" id="MJXp-Span-657"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-658" style="margin-right: 0.05em;">v</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-659" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-660">H</span><span class="MJXp-mo" id="MJXp-Span-661" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-74-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="43.101ex" height="2.812ex" viewBox="0 -824.7 18557.2 1210.6" role="img" focusable="false" style="vertical-align: -0.896ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-46" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAINI-69" x="1024" y="-213"></use><use xlink:href="#MJMAIN-3D" x="1319" y="0"></use><use xlink:href="#MJMAIN-5B" x="2375" y="0"></use><use xlink:href="#MJMATHI-3C1" x="2654" y="0"></use><g transform="translate(3171,0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="686" y="-213"></use></g><use xlink:href="#MJMAIN-2C" x="4001" y="0"></use><use xlink:href="#MJMATHI-3C1" x="4446" y="0"></use><g transform="translate(4964,0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="686" y="-213"></use></g><g transform="translate(5903,0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="686" y="-213"></use></g><use xlink:href="#MJMAIN-2B" x="6955" y="0"></use><use xlink:href="#MJMATHI-70" x="7956" y="0"></use><g transform="translate(8460,0)"><use xlink:href="#MJMATHI-3B4" x="0" y="0"></use><g transform="translate(444,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-30" x="624" y="0"></use></g></g><use xlink:href="#MJMAIN-2C" x="9799" y="0"></use><use xlink:href="#MJMATHI-3C1" x="10244" y="0"></use><g transform="translate(10762,0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="686" y="-213"></use></g><g transform="translate(11701,0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="686" y="-213"></use></g><use xlink:href="#MJMAIN-2B" x="12753" y="0"></use><use xlink:href="#MJMATHI-70" x="13754" y="0"></use><g transform="translate(14258,0)"><use xlink:href="#MJMATHI-3B4" x="0" y="0"></use><g transform="translate(444,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="624" y="0"></use></g></g><use xlink:href="#MJMAIN-2C" x="15597" y="0"></use><use xlink:href="#MJMATHI-3C1" x="16042" y="0"></use><g transform="translate(16560,0)"><use xlink:href="#MJMATHI-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="686" y="-213"></use></g><use xlink:href="#MJMATHI-48" x="17390" y="0"></use><use xlink:href="#MJMAIN-5D" x="18278" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-74">{\bf F_{\it i}} = [ \rho v_i, \rho v_0 v_i + p \delta_{i,0}, \rho v_1 v_i + p \delta_{i,1}, \rho v_i H ]</script></p>
<p>Specifically, the example solves for an exact solution of the equations whereby
a vortex is transported by a uniform flow. Since all boundaries are periodic
here, the method's accuracy can be assessed by measuring the difference between
the solution and the initial condition at a later time when the vortex returns
to its initial location.</p>
<p>Note that as the order of the spatial discretization increases, the timestep
must become smaller. This example currently uses a simple estimate derived by
<a href="https://link.springer.com/article/10.1023/A:1012873910884">Cockburn and Shu</a>
for the 1D RKDG method. An additional factor can be tuned by passing the <code>--cfl</code>
(or <code>-c</code> shorter) flag.</p>
<p>The example demonstrates user-defined bilinear and nonlinear form integrators
for systems of equations that are defined with block vectors, and how these are
used with an operator for explicit time integrators. In this case the system
also involves an external approximate Riemann solver for the DG interface flux.
It also demonstrates how to use GLVis for in-situ visualization of vector grid
functions.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex18.cpp">ex18.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex18p.cpp">ex18p.cpp</a>) version.
We recommend viewing examples 9, 14 and 17 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex19" style="display: block;">
<h2 id="example-19-incompressible-nonlinear-elasticity">Example 19: Incompressible Nonlinear Elasticity</h2>
<p><img class="floatright" src="../doc/web/examples/ex19.png"></p>
<p>This example code solves the quasi-static incompressible nonlinear
hyperelasticity equations. Specifically, it solves the nonlinear equation
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-662"><span class="MJXp-mi" id="MJXp-Span-663">∇</span><span class="MJXp-mo" id="MJXp-Span-664" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-665">σ</span><span class="MJXp-mo" id="MJXp-Span-666" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-667">F</span><span class="MJXp-mo" id="MJXp-Span-668" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-669" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-670">0</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-75-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.756ex" height="2.684ex" viewBox="0 -824.7 5492 1155.8" role="img" focusable="false" style="vertical-align: -0.769ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2207" x="0" y="0"></use><use xlink:href="#MJMAIN-22C5" x="1055" y="0"></use><use xlink:href="#MJMATHI-3C3" x="1556" y="0"></use><use xlink:href="#MJMAIN-28" x="2128" y="0"></use><use xlink:href="#MJMATHI-46" x="2518" y="0"></use><use xlink:href="#MJMAIN-29" x="3267" y="0"></use><use xlink:href="#MJMAIN-3D" x="3935" y="0"></use><use xlink:href="#MJMAIN-30" x="4991" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-75">
\nabla \cdot \sigma(F) = 0
</script>
subject to the constraint
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-671"><span class="MJXp-mtext" id="MJXp-Span-672">det&nbsp;</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-673">F</span><span class="MJXp-mo" id="MJXp-Span-674" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-675">1</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processed" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-76-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.812ex" height="2.048ex" viewBox="0 -769.9 4224.6 881.6" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-64"></use><use xlink:href="#MJMAIN-65" x="556" y="0"></use><use xlink:href="#MJMAIN-74" x="1001" y="0"></use><use xlink:href="#MJMATHI-46" x="1640" y="0"></use><use xlink:href="#MJMAIN-3D" x="2667" y="0"></use><use xlink:href="#MJMAIN-31" x="3724" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-76">
\text{det } F = 1
</script>
where <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-676"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-677">σ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-77-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.411ex" viewBox="0 -495.6 572.5 607.3" role="img" focusable="false" style="vertical-align: -0.259ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-3C3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-77">\sigma</script> is the Cauchy stress and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-678"><span class="MJXp-msubsup" id="MJXp-Span-679"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-680" style="margin-right: 0.05em;">F</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-681" style="vertical-align: -0.4em;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-682">i</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-683">j</span></span></span><span class="MJXp-mo" id="MJXp-Span-684" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-msubsup" id="MJXp-Span-685"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-686" style="margin-right: 0.05em;">δ</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-687" style="vertical-align: -0.4em;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-688">i</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-689">j</span></span></span><span class="MJXp-mo" id="MJXp-Span-690" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-msubsup" id="MJXp-Span-691"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-692" style="margin-right: 0.05em;">u</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-693" style="vertical-align: -0.4em;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-694">i</span><span class="MJXp-mo" id="MJXp-Span-695">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-696">j</span></span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-78-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.684ex" height="2.812ex" viewBox="0 -824.7 6322.4 1210.6" role="img" focusable="false" style="vertical-align: -0.896ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-46" x="0" y="0"></use><g transform="translate(643,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="345" y="0"></use></g><use xlink:href="#MJMAIN-3D" x="1557" y="0"></use><g transform="translate(2613,0)"><use xlink:href="#MJMATHI-3B4" x="0" y="0"></use><g transform="translate(444,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="345" y="0"></use></g></g><use xlink:href="#MJMAIN-2B" x="3916" y="0"></use><g transform="translate(4916,0)"><use xlink:href="#MJMATHI-75" x="0" y="0"></use><g transform="translate(572,-150)"><use transform="scale(0.707)" xlink:href="#MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMAIN-2C" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#MJMATHI-6A" x="624" y="0"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-78">F_{ij} = \delta_{ij} + u_{i,j}</script> is the deformation
gradient. To handle the incompressibility constraint, pressure is included as
an independent unknown <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-697"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-698">p</span></span></span><span class="MathJax_SVG MathJax_SVG_Processed" id="MathJax-Element-79-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.259ex" height="1.92ex" viewBox="-38.5 -550.5 542 826.7" role="img" focusable="false" style="vertical-align: -0.642ex; margin-left: -0.089ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-70" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-79">p</script> and the stress response is modeled as an <a href="http://solidmechanics.org/text/Chapter3_5/Chapter3_5.htm">incompressible
neo-Hookean hyperelastic solid</a>.
