From 598305bc72839433d7724094e4272630666938d0 Mon Sep 17 00:00:00 2001
From: Joachim Schoeberl <joachim.schoeberl@tuwien.ac.at>
Date: Sat, 2 Mar 2024 10:42:32 +0100
Subject: [PATCH] Update documentation

---
 _sources/intro.md | 13 ++++++++++++-
 intro.html        | 16 ++++++++++++++--
 searchindex.js    |  2 +-
 3 files changed, 27 insertions(+), 4 deletions(-)

diff --git a/_sources/intro.md b/_sources/intro.md
index 6fb2e798..1ceda499 100644
--- a/_sources/intro.md
+++ b/_sources/intro.md
@@ -32,7 +32,7 @@ If you like the material, show it by giving a star on github.
 
 ## Literature
 
-* Lecture notes Schöberl and Faustmann+Schöberl 
+* Lecture notes Schöberl and Faustmann+Schöberl (available in TU-Wien TUWEL)
 
 Books:
 
@@ -68,6 +68,17 @@ Known issues are
 - If you get an error like `externally-managed-environment`, then either use
 virtual environments, or add the flag `--break-system-packages` to the pip command, see [explanation](https://veronneau.org/python-311-pip-and-breaking-system-packages.html)
 
+- If you have conflicts with other packages, you may install NGSolve in a [virtual environment](https://docs.python.org/3/library/venv.html#creating-virtual-environments). For example I did
+
+      python3 -m venv /Users/joachim/numpde
+      source /Users/joachim/numpde/bin/activate
+
+- If NGSolve compuatations are working, but you don't get the rendering: For jupyter notebook version < 7.0.0 you have to run additionally
+
+      jupyter nbextension install --user --py webgui_jupyter_widgets
+      jupyter nbextension enable --user --py webgui_jupyter_widgets
+  
+
 
 If local installation does not work, there are alternatives:
 