The geometry of the domain is assumed to be as follows:</p>
<p><img alt="" src="../doc/web/examples/ex19-domain.png"></p>
<p>This formulation requires solving the saddle point system
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-699"><span class="MJXp-mrow" id="MJXp-Span-700"><span class="MJXp-mo" id="MJXp-Span-701" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.617em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.467em; margin-left: -0.17em;">[</span></span><span class="MJXp-mtable" id="MJXp-Span-702"><span><span class="MJXp-mtr" id="MJXp-Span-703" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-704" style="text-align: center;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-705">K</span></span><span class="MJXp-mtd" id="MJXp-Span-706" style="padding-left: 1em; text-align: center;"><span class="MJXp-msubsup" id="MJXp-Span-707"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-708" style="margin-right: 0.05em;">B</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-709" style="vertical-align: 0.5em;">T</span></span></span></span><span class="MJXp-mtr" id="MJXp-Span-710" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-711" style="padding-top: 0.4em; text-align: center;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-712">B</span></span><span class="MJXp-mtd" id="MJXp-Span-713" style="padding-left: 1em; padding-top: 0.4em; text-align: center;"><span class="MJXp-mn" id="MJXp-Span-714">0</span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-715" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.617em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.467em; margin-left: -0.17em;">]</span></span></span><span class="MJXp-mrow" id="MJXp-Span-716"><span class="MJXp-mo" id="MJXp-Span-717" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.528em;"><span class="MJXp-right MJXp-scale5" style="font-size: 3.111em; margin-left: -0.13em;">[</span></span><span class="MJXp-mtable" id="MJXp-Span-718"><span><span class="MJXp-mtr" id="MJXp-Span-719" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-720" style="text-align: center;"><span class="MJXp-mi" id="MJXp-Span-721">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-722">u</span></span></span><span class="MJXp-mtr" id="MJXp-Span-723" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-724" style="padding-top: 0.4em; text-align: center;"><span class="MJXp-mi" id="MJXp-Span-725">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-726">p</span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-727" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.528em;"><span class="MJXp-right MJXp-scale5" style="font-size: 3.111em; margin-left: -0.13em;">]</span></span></span><span class="MJXp-mo" id="MJXp-Span-728" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-729"><span class="MJXp-mo" id="MJXp-Span-730" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.717em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.867em; margin-left: -0.17em;">[</span></span><span class="MJXp-mtable" id="MJXp-Span-731"><span><span class="MJXp-mtr" id="MJXp-Span-732" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-733" style="text-align: center;"><span class="MJXp-msubsup" id="MJXp-Span-734"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-735" style="margin-right: 0.05em;">R</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-736" style="vertical-align: -0.4em;">u</span></span></span></span><span class="MJXp-mtr" id="MJXp-Span-737" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-738" style="padding-top: 0.4em; text-align: center;"><span class="MJXp-msubsup" id="MJXp-Span-739"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-740" style="margin-right: 0.05em;">R</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-741" style="vertical-align: -0.4em;">p</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-742" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.717em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.867em; margin-left: -0.17em;">]</span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-80-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-80"> \left[ \begin{array}{cc}
   K &B^T \\
   B & 0
\end{array} \right]
\left[\begin{array}{c} \Delta u \\ \Delta p \end{array} \right] =
\left[\begin{array}{c} R_u \\ R_p \end{array} \right]
</script>
at each Newton step. To solve this linear system, we implement a specialized block
preconditioner of the form
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-743"><span class="MJXp-msubsup" id="MJXp-Span-744"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-745" style="margin-right: 0.05em;">P</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-746" style="vertical-align: 0.5em;"><span class="MJXp-mo" id="MJXp-Span-747">−</span><span class="MJXp-mn" id="MJXp-Span-748">1</span></span></span><span class="MJXp-mo" id="MJXp-Span-749" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-750"><span class="MJXp-mo" id="MJXp-Span-751" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.679em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.716em; margin-left: -0.17em;">[</span></span><span class="MJXp-mtable" id="MJXp-Span-752"><span><span class="MJXp-mtr" id="MJXp-Span-753" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-754" style="text-align: center;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-755">I</span></span><span class="MJXp-mtd" id="MJXp-Span-756" style="padding-left: 1em; text-align: center;"><span class="MJXp-mo" id="MJXp-Span-757" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-msubsup" id="MJXp-Span-758"><span class="MJXp-mrow" id="MJXp-Span-759" style="margin-right: 0.05em;"><span class="MJXp-munderover" id="MJXp-Span-760"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.92em;"><span class="MJXp-mo" id="MJXp-Span-762" style="margin-left: 0px; margin-right: 0px;">˜</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-761">K</span></span></span></span></span></span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-763" style="vertical-align: 0.724em;"><span class="MJXp-mo" id="MJXp-Span-764">−</span><span class="MJXp-mn" id="MJXp-Span-765">1</span></span></span><span class="MJXp-msubsup" id="MJXp-Span-766"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-767" style="margin-right: 0.05em;">B</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-768" style="vertical-align: 0.5em;">T</span></span></span></span><span class="MJXp-mtr" id="MJXp-Span-769" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-770" style="padding-top: 0.4em; text-align: center;"><span class="MJXp-mn" id="MJXp-Span-771">0</span></span><span class="MJXp-mtd" id="MJXp-Span-772" style="padding-left: 1em; padding-top: 0.4em; text-align: center;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-773">I</span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-774" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.679em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.716em; margin-left: -0.17em;">]</span></span></span><span class="MJXp-mrow" id="MJXp-Span-775"><span class="MJXp-mo" id="MJXp-Span-776" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.83em;"><span class="MJXp-right MJXp-scale3" style="font-size: 4.32em; margin-left: -0.21em;">[</span></span><span class="MJXp-mtable" id="MJXp-Span-777"><span><span class="MJXp-mtr" id="MJXp-Span-778" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-779" style="text-align: center;"><span class="MJXp-msubsup" id="MJXp-Span-780"><span class="MJXp-mrow" id="MJXp-Span-781" style="margin-right: 0.05em;"><span class="MJXp-munderover" id="MJXp-Span-782"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.92em;"><span class="MJXp-mo" id="MJXp-Span-784" style="margin-left: 0px; margin-right: 0px;">˜</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-783">K</span></span></span></span></span></span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-785" style="vertical-align: 0.724em;"><span class="MJXp-mo" id="MJXp-Span-786">−</span><span class="MJXp-mn" id="MJXp-Span-787">1</span></span></span></span><span class="MJXp-mtd" id="MJXp-Span-788" style="padding-left: 1em; text-align: center;"><span class="MJXp-mn" id="MJXp-Span-789">0</span></span></span><span class="MJXp-mtr" id="MJXp-Span-790" style="vertical-align: baseline;"><span class="MJXp-mtd" id="MJXp-Span-791" style="padding-top: 0.4em; text-align: center;"><span class="MJXp-mn" id="MJXp-Span-792">0</span></span><span class="MJXp-mtd" id="MJXp-Span-793" style="padding-left: 1em; padding-top: 0.4em; text-align: center;"><span class="MJXp-mo" id="MJXp-Span-794" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-795">γ</span><span class="MJXp-msubsup" id="MJXp-Span-796"><span class="MJXp-mrow" id="MJXp-Span-797" style="margin-right: 0.05em;"><span class="MJXp-munderover" id="MJXp-Span-798"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.92em;"><span class="MJXp-mo" id="MJXp-Span-800" style="margin-left: 0px; margin-right: 0px;">˜</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-799">S</span></span></span></span></span></span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-801" style="vertical-align: 0.724em;"><span class="MJXp-mo" id="MJXp-Span-802">−</span><span class="MJXp-mn" id="MJXp-Span-803">1</span></span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-804" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.83em;"><span class="MJXp-right MJXp-scale3" style="font-size: 4.32em; margin-left: -0.21em;">]</span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-81-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-81">
P^{-1} =
\left[\begin{array}{cc} I & -\tilde{K}^{-1}B^T \\ 0 & I \end{array} \right]
\left[\begin{array}{cc} \tilde{K}^{-1} & 0 \\ 0 & -\gamma \tilde{S}^{-1} \end{array} \right]
</script>
where <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-805"><span class="MJXp-msubsup" id="MJXp-Span-806"><span class="MJXp-mrow" id="MJXp-Span-807" style="margin-right: 0.05em;"><span class="MJXp-munderover" id="MJXp-Span-808"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.92em;"><span class="MJXp-mo" id="MJXp-Span-810" style="margin-left: 0px; margin-right: 0px;">˜</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-809">K</span></span></span></span></span></span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-811" style="vertical-align: 0.724em;"><span class="MJXp-mo" id="MJXp-Span-812">−</span><span class="MJXp-mn" id="MJXp-Span-813">1</span></span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-82-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-82">\tilde{K}^{-1}</script> is an approximation of the inverse of the stiffness matrix <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-814"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-815">K</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-83-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-83">K</script> and
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-816"><span class="MJXp-msubsup" id="MJXp-Span-817"><span class="MJXp-mrow" id="MJXp-Span-818" style="margin-right: 0.05em;"><span class="MJXp-munderover" id="MJXp-Span-819"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.92em;"><span class="MJXp-mo" id="MJXp-Span-821" style="margin-left: 0px; margin-right: 0px;">˜</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-820">S</span></span></span></span></span></span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-822" style="vertical-align: 0.724em;"><span class="MJXp-mo" id="MJXp-Span-823">−</span><span class="MJXp-mn" id="MJXp-Span-824">1</span></span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-84-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-84">\tilde{S}^{-1}</script> is an approximation of the inverse of the Schur complement <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-825"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-826">S</span><span class="MJXp-mo" id="MJXp-Span-827" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-828">B</span><span class="MJXp-msubsup" id="MJXp-Span-829"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-830" style="margin-right: 0.05em;">K</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-831" style="vertical-align: 0.5em;"><span class="MJXp-mo" id="MJXp-Span-832">−</span><span class="MJXp-mn" id="MJXp-Span-833">1</span></span></span><span class="MJXp-msubsup" id="MJXp-Span-834"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-835" style="margin-right: 0.05em;">B</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-836" style="vertical-align: 0.5em;">T</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-85-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-85">S = BK^{-1}B^T</script>.
To approximate the Schur complement, we use the mass matrix for the pressure variable <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-837"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-838">p</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-86-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-86">p</script>.</p>
<p>The example demonstrates how to solve nonlinear systems of equations that are defined with
block vectors as well as how to implement specialized block preconditioners for use in
iterative solvers.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex19.cpp">ex19.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex19p.cpp">ex19p.cpp</a>) version.
We recommend viewing examples 2, 5 and 10 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex20" style="display: block;">
<h2 id="example-20-symplectic-integration-of-hamiltonian-systems">Example 20: Symplectic Integration of Hamiltonian Systems</h2>
<p><img class="floatright" src="../doc/web/examples/ex20.png"></p>
<p>This example demonstrates the use of the variable order, symplectic time
integration algorithm. Symplectic integration algorithms are designed to
conserve energy when integrating systems of ODEs which are derived from
Hamiltonian systems.</p>
<p>Hamiltonian systems define the energy of a system as a function of
time (t), a set of generalized coordinates (q), and their corresponding
generalized momenta (p).
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-839"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-840">H</span><span class="MJXp-mo" id="MJXp-Span-841" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-842">q</span><span class="MJXp-mo" id="MJXp-Span-843" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-844">p</span><span class="MJXp-mo" id="MJXp-Span-845" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-846">t</span><span class="MJXp-mo" id="MJXp-Span-847" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-848" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-849">T</span><span class="MJXp-mo" id="MJXp-Span-850" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-851">p</span><span class="MJXp-mo" id="MJXp-Span-852" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-853" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-854">V</span><span class="MJXp-mo" id="MJXp-Span-855" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-856">q</span><span class="MJXp-mo" id="MJXp-Span-857" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-858">t</span><span class="MJXp-mo" id="MJXp-Span-859" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-87-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-87">
H(q,p,t) = T(p) + V(q,t)
</script>
Hamilton's equations then specify how q and p evolve in time:
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-860"><span class="MJXp-mfrac" id="MJXp-Span-861" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-862">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-863">q</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-864">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-865">t</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-866" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mfrac" id="MJXp-Span-867" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-868">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-869">H</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-870">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-871">p</span></span></span></span></span></span><span class="MJXp-mspace" id="MJXp-Span-872" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mo" id="MJXp-Span-873" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mspace" id="MJXp-Span-874" style="width: 2em; height: 0em;"></span><span class="MJXp-mfrac" id="MJXp-Span-875" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-876">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-877">p</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-878">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-879">t</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-880" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mo" id="MJXp-Span-881" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mfrac" id="MJXp-Span-882" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-883">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-884">H</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-885">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-886">q</span></span></span></span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-88-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-88">
\frac{dq}{dt} =  \frac{dH}{dp}\,,\qquad
\frac{dp}{dt} = -\frac{dH}{dq}
</script></p>
<p>To use the symplectic integration classes we need to define an <code>mfem::Operator</code>
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-887"><span class="MJXp-mrow" id="MJXp-Span-888"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-889">P</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-89-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-89">{\bf P}</script> which evaluates the action of dH/dp, and an
<code>mfem::TimeDependentOperator</code> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-890"><span class="MJXp-mrow" id="MJXp-Span-891"><span class="MJXp-mi MJXp-bold" id="MJXp-Span-892">F</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-90-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-90">{\bf F}</script> which computes -dH/dq.</p>
<p>This example visualizes its results as an evolution in phase space by defining
the axes to be <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-893"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-894">q</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-91-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-91">q</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-895"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-896">p</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-92-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-92">p</script>, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-897"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-898">t</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-93-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-93">t</script> rather than <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-899"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-900">x</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-94-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-94">x</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-901"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-902">y</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-95-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-95">y</script>, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-903"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-904">z</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-96-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-96">z</script>.  In this space
we build a ribbon-like mesh with nodes at <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-905"><span class="MJXp-mo" id="MJXp-Span-906" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mn" id="MJXp-Span-907">0</span><span class="MJXp-mo" id="MJXp-Span-908" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-909">0</span><span class="MJXp-mo" id="MJXp-Span-910" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-911">t</span><span class="MJXp-mo" id="MJXp-Span-912" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-97-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-97">(0,0,t)</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-913"><span class="MJXp-mo" id="MJXp-Span-914" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-915">q</span><span class="MJXp-mo" id="MJXp-Span-916" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-917">p</span><span class="MJXp-mo" id="MJXp-Span-918" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-919">t</span><span class="MJXp-mo" id="MJXp-Span-920" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-98-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-98">(q,p,t)</script>. Finally we
plot the energy as a function of time as a scalar field on this ribbon-like
mesh.  This scheme highlights any variations in the energy of the system.</p>
<p>This example offers five simple 1D Hamiltonians:</p>
<ul>
<li>Simple Harmonic Oscillator (mass on a spring)
  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-921"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-922">H</span><span class="MJXp-mo" id="MJXp-Span-923" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mfrac" id="MJXp-Span-924" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-925">1</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-926">2</span></span></span></span></span></span><span class="MJXp-mrow" id="MJXp-Span-927"><span class="MJXp-mo" id="MJXp-Span-928" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">(</span></span><span class="MJXp-mfrac" id="MJXp-Span-929" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-msubsup" id="MJXp-Span-930"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-931" style="margin-right: 0.05em;">p</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-932" style="vertical-align: 0.5em;">2</span></span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-933">m</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-934" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mfrac" id="MJXp-Span-935" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-msubsup" id="MJXp-Span-936"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-937" style="margin-right: 0.05em;">q</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-938" style="vertical-align: 0.5em;">2</span></span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-939">k</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-940" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">)</span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-99-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-99">H = \frac{1}{2}\left( \frac{p^2}{m} + \frac{q^2}{k} \right)</script></li>