diff --git a/intro.html b/intro.html
index 918dc762..7e84576a 100644
--- a/intro.html
+++ b/intro.html
@@ -487,7 +487,7 @@ <h1>An Interactive Introduction to the Finite Element Method<a class="headerlink
 <section id="literature">
 <h2>Literature<a class="headerlink" href="#literature" title="Link to this heading">#</a></h2>
 <ul class="simple">
-<li><p>Lecture notes Schöberl and Faustmann+Schöberl</p></li>
+<li><p>Lecture notes Schöberl and Faustmann+Schöberl (available in TU-Wien TUWEL)</p></li>
 </ul>
 <p>Books:</p>
 <ul class="simple">
@@ -517,10 +517,22 @@ <h2>Installing NGSolve<a class="headerlink" href="#installing-ngsolve" title="Li
 </pre></div>
 </div>
 <p>Known issues are</p>
-<ul class="simple">
+<ul>
 <li><p>Use pip3 instead of pip if there is no pip</p></li>
 <li><p>If you get an error like <code class="docutils literal notranslate"><span class="pre">externally-managed-environment</span></code>, then either use
 virtual environments, or add the flag <code class="docutils literal notranslate"><span class="pre">--break-system-packages</span></code> to the pip command, see <a class="reference external" href="https://veronneau.org/python-311-pip-and-breaking-system-packages.html">explanation</a></p></li>
+<li><p>If you have conflicts with other packages, you may install NGSolve in a <a class="reference external" href="https://docs.python.org/3/library/venv.html#creating-virtual-environments">virtual environment</a>. For example I did</p>
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>python3 -m venv /Users/joachim/numpde
+source /Users/joachim/numpde/bin/activate
+</pre></div>
+</div>
+</li>
+<li><p>If NGSolve compuatations are working, but you don’t get the rendering: For jupyter notebook version &lt; 7.0.0 you have to run additionally</p>
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>jupyter nbextension install --user --py webgui_jupyter_widgets
+jupyter nbextension enable --user --py webgui_jupyter_widgets
+</pre></div>
+</div>
+</li>
 </ul>
 <p>If local installation does not work, there are alternatives:</p>
 <ul>
diff --git a/searchindex.js b/searchindex.js
index 3b7cc926..a7aba446 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"docnames": ["DG/Nitsche", "DG/elliptic", "DG/elliptic_stdDG", "DG/fourthorder", "DG/instationary", "DG/splitting", "DG/stationary", "DG/stokes", "MPIparallel/PETSc_interface", "MPIparallel/distmesh", "MPIparallel/hdivnstsimple/NavierStokesMT", "MPIparallel/hdivnstsimple/TestConvert", "MPIparallel/hdivnstsimple/veclaplace", "MPIparallel/hdivnstsimple/veclaplace-auxiliary", "MPIparallel/intro", "MPIparallel/paralleliteration", "MPIparallel/parallelstokes", "MPIparallel/petsc", "MPIparallel/vectypes", "README", "abstracttheory/BasicProperties", "abstracttheory/Coercive", "abstracttheory/RieszRepresentation", "abstracttheory/infsup", "abstracttheory/subspaceprojection", "domaindecomposition/BDDC", "domaindecomposition/FETI", "domaindecomposition/FETI-DP", "domaindecomposition/introduction", "domaindecomposition/traces", "ex20210609", "helmholtz/absorbing", "helmholtz/grating", "iFEM", "intro", "iterative/Chebyshev", "iterative/Richardson", "iterative/conjugategradients", "iterative/gradientmethod", "iterative/preconditioning", "iterative/simple", "mixed/abstract", "mixed/abstractfem", "mixed/boundary", "mixed/pardep", "mixed/secondorder", "mixed/stokes", "mixedelasticity/dynamics", "mixedelasticity/hellingerreissner", "mixedelasticity/reducedsymmetry", "mixedelasticity/tdnns", "multigrid/algorithms", "multigrid/analysisMG", "multigrid/analysisML", "multigrid/multilevel-extension", "plates/hhj", "plates/hhj-tdnns", "plates/reissnermindlin", "plates/shells", "plates/tdnnsplate", "preamble", "primal/boundary_conditions", "primal/elasticity3D", "primal/exercises", "primal/first_example", "primal/solvers", "primal/subdomains", "saddlepoint/bramblepasciak", "saddlepoint/fewconstraints", "saddlepoint/parameterdependent", "saddlepoint/structure", "saddlepoint/useBP", "secondorder/erroranalysis", "secondorder/erroranalysisl2h1", "secondorder/finiteelements", "secondorder/framework", "secondorder/hdiv", "secondorder/hybridization", "sobolev/Friedrichs", "sobolev/Untitled", "sobolev/Untitled1", "sobolev/Untitled2", "sobolev/poincare", "sobolev/trace", "sobolevspaces/GeneralizedDerivatives", "sobolevspaces/SobolevSpaces", "sobolevspaces/Traces", "sobolevspaces/equivalentnorms", "subspacecorrection/asm", "subspacecorrection/examples", "subspacecorrection/exercise", "subspacecorrection/minimaldd", "subspacecorrection/overlapping", "timedependent/ODEs/runge_kutta", "timedependent/ODEs/singlestep", "timedependent/intro/Untitled", "timedependent/intro/heatequation", "timedependent/intro/waveequation", "timedependent/parabolic/exponential", "timedependent/waves/Maxwell-DG", "timedependent/waves/Untitled", "timedependent/waves/lts", "timedependent/waves/ringresonator", "timedependent/waves/testpml/ringresonator", "timedependent/waves/wave-leapfrogDG"], "filenames": ["DG/Nitsche.ipynb", "DG/elliptic.ipynb", "DG/elliptic_stdDG.ipynb", "DG/fourthorder.ipynb", "DG/instationary.ipynb", "DG/splitting.ipynb", "DG/stationary.ipynb", "DG/stokes.ipynb", "MPIparallel/PETSc_interface.ipynb", "MPIparallel/distmesh.ipynb", "MPIparallel/hdivnstsimple/NavierStokesMT.ipynb", "MPIparallel/hdivnstsimple/TestConvert.ipynb", "MPIparallel/hdivnstsimple/veclaplace.ipynb", "MPIparallel/hdivnstsimple/veclaplace-auxiliary.ipynb", "MPIparallel/intro.ipynb", "MPIparallel/paralleliteration.ipynb", "MPIparallel/parallelstokes.ipynb", "MPIparallel/petsc.ipynb", "MPIparallel/vectypes.ipynb", "README.md", "abstracttheory/BasicProperties.ipynb", "abstracttheory/Coercive.ipynb", "abstracttheory/RieszRepresentation.ipynb", "abstracttheory/infsup.ipynb", "abstracttheory/subspaceprojection.ipynb", "domaindecomposition/BDDC.ipynb", "domaindecomposition/FETI.ipynb", "domaindecomposition/FETI-DP.ipynb", "domaindecomposition/introduction.ipynb", "domaindecomposition/traces.ipynb", "ex20210609.ipynb", "helmholtz/absorbing.ipynb", "helmholtz/grating.ipynb", "iFEM.ipynb", "intro.md", "iterative/Chebyshev.ipynb", "iterative/Richardson.ipynb", "iterative/conjugategradients.ipynb", "iterative/gradientmethod.ipynb", "iterative/preconditioning.ipynb", "iterative/simple.ipynb", "mixed/abstract.ipynb", "mixed/abstractfem.ipynb", "mixed/boundary.ipynb", "mixed/pardep.ipynb", "mixed/secondorder.ipynb", "mixed/stokes.ipynb", "mixedelasticity/dynamics.ipynb", "mixedelasticity/hellingerreissner.ipynb", "mixedelasticity/reducedsymmetry.ipynb", "mixedelasticity/tdnns.ipynb", "multigrid/algorithms.ipynb", "multigrid/analysisMG.ipynb", "multigrid/analysisML.ipynb", "multigrid/multilevel-extension.ipynb", "plates/hhj.ipynb", "plates/hhj-tdnns.ipynb", "plates/reissnermindlin.ipynb", "plates/shells.ipynb", "plates/tdnnsplate.ipynb", "preamble.ipynb", "primal/boundary_conditions.ipynb", "primal/elasticity3D.ipynb", "primal/exercises.ipynb", "primal/first_example.ipynb", "primal/solvers.ipynb", "primal/subdomains.ipynb", "saddlepoint/bramblepasciak.ipynb", "saddlepoint/fewconstraints.ipynb", "saddlepoint/parameterdependent.ipynb", "saddlepoint/structure.ipynb", "saddlepoint/useBP.ipynb", "secondorder/erroranalysis.ipynb", "secondorder/erroranalysisl2h1.ipynb", "secondorder/finiteelements.ipynb", "secondorder/framework.ipynb", "secondorder/hdiv.ipynb", "secondorder/hybridization.ipynb", "sobolev/Friedrichs.ipynb", "sobolev/Untitled.ipynb", "sobolev/Untitled1.ipynb", "sobolev/Untitled2.ipynb", "sobolev/poincare.ipynb", "sobolev/trace.ipynb", "sobolevspaces/GeneralizedDerivatives.ipynb", "sobolevspaces/SobolevSpaces.ipynb", "sobolevspaces/Traces.ipynb", "sobolevspaces/equivalentnorms.ipynb", "subspacecorrection/asm.ipynb", "subspacecorrection/examples.ipynb", "subspacecorrection/exercise.ipynb", "subspacecorrection/minimaldd.ipynb", "subspacecorrection/overlapping.ipynb", "timedependent/ODEs/runge_kutta.ipynb", "timedependent/ODEs/singlestep.ipynb", "timedependent/intro/Untitled.ipynb", "timedependent/intro/heatequation.ipynb", "timedependent/intro/waveequation.ipynb", "timedependent/parabolic/exponential.ipynb", "timedependent/waves/Maxwell-DG.ipynb", "timedependent/waves/Untitled.ipynb", "timedependent/waves/lts.ipynb", "timedependent/waves/ringresonator.ipynb", "timedependent/waves/testpml/ringresonator.ipynb", "timedependent/waves/wave-leapfrogDG.ipynb"], "titles": ["<span class=\"section-number\">20. </span>Nitsche\u2019s Method for boundary and interface conditions", "<span class=\"section-number\">22. </span>Hybrid DG for elliptic equations", "DG - Methods for elliptic problems", "<span class=\"section-number\">24. </span>Fourth Order Equation", "<span class=\"section-number\">19. </span>Instationary Transport Equation", "<span class=\"section-number\">23. </span>Splitting Methods for the time-dependent convection diffusion equation", "<span class=\"section-number\">18. </span>Stationary Transport Equation", "<span class=\"section-number\">25. </span>H(div)-conforming Stokes", "<span class=\"section-number\">66. </span>NGSolve - PETSc interface", "<span class=\"section-number\">61. </span>Distributed Meshes and Spaces", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "<span class=\"section-number\">60. </span>Introduction to MPI with mpi4py", "<span class=\"section-number\">64. </span>Iteration methods in parallel", "<span class=\"section-number\">67. </span>Solving Stokes in parallel", "<span class=\"section-number\">65. </span>Using PETSc", "<span class=\"section-number\">63. </span>Consistent and Distributed Vectors", "This repository contains an interactive introduction to the Finite Element Method", "<span class=\"section-number\">7. </span>Basic properties", "<span class=\"section-number\">10. </span>Coercive variational problems and their approximation", "<span class=\"section-number\">9. </span>Riesz representation theorem and symmetric variational problems", "<span class=\"section-number\">11. </span>Inf-sup stable variational problems", "<span class=\"section-number\">8. </span>Projection onto subspaces", "<span class=\"section-number\">59. </span>BDDC - Preconditioner", "<span class=\"section-number\">57. </span>FETI methods", "<span class=\"section-number\">58. </span>FETI-DP", "<span class=\"section-number\">54. </span>Introduction to Non-overlapping Domain Decomposition", "<span class=\"section-number\">55. </span>Traces spaces", "Implement a parallel BDDC preconditioner", "Helmholtz Equation", "Grating", "An Interactive Introduction to the Finite Element Method", "An Interactive Introduction to the Finite Element Method", "<span class=\"section-number\">37. </span>The Chebyshev Method", "<span class=\"section-number\">33. </span>The Richardson Iteration", "<span class=\"section-number\">38. </span>Conjugate Gradients", "<span class=\"section-number\">34. </span>The Gradient Method", "<span class=\"section-number\">35. </span>Preconditioning", "<span class=\"section-number\">32. </span>Basic Iterative Methods", "<span class=\"section-number\">15. </span>Abstract Theory", "<span class=\"section-number\">16. </span>Abstract theory for mixed finite element methods", "<span class=\"section-number\">13. </span>Boundary Conditions", "<span class=\"section-number\">17. </span>Parameter Dependent Problems", "<span class=\"section-number\">14. </span>Mixed Methods for second order equations", "<span class=\"section-number\">12. </span>Stokes Equation", "Non-linear dynamics", "<span class=\"math notranslate nohighlight\">\\(\\DeclareMathOperator{\\opdiv}{div}\\)</span>\nHellinger Reissner mixed formulation", "<span class=\"math notranslate nohighlight\">\\(\\DeclareMathOperator{\\opdiv}{div}\\)</span>\n<span class=\"math notranslate nohighlight\">\\(\\DeclareMathOperator{\\opcurl}{curl}\\)</span>\n<span class=\"math notranslate nohighlight\">\\(\\DeclareMathOperator{\\eps}{\\varepsilon}\\)</span>\nReduced symmetry methods", "Tangential displacement normal normal stress continuous finite elements", "<span class=\"section-number\">44. </span>Multigrid and Multilevel Methods", "<span class=\"section-number\">46. </span>Analysis of the Multigrid Iteration", "<span class=\"section-number\">45. </span>Analysis of the multi-level preconditioner", "<span class=\"section-number\">47. </span>Multi-level Extension", "The Hellan Herrmann Johnson Method for Kirchhoff plates", "Relationship between HHJ and TDNNS", "Reissner Mindlin Plates", "Nonlinear Shells", "3D-TDNNS Elasticity and Reissner Mindlin Plate elements", "Preamble", "<span class=\"section-number\">2. </span>Boundary Conditions", "<span class=\"section-number\">5. </span>3D Solid Mechanics", "<span class=\"section-number\">6. </span>Exercises", "<span class=\"section-number\">1. </span>Solving the Poisson Equation", "<span class=\"section-number\">4. </span>Iterative Solvers", "<span class=\"section-number\">3. </span>Variable Coefficients", "<span class=\"section-number\">49. </span>The Bramble-Pasciak Transformation", "<span class=\"section-number\">50. </span>A Small Number of Constraints", "<span class=\"section-number\">52. </span>Parameter Dependent Problems", "<span class=\"section-number\">48. </span>Structure of Saddle-point Problems", "&lt;no title&gt;", "<span class=\"section-number\">29. </span>Finite Element Error Analysis", "<span class=\"section-number\">30. </span>Error Analysis in <span class=\"math notranslate nohighlight\">\\(L_2 \\times H^1\\)</span>", "<span class=\"section-number\">28. </span>Finite Elements in <span class=\"math notranslate nohighlight\">\\(H(\\operatorname{div})\\)</span>", "<span class=\"section-number\">26. </span>Application of the abstract theory", "<span class=\"section-number\">27. </span>The function space <span class=\"math notranslate nohighlight\">\\(H(\\operatorname{div})\\)</span>", "<span class=\"section-number\">31. </span>Hybridization Techniques", "Friedrichs\u2019 Inequality", "&lt;no title&gt;", "&lt;no title&gt;", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "Generalized derivatives", "Sobolev spaces", "Trace theorems and their applications", "Equivalent norms on <span class=\"math notranslate nohighlight\">\\(H^1\\)</span> and on sub-spaces", "<span class=\"section-number\">39. </span>Additive Schwarz Methods", "<span class=\"section-number\">40. </span>Same Examples of ASM preconditioners", "<span class=\"section-number\">43. </span>Exercise: Robust preconditioners", "<span class=\"section-number\">41. </span>Domain Decomposition with minimal overlap", "<span class=\"section-number\">42. </span>Overlapping Domain Decomposition Methods", "Runge Kutta Methods", "Single-step methods", "&lt;no title&gt;", "Heat Equation", "Wave Equation", "Exponential Integrators for Parabolic Equations", "&lt;no title&gt;", "&lt;no title&gt;", "Mass-lumping and Local time-stepping", "Nano-optics: A ring-resonator", "Nano-optics: A ring-resonator", "Discontinuous Galerkin for the Wave Equation"], "terms": {"from": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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"stationary-transport-equation"]], "Discontinuous Galerkin method": [[6, "discontinuous-galerkin-method"]], "H(div)-conforming Stokes": [[7, "h-div-conforming-stokes"]], "NGSolve - PETSc interface": [[8, "ngsolve-petsc-interface"]], "PETSc preconditioner for NGSolve": [[8, "petsc-preconditioner-for-ngsolve"]], "Distributed