<li>Pendulum
  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-941"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-942">H</span><span class="MJXp-mo" id="MJXp-Span-943" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mfrac" id="MJXp-Span-944" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-945">1</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-946">2</span></span></span></span></span></span><span class="MJXp-mrow" id="MJXp-Span-947"><span class="MJXp-mo" id="MJXp-Span-948" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">[</span></span><span class="MJXp-mfrac" id="MJXp-Span-949" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-msubsup" id="MJXp-Span-950"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-951" style="margin-right: 0.05em;">p</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-952" style="vertical-align: 0.5em;">2</span></span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-953">m</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-954" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-955">k</span><span class="MJXp-mrow" id="MJXp-Span-956"><span class="MJXp-mo" id="MJXp-Span-957" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mn" id="MJXp-Span-958">1</span><span class="MJXp-mo" id="MJXp-Span-959" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-960">c</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-961">o</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-962">s</span><span class="MJXp-mo" id="MJXp-Span-963" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-964">q</span><span class="MJXp-mo" id="MJXp-Span-965" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-966" style="margin-left: 0em; margin-right: 0em;">)</span></span><span class="MJXp-mo" id="MJXp-Span-967" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">]</span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-100-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-100">H = \frac{1}{2}\left[ \frac{p^2}{m} - k \left( 1 - cos(q) \right) \right]</script></li>
<li>Gaussian Potential Well
  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-968"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-969">H</span><span class="MJXp-mo" id="MJXp-Span-970" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mfrac" id="MJXp-Span-971" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-msubsup" id="MJXp-Span-972"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-973" style="margin-right: 0.05em;">p</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-974" style="vertical-align: 0.5em;">2</span></span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-975">2</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-976">m</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-977" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-978">k</span><span class="MJXp-msubsup" id="MJXp-Span-979"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-980" style="margin-right: 0.05em;">e</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-981" style="vertical-align: 0.5em;"><span class="MJXp-mo" id="MJXp-Span-982">−</span><span class="MJXp-msubsup" id="MJXp-Span-983"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-984" style="margin-right: 0.05em;">q</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-985" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mrow" id="MJXp-Span-986"><span class="MJXp-mo" id="MJXp-Span-987">/</span></span><span class="MJXp-mn" id="MJXp-Span-988">2</span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-101-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-101">H = \frac{p^2}{2m} - k e^{-q^2 / 2}</script></li>
<li>Quartic Potential
  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-989"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-990">H</span><span class="MJXp-mo" id="MJXp-Span-991" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mfrac" id="MJXp-Span-992" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-993">1</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-994">2</span></span></span></span></span></span><span class="MJXp-mrow" id="MJXp-Span-995"><span class="MJXp-mo" id="MJXp-Span-996" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">[</span></span><span class="MJXp-mfrac" id="MJXp-Span-997" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-msubsup" id="MJXp-Span-998"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-999" style="margin-right: 0.05em;">p</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1000" style="vertical-align: 0.5em;">2</span></span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1001">m</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1002" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1003">k</span><span class="MJXp-mrow" id="MJXp-Span-1004"><span class="MJXp-mo" id="MJXp-Span-1005" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.289em;"><span class="MJXp-right MJXp-scale6" style="font-size: 2.156em; margin-left: -0.09em;">(</span></span><span class="MJXp-mn" id="MJXp-Span-1006">1</span><span class="MJXp-mo" id="MJXp-Span-1007" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-msubsup" id="MJXp-Span-1008"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1009" style="margin-right: 0.05em;">q</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1010" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1011" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.289em;"><span class="MJXp-right MJXp-scale6" style="font-size: 2.156em; margin-left: -0.09em;">)</span></span></span><span class="MJXp-msubsup" id="MJXp-Span-1012"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1013" style="margin-right: 0.05em;">q</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1014" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1015" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">]</span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-102-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-102">H = \frac{1}{2}\left[ \frac{p^2}{m} + k \left( 1 + q^2 \right) q^2 \right]</script></li>
<li>Negative Quartic Potential
  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-1016"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1017">H</span><span class="MJXp-mo" id="MJXp-Span-1018" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mfrac" id="MJXp-Span-1019" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-1020">1</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-1021">2</span></span></span></span></span></span><span class="MJXp-mrow" id="MJXp-Span-1022"><span class="MJXp-mo" id="MJXp-Span-1023" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">[</span></span><span class="MJXp-mfrac" id="MJXp-Span-1024" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-msubsup" id="MJXp-Span-1025"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1026" style="margin-right: 0.05em;">p</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1027" style="vertical-align: 0.5em;">2</span></span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1028">m</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1029" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1030">k</span><span class="MJXp-mrow" id="MJXp-Span-1031"><span class="MJXp-mo" id="MJXp-Span-1032" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">(</span></span><span class="MJXp-mn" id="MJXp-Span-1033">1</span><span class="MJXp-mo" id="MJXp-Span-1034" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mfrac" id="MJXp-Span-1035" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-msubsup" id="MJXp-Span-1036"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1037" style="margin-right: 0.05em;">q</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1038" style="vertical-align: 0.5em;">2</span></span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mn" id="MJXp-Span-1039">8</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1040" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">)</span></span></span><span class="MJXp-msubsup" id="MJXp-Span-1041"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1042" style="margin-right: 0.05em;">q</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1043" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1044" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.678em;"><span class="MJXp-right MJXp-scale4" style="font-size: 3.711em; margin-left: -0.17em;">]</span></span></span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-103-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-103">H = \frac{1}{2}\left[ \frac{p^2}{m} + k \left( 1 - \frac{q^2}{8} \right) q^2 \right]</script></li>
</ul>
<p>In all cases these Hamiltonians are shifted by constant values so that the
energy will remain positive. The mean and standard deviation of the computed
energies at each time step are displayed upon completion.</p>
<p>When run in parallel, each processor integrates the same Hamiltonian
system but starting from different initial conditions.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex20.cpp">ex20.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex20p.cpp">ex20p.cpp</a>) version.
See the <a href="#maxwell-miniapp-transient-full-wave-electromagnetics">Maxwell</a> miniapp for another
application of symplectic integration.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex21" style="display: block;">
<h2 id="example-21-adaptive-mesh-refinement-for-linear-elasticity">Example 21: Adaptive mesh refinement for linear elasticity</h2>
<p><img class="floatright" src="../doc/web/examples/ex21.png"></p>
<p>This is a version of Example 2 with a simple adaptive mesh
refinement loop. The problem being solved is again linear
elasticity describing a multi-material cantilever beam.
The problem is solved on a sequence of meshes which
are locally refined in a conforming (triangles, tetrahedrons)
or non-conforming (quadrilaterals, hexahedra) manner according
to a simple ZZ error estimator.</p>
<p>The example demonstrates MFEM's capability to work with both
conforming and nonconforming refinements, in 2D and 3D, on
linear and curved meshes. Interpolation of functions from
coarse to fine meshes, as well as persistent GLVis
visualization are also illustrated.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex21.cpp">ex21.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex21p.cpp">ex21p.cpp</a>) version.
We recommend viewing Examples 2 and 6 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex22" style="display: block;">
<h2 id="example-22-complex-linear-systems">Example 22: Complex Linear Systems</h2>
<p><img class="floatright" src="../doc/web/examples/ex22.gif"></p>
<p>This example code demonstrates the use of MFEM to define and
solve a complex-valued linear system. It implements three variants
of a damped harmonic oscillator:</p>
<ul>
<li>
<p>A scalar <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1045"><span class="MJXp-msubsup" id="MJXp-Span-1046"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1047" style="margin-right: 0.05em;">H</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1048" style="vertical-align: 0.5em;">1</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-104-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-104">H^1</script> field:
  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-1049"><span class="MJXp-mo" id="MJXp-Span-1050" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi" id="MJXp-Span-1051">∇</span><span class="MJXp-mo" id="MJXp-Span-1052" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mrow" id="MJXp-Span-1053"><span class="MJXp-mo" id="MJXp-Span-1054" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1055">a</span><span class="MJXp-mi" id="MJXp-Span-1056">∇</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1057">u</span><span class="MJXp-mo" id="MJXp-Span-1058" style="margin-left: 0em; margin-right: 0em;">)</span></span><span class="MJXp-mo" id="MJXp-Span-1059" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-msubsup" id="MJXp-Span-1060"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1061" style="margin-right: 0.05em;">ω</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1062" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1063">b</span><span class="MJXp-mspace" id="MJXp-Span-1064" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1065">u</span><span class="MJXp-mo" id="MJXp-Span-1066" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1067">i</span><span class="MJXp-mspace" id="MJXp-Span-1068" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1069">ω</span><span class="MJXp-mspace" id="MJXp-Span-1070" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1071">c</span><span class="MJXp-mspace" id="MJXp-Span-1072" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1073">u</span><span class="MJXp-mo" id="MJXp-Span-1074" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-1075">0</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-105-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-105">-\nabla\cdot\left(a \nabla u\right) - \omega^2 b\,u + i\,\omega\,c\,u = 0</script></p>
</li>
<li>
<p>A vector <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1076"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1077">H</span><span class="MJXp-mo" id="MJXp-Span-1078" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1079">C</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1080">u</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1081">r</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1082">l</span><span class="MJXp-mo" id="MJXp-Span-1083" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-106-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-106">H(Curl)</script> field:
  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-1084"><span class="MJXp-mi" id="MJXp-Span-1085">∇</span><span class="MJXp-mo" id="MJXp-Span-1086" style="margin-left: 0.267em; margin-right: 0.267em;">×</span><span class="MJXp-mrow" id="MJXp-Span-1087"><span class="MJXp-mo" id="MJXp-Span-1088" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.129em;"><span class="MJXp-right MJXp-scale9" style="font-size: 1.516em; margin-left: 0.03em;">(</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1089">a</span><span class="MJXp-mi" id="MJXp-Span-1090">∇</span><span class="MJXp-mo" id="MJXp-Span-1091" style="margin-left: 0.267em; margin-right: 0.267em;">×</span><span class="MJXp-mrow" id="MJXp-Span-1092"><span class="MJXp-munderover" id="MJXp-Span-1093"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.68em;"><span class="MJXp-mo" id="MJXp-Span-1095" style="margin-left: 0px; margin-right: 0px; font-size: 60%;">→</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1094">u</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1096" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.129em;"><span class="MJXp-right MJXp-scale9" style="font-size: 1.516em; margin-left: 0.03em;">)</span></span></span><span class="MJXp-mo" id="MJXp-Span-1097" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-msubsup" id="MJXp-Span-1098"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1099" style="margin-right: 0.05em;">ω</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1100" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1101">b</span><span class="MJXp-mspace" id="MJXp-Span-1102" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-1103"><span class="MJXp-munderover" id="MJXp-Span-1104"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.68em;"><span class="MJXp-mo" id="MJXp-Span-1106" style="margin-left: 0px; margin-right: 0px; font-size: 60%;">→</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1105">u</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1107" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1108">i</span><span class="MJXp-mspace" id="MJXp-Span-1109" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1110">ω</span><span class="MJXp-mspace" id="MJXp-Span-1111" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1112">c</span><span class="MJXp-mspace" id="MJXp-Span-1113" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-1114"><span class="MJXp-munderover" id="MJXp-Span-1115"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.68em;"><span class="MJXp-mo" id="MJXp-Span-1117" style="margin-left: 0px; margin-right: 0px; font-size: 60%;">→</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1116">u</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1118" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-1119">0</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-107-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-107">\nabla\times\left(a\nabla\times\vec{u}\right) - \omega^2 b\,\vec{u} + i\,\omega\,c\,\vec{u} = 0</script></p>
</li>
<li>
<p>A vector <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1120"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1121">H</span><span class="MJXp-mo" id="MJXp-Span-1122" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1123">D</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1124">i</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1125">v</span><span class="MJXp-mo" id="MJXp-Span-1126" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-108-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-108">H(Div)</script> field:
  <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-1127"><span class="MJXp-mo" id="MJXp-Span-1128" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mi" id="MJXp-Span-1129">∇</span><span class="MJXp-mrow" id="MJXp-Span-1130"><span class="MJXp-mo" id="MJXp-Span-1131" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.129em;"><span class="MJXp-right MJXp-scale9" style="font-size: 1.516em; margin-left: 0.03em;">(</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1132">a</span><span class="MJXp-mi" id="MJXp-Span-1133">∇</span><span class="MJXp-mo" id="MJXp-Span-1134" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mrow" id="MJXp-Span-1135"><span class="MJXp-munderover" id="MJXp-Span-1136"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.68em;"><span class="MJXp-mo" id="MJXp-Span-1138" style="margin-left: 0px; margin-right: 0px; font-size: 60%;">→</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1137">u</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1139" style="margin-left: 0em; margin-right: 0em; vertical-align: -0.129em;"><span class="MJXp-right MJXp-scale9" style="font-size: 1.516em; margin-left: 0.03em;">)</span></span></span><span class="MJXp-mo" id="MJXp-Span-1140" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-msubsup" id="MJXp-Span-1141"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1142" style="margin-right: 0.05em;">ω</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1143" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1144">b</span><span class="MJXp-mspace" id="MJXp-Span-1145" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-1146"><span class="MJXp-munderover" id="MJXp-Span-1147"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.68em;"><span class="MJXp-mo" id="MJXp-Span-1149" style="margin-left: 0px; margin-right: 0px; font-size: 60%;">→</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1148">u</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1150" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1151">i</span><span class="MJXp-mspace" id="MJXp-Span-1152" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1153">ω</span><span class="MJXp-mspace" id="MJXp-Span-1154" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1155">c</span><span class="MJXp-mspace" id="MJXp-Span-1156" style="width: 0.167em; height: 0em;"></span><span class="MJXp-mrow" id="MJXp-Span-1157"><span class="MJXp-munderover" id="MJXp-Span-1158"><span><span class="MJXp-over"><span class="" style="margin-bottom: -0.68em;"><span class="MJXp-mo" id="MJXp-Span-1160" style="margin-left: 0px; margin-right: 0px; font-size: 60%;">→</span></span><span class=""><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1159">u</span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1161" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-1162">0</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-109-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-109">-\nabla\left(a \nabla\cdot\vec{u}\right) - \omega^2 b\,\vec{u} + i\,\omega\,c\,\vec{u} = 0</script></p>
</li>
</ul>
<p>In each case the field is driven by a forced oscillation, with
angular frequency <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1163"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1164">ω</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-110-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-110">\omega</script>, imposed at the boundary or a portion
of the boundary.</p>
<p>The example also demonstrates how to display a time-varying solution as
a sequence of fields sent to a single GLVis socket.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex22.cpp">ex22.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex22p.cpp">ex22p.cpp</a>) version.
We recommend viewing examples 1, 3, and 4 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex23" style="display: block;">
<h2 id="example-23-wave-problem">Example 23: Wave Problem</h2>
<p><img class="floatright" src="../doc/web/examples/ex23.png"></p>
<p>This example code solves a simple 2D/3D  wave
equation with a second order time derivative:
<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-1165"><span class="MJXp-mfrac" id="MJXp-Span-1166" style="vertical-align: 0.25em;"><span class="MJXp-box"><span class="MJXp-msubsup" id="MJXp-Span-1167"><span class="MJXp-mi" id="MJXp-Span-1168" style="margin-right: 0.05em;">∂</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1169" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1170">u</span></span><span class="MJXp-box" style="margin-top: -0.9em;"><span class="MJXp-denom"><span><span class="MJXp-rule" style="height: 1em; border-top-style: none; border-bottom-width: 1px; border-bottom-style: solid; margin: 0.1em 0px;"></span></span><span><span class="MJXp-box"><span class="MJXp-mi" id="MJXp-Span-1171">∂</span><span class="MJXp-msubsup" id="MJXp-Span-1172"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1173" style="margin-right: 0.05em;">t</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1174" style="vertical-align: 0.5em;">2</span></span></span></span></span></span></span><span class="MJXp-mo" id="MJXp-Span-1175" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-msubsup" id="MJXp-Span-1176"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1177" style="margin-right: 0.05em;">c</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1178" style="vertical-align: 0.5em;">2</span></span><span class="MJXp-mi" id="MJXp-Span-1179">Δ</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1180">u</span><span class="MJXp-mo" id="MJXp-Span-1181" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-1182">0</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-111-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-111">\frac{\partial^2 u}{\partial t^2} - c^2\Delta u = 0</script>
The boundary conditions are either Dirichlet or Neumann.</p>
<p>The example demonstrates the use of time dependent operators,
implicit solvers and second order time integration.</p>
<p><em>The example has only a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex23.cpp">ex23.cpp</a>) version.
We recommend viewing examples 9 and 10 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="ex24" style="display: block;">
<h2 id="example-24-mixed-finite-element-spaces">Example 24: Mixed finite element spaces</h2>
<p><img class="floatright" src="../doc/web/examples/ex24.png"></p>
<p>This example code illustrates usage of mixed finite element
spaces. Using two different approaches, we project a gradient
of a function in <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1183"><span class="MJXp-msubsup" id="MJXp-Span-1184"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1185" style="margin-right: 0.05em;">H</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1186" style="vertical-align: 0.5em;">1</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-112-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-112">H^1</script> to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1187"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1188">H</span><span class="MJXp-mo" id="MJXp-Span-1189" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1190">c</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1191">u</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1192">r</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1193">l</span><span class="MJXp-mo" id="MJXp-Span-1194" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-113-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-113">H(curl)</script>. Other spaces and example
computations are to be added in the future.</p>
<p>We also illustrate usage of a DiscreteLinearOperator and a
DiscreteInterpolator to interpolate a gradient in an <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1195"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1196">H</span><span class="MJXp-mo" id="MJXp-Span-1197" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1198">c</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1199">u</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1200">r</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1201">l</span><span class="MJXp-mo" id="MJXp-Span-1202" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-114-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-114">H(curl)</script>
finite element space.</p>
<p><em>The example has a serial (<a href="https://github.com/mfem/mfem/blob/master/examples/ex24.cpp">ex24.cpp</a>)
and a parallel (<a href="https://github.com/mfem/mfem/blob/master/examples/ex24p.cpp">ex24p.cpp</a>) version.
Partial assembly and GPU devices are supported.
We recommend viewing examples 1 and 3 before viewing this example.</em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="volta" style="display: block;">
<h2 id="volta-miniapp-electrostatics">Volta Miniapp: Electrostatics</h2>
<p><img class="floatright" src="../doc/web/examples/volta.png"></p>
<p>This miniapp demonstrates the use of MFEM to solve realistic problems
in the field of linear electrostatics.  Its features include:</p>
<ul>
<li>dielectric materials</li>
<li>charge densities</li>
<li>surface charge densities</li>
<li>prescribed voltages</li>
<li>applied polarizations</li>
<li>high order meshes</li>
<li>high order basis functions</li>
<li>adaptive mesh refinement</li>
<li>advanced visualization</li>
</ul>
<p>For more details, please see the <a href="http://mfem.org/electromagnetics/">documentation</a> in the
<code>miniapps/electromagnetics</code> directory.</p>
<p><em>The miniapp has only a parallel
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/electromagnetics/volta.cpp">volta.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="tesla" style="display: block;">
<h2 id="tesla-miniapp-magnetostatics">Tesla Miniapp: Magnetostatics</h2>
<p><img class="floatright" src="../doc/web/examples/tesla.png"></p>
<p>This miniapp showcases many of MFEM's features while solving a variety
of realistic magnetostatics problems.  Its features include:</p>
<ul>
<li>diamagnetic and/or paramagnetic materials</li>
<li>ferromagnetic materials</li>
<li>volumetric current densities</li>
<li>surface current densities</li>
<li>external fields</li>
<li>high order meshes</li>
<li>high order basis functions</li>
<li>adaptive mesh refinement</li>
<li>advanced visualization</li>
</ul>
<p>For more details, please see the <a href="http://mfem.org/electromagnetics/">documentation</a> in the
<code>miniapps/electromagnetics</code> directory.</p>
<p><em>The miniapp has only a parallel
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/electromagnetics/tesla.cpp">tesla.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="maxwell" style="display: block;">
<h2 id="maxwell-miniapp-transient-full-wave-electromagnetics">Maxwell Miniapp: Transient Full-Wave Electromagnetics</h2>
<p><img class="floatright" src="../doc/web/examples/maxwell.png"></p>
<p>This miniapp solves the equations of transient full-wave electromagnetics.</p>
<p>Its features include:</p>
<ul>
<li>mixed formulation of the coupled first-order Maxwell equations</li>
<li><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1203"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1204">H</span><span class="MJXp-mo" id="MJXp-Span-1205" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-1206"><span class="MJXp-mi" id="MJXp-Span-1207">c</span><span class="MJXp-mi" id="MJXp-Span-1208">u</span><span class="MJXp-mi" id="MJXp-Span-1209">r</span><span class="MJXp-mi" id="MJXp-Span-1210">l</span></span><span class="MJXp-mo" id="MJXp-Span-1211" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-115-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-115">H(\mathrm{curl})</script> discretization of the electric field</li>
<li><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1212"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1213">H</span><span class="MJXp-mo" id="MJXp-Span-1214" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-1215"><span class="MJXp-mi" id="MJXp-Span-1216">d</span><span class="MJXp-mi" id="MJXp-Span-1217">i</span><span class="MJXp-mi" id="MJXp-Span-1218">v</span></span><span class="MJXp-mo" id="MJXp-Span-1219" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-116-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-116">H(\mathrm{div})</script> discretization of the magnetic flux</li>
<li>energy conserving, variable order, implicit time integration</li>
<li>dielectric materials</li>
<li>diamagnetic and/or paramagnetic materials</li>
<li>conductive materials</li>
<li>volumetric current densities</li>
<li>Sommerfeld absorbing boundary conditions</li>
<li>high order meshes</li>
<li>high order basis functions</li>
<li>advanced visualization</li>
</ul>
<p>For more details, please see the <a href="http://mfem.org/electromagnetics/">documentation</a> in the
<code>miniapps/electromagnetics</code> directory.</p>
<p><em>The miniapp has only a parallel
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/electromagnetics/maxwell.cpp">maxwell.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="joule" style="display: block;">
<h2 id="joule-miniapp-transient-magnetics-and-joule-heating">Joule Miniapp: Transient Magnetics and Joule Heating</h2>
<p><img class="floatright" src="../doc/web/examples/joule.png"></p>
<p>This miniapp solves the equations of transient low-frequency (a.k.a. eddy current)
electromagnetics, and simultaneously computes transient heat transfer with the heat source given
by the electromagnetic Joule heating.</p>
<p>Its features include:</p>
<ul>
<li><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1220"><span class="MJXp-msubsup" id="MJXp-Span-1221"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1222" style="margin-right: 0.05em;">H</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1223" style="vertical-align: 0.5em;">1</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-117-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-117">H^1</script> discretization of the electrostatic potential</li>
<li><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1224"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1225">H</span><span class="MJXp-mo" id="MJXp-Span-1226" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-1227"><span class="MJXp-mi" id="MJXp-Span-1228">c</span><span class="MJXp-mi" id="MJXp-Span-1229">u</span><span class="MJXp-mi" id="MJXp-Span-1230">r</span><span class="MJXp-mi" id="MJXp-Span-1231">l</span></span><span class="MJXp-mo" id="MJXp-Span-1232" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-118-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-118">H(\mathrm{curl})</script> discretization of the electric field</li>
<li><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1233"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1234">H</span><span class="MJXp-mo" id="MJXp-Span-1235" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-1236"><span class="MJXp-mi" id="MJXp-Span-1237">d</span><span class="MJXp-mi" id="MJXp-Span-1238">i</span><span class="MJXp-mi" id="MJXp-Span-1239">v</span></span><span class="MJXp-mo" id="MJXp-Span-1240" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-119-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-119">H(\mathrm{div})</script> discretization of the magnetic field</li>
<li><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1241"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1242">H</span><span class="MJXp-mo" id="MJXp-Span-1243" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mrow" id="MJXp-Span-1244"><span class="MJXp-mi" id="MJXp-Span-1245">d</span><span class="MJXp-mi" id="MJXp-Span-1246">i</span><span class="MJXp-mi" id="MJXp-Span-1247">v</span></span><span class="MJXp-mo" id="MJXp-Span-1248" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-120-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-120">H(\mathrm{div})</script> discretization of the heat flux</li>
<li><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1249"><span class="MJXp-msubsup" id="MJXp-Span-1250"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1251" style="margin-right: 0.05em;">L</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1252" style="vertical-align: 0.5em;">2</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-121-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-121">L^2</script> discretization of the temperature</li>
<li>implicit transient time integration</li>
<li>high order meshes</li>
<li>high order basis functions</li>
<li>adaptive mesh refinement</li>
<li>advanced visualization</li>
</ul>
<p>For more details, please see the <a href="http://mfem.org/electromagnetics/">documentation</a> in the
<code>miniapps/electromagnetics</code> directory.</p>
<p><em>The miniapp has only a parallel
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/electromagnetics/joule.cpp">joule.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="mobius-strip" style="display: block;">
<h2 id="mobius-strip-miniapp">Mobius Strip Miniapp</h2>
<p><img class="floatright" src="../doc/web/examples/mobius-strip.png"></p>
<p>This miniapp generates various Mobius strip-like surface meshes. It is a good
way to generate complex surface meshes.</p>
<p>Manipulating the mesh topology and performing mesh transformation are demonstrated.</p>
<p>The <code>mobius-strip</code> mesh in the <code>data</code> directory was generated with this miniapp.</p>
<p>For more details, please see the <a href="http://mfem.org/meshing/">documentation</a> in the
<code>miniapps/meshing</code> directory.</p>
<p><em>The miniapp has only a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/meshing/mobius-strip.cpp">mobius-strip.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="klein-bottle" style="display: block;">
<h2 id="klein-bottle-miniapp">Klein Bottle Miniapp</h2>
<p><img class="floatright" src="../doc/web/examples/klein-bottle.png"></p>
<p>This miniapp generates three types of Klein bottle surfaces. It is similar to
the mobius-strip miniapp.</p>
<p>Manipulating the mesh topology and performing mesh transformation are demonstrated.</p>
<p>The <code>klein-bottle</code> and <code>klein-donut</code> meshes in the <code>data</code> directory were generated with this miniapp.</p>
<p>For more details, please see the <a href="http://mfem.org/meshing/">documentation</a> in the
<code>miniapps/meshing</code> directory.</p>
<p><em>The miniapp has only a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/meshing/klein-bottle.cpp">klein-bottle.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="toroid" style="display: block;">
<h2 id="toroid-miniapp">Toroid Miniapp</h2>
<p><img class="floatright" src="../doc/web/examples/toroid-wedge.png"></p>
<p>This miniapp generates two types of toroidal volume meshes; one with
triangular cross sections and one with square cross sections.  It
works by defining a stack of individual elements and bending them so
that the bottom and top of the stack can be joined to form a torus. It
supports various options including:</p>
<ul>
<li>The element type: 0 - Wedge, 1 - Hexahedron</li>
<li>The geometric order of the elements</li>
<li>The major and minor radii</li>
<li>The number of elements in the azimuthal direction</li>
<li>The number of nodes to offset by before rejoining the stack</li>
<li>The initial angle of the cross sectional shape</li>
<li>The number of uniform refinement steps to apply</li>
</ul>
<p>Along with producing some visually interesting meshes, this miniapp
demonstrates how simple 3D meshes can be constructed and transformed
in MFEM.  It also produces a family of meshes with simple but
non-trivial topology for testing various features in MFEM.</p>
<p><em>This miniapp has only a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/meshing/toroid.cpp">toroid.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="extruder" style="display: block;">
<h2 id="extruder-miniapp">Extruder Miniapp</h2>
<p><img class="floatright" src="../doc/web/examples/extruded-star.png"></p>
<p>This miniapp creates higher dimensional meshes from lower dimensional meshes
by extrusion.  Simple coordinate transformations can also be applied if desired.</p>
<ul>
<li>The initial mesh can be 1D or 2D</li>
<li>1D meshes can be extruded in both the y and z directions</li>
<li>2D meshes can be triangular, quadrilateral, or contain both element types</li>
<li>Meshes with high order geometry are supported</li>
<li>User can specify the number of elements and the distance to extrude</li>
<li>Geometric order of the transformed mesh can be user selected or automatic</li>
</ul>
<p>This miniapp provides another demonstration of how simple meshes can be
constructed and transformed in MFEM.</p>
<p><em>This miniapp has only a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/meshing/extruder.cpp">extruder.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="shaper" style="display: block;">
<h2 id="shaper-miniapp">Shaper Miniapp</h2>
<p><img class="floatright" src="../doc/web/examples/shaper.png"></p>
<p>This miniapp performs multiple levels of adaptive mesh refinement to resolve the
interfaces between different "materials" in the mesh, as specified by a given
material function.</p>
<p>It can be used as a simple initial mesh generator, for example in the case when
the interface is too complex to describe without local refinement. Both
conforming and non-conforming refinements are supported.</p>
<p>For more details, please see the <a href="http://mfem.org/meshing/">documentation</a> in the
<code>miniapps/meshing</code> directory.</p>
<p><em>The miniapp has only a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/meshing/shaper.cpp">shaper.cpp</a>) version.
<strong>We recommend that new users start with the example codes before
moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="mesh-explorer" style="display: block;">
<h2 id="mesh-explorer-miniapp">Mesh Explorer Miniapp</h2>
<p><img class="floatright" src="../doc/web/examples/mesh-explorer.png"></p>
<p>This miniapp is a handy tool to examine, visualize and manipulate a given
mesh. Some of its features are:</p>
<ul>
<li>visualizing of mesh materials and individual mesh elements</li>
<li>mesh scaling, randomization, and general transformation</li>
<li>manipulation of the mesh curvature</li>
<li>the ability to simulate parallel partitioning</li>
<li>quantitative and visual reports of mesh quality</li>
</ul>
<p>For more details, please see the <a href="http://mfem.org/meshing/">documentation</a> in the
<code>miniapps/meshing</code> directory.</p>
<p><em>The miniapp has only a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/meshing/mesh-explorer.cpp">mesh-explorer.cpp</a>) version.
<strong>We recommend that new users start with the example codes before moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="mesh-optimizer" style="display: block;">
<h2 id="mesh-optimizer-miniapp">Mesh Optimizer Miniapp</h2>
<p><img class="floatright" src="../doc/web/examples/mesh-optimizer.png"></p>
<p>This miniapp performs mesh optimization using the Target-Matrix Optimization
Paradigm (TMOP) by P.Knupp et al., and a global variational minimization
approach. It minimizes the quantity</p>
<p><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math MJXp-display" id="MJXp-Span-1253"><span class="MJXp-munderover" id="MJXp-Span-1254"><span class=""><span class="MJXp-mo" id="MJXp-Span-1255" style="margin-left: 0.111em; margin-right: 0.167em;"><span class="MJXp-largeop">∑</span></span></span><span class=" MJXp-script"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1256" style="margin-left: 0px;">T</span></span></span><span class="MJXp-msubsup" id="MJXp-Span-1257"><span class="MJXp-mo" id="MJXp-Span-1258" style="margin-left: 0em; margin-right: 0.05em;"><span class="MJXp-largeop MJXp-int">∫</span></span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1259" style="vertical-align: -0.46em;">T</span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1260">μ</span><span class="MJXp-mo" id="MJXp-Span-1261" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1262">J</span><span class="MJXp-mo" id="MJXp-Span-1263" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1264">x</span><span class="MJXp-mo" id="MJXp-Span-1265" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1266" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1267" style="margin-left: 0em; margin-right: 0.222em;">,</span></span></span><div class="MathJax_SVG_Display MathJax_SVG_Processing"><span class="MathJax_SVG" id="MathJax-Element-122-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span></div><script type="math/tex; mode=display" id="MathJax-Element-122">\sum_T \int_T \mu(J(x)),</script></p>
<p>where <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1268"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1269">T</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-123-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-123">T</script> are the target (ideal) elements, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1270"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1271">J</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-124-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-124">J</script> is the Jacobian of the
transformation from the target to the physical element, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1272"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1273">μ</span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-125-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-125">\mu</script> is the mesh
quality metric.</p>
<p>This metric can measure shape, size or alignment of the region around each
quadrature point. The combination of targets and quality metrics is used to
optimize the physical node positions, i.e., they must be as close as possible to
the shape / size / alignment of their targets.</p>
<p>This code also demonstrates a possible use of nonlinear operators, as well as
their coupling to Newton methods for solving minimization problems. Note that
the utilized Newton methods are oriented towards avoiding invalid meshes with
negative Jacobian determinants. Each Newton step requires the inversion of a
Jacobian matrix, which is done through an inner linear solver.</p>
<p>For more details, please see the <a href="http://mfem.org/meshing/">documentation</a> in the
<code>miniapps/meshing</code> directory.</p>
<p><em>The miniapp has a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/meshing/mesh-optimizer.cpp">mesh-optimizer.cpp</a>) and a
parallel (<a href="https://github.com/mfem/mfem/blob/master/miniapps/meshing/pmesh-optimizer.cpp">pmesh-optimizer.cpp</a>)
version.
<strong>We recommend that new users start with the example codes before moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="lor-transfer" style="display: block;">
<h2 id="low-order-refined-transfer-miniapp">Low-Order Refined Transfer Miniapp</h2>
<p><img class="floatright" width="450" src="../doc/web/examples/lor-transfer.png"></p>
<p>The <code>lor-transfer</code> miniapp, found under <code>miniapps/tools</code> demonstrates the
capability to generate a <em>low-order refined</em> mesh from a high-order mesh, and to
transfer solutions between these meshes.</p>
<p>Grid functions can be transferred between the coarse, high-order mesh and the
low-order refined mesh using either <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1274"><span class="MJXp-msubsup" id="MJXp-Span-1275"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1276" style="margin-right: 0.05em;">L</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1277" style="vertical-align: 0.5em;">2</span></span></span></span><span class="MathJax_SVG MathJax_SVG_Processing" id="MathJax-Element-126-Frame" tabindex="0" style="font-size: 100%; display: inline-block;"></span><script type="math/tex" id="MathJax-Element-126">L^2</script> projection or pointwise evaluation.
These transfer operators can be designed to discretely conserve mass and to
recover the original high-order solution when transferring a low-order grid
function that was obtained by restricting a high-order grid function to the
low-order refined space.</p>
<p><em>The miniapp has only a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/tools/lor-transfer.cpp">lor-transfer.cpp</a>) version.
<strong>We recommend that new users start with the example codes before moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="gslib-interpolation" style="display: block;">
<h2 id="interpolation-miniapps">Interpolation Miniapps</h2>
<p><img class="floatright" width="450" src="../doc/web/examples/gslib.png"></p>
<p>The interpolation miniapp, found under <code>miniapps/gslib</code>, demonstrate the
capability to interpolate high-order finite element functions at given set of
points in physical space.</p>
<p>These miniapps utilize the <a href="https://github.com/gslib/gslib"><code>gslib</code></a> library's
high-order  interpolation utility for quad and hex meshes.
The <em>Find Points</em> miniapp has a serial
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/gslib/findpts.cpp">findpts.cpp</a>)
and a parallel
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/gslib/pfindpts.cpp">pfindpts.cpp</a>)
version that demonstrate the basic procedures for point search and evaluation
of grid functions.
The <em>Field Diff</em> miniapp
(<a href="https://github.com/mfem/mfem/blob/master/miniapps/gslib/field-diff.cpp">field-diff.cpp</a>)
demonstrates how grid functions on two different meshes can be compared with
each other.</p>
<p><em><strong>These miniapps require installation of the <a href="https://github.com/gslib/gslib"><code>gslib</code></a> library. We recommend that new users start with the example codes before moving to the miniapps.</strong></em>
</p><div style="clear:both;"></div>
<br><p></p>
</div>
<div id="laghos" style="display: block;">
<h2 id="laghos-miniapp">Laghos Miniapp</h2>
<p><img class="floatright" src="../doc/web/examples/laghos.png"></p>
<p><strong>Laghos</strong> (LAGrangian High-Order Solver) is a miniapp that solves the
time-dependent Euler equations of compressible gas dynamics in a moving
Lagrangian frame using unstructured high-order finite element spatial
discretization and explicit high-order time-stepping.</p>
<p>The computational motives captured in Laghos include:</p>
<ul>
<li>Support for unstructured meshes, in 2D and 3D, with quadrilateral and
  hexahedral elements (triangular and tetrahedral elements can also be used, but
  with the less efficient full assembly option). Serial and parallel mesh
  refinement options can be set via a command-line flag.</li>
<li>Explicit time-stepping loop with a variety of time integrator options. Laghos
  supports Runge-Kutta ODE solvers of orders 1, 2, 3, 4 and 6.</li>
<li>Continuous and discontinuous high-order finite element discretization spaces
  of runtime-specified order.</li>
<li>Moving (high-order) meshes.</li>
<li>Separation between the assembly and the quadrature point-based computations.</li>
<li>Point-wise definition of mesh size, time-step estimate and artificial
  viscosity coefficient.</li>
<li>Constant-in-time velocity mass operator that is inverted iteratively on
  each time step. This is an example of an operator that is prepared once (fully
  or partially assembled), but is applied many times. The application cost is
  dominant for this operator.</li>
<li>Time-dependent force matrix that is prepared every time step (fully or
  partially assembled) and is applied just twice per "assembly". Both the
  preparation and the application costs are important for this operator.</li>
<li>Domain-decomposed MPI parallelism.</li>
<li>Optional in-situ visualization with <a href="http:/glvis.org">GLVis</a> and data output
  for visualization / data analysis with <a href="http://visit.llnl.gov">VisIt</a>.</li>
</ul>
<p>The Laghos miniapp is part of the <a href="http://ceed.exascaleproject.org/software">CEED software suite</a>,
a collection of software benchmarks, miniapps, libraries and APIs for
efficient exascale discretizations based on high-order finite element
and spectral element methods. See http://github.com/ceed for more
information and source code availability.</p>
<p><em>This is an external miniapp, available at <a href="https://github.com/CEED/Laghos">https://github.com/CEED/Laghos</a>.</em>
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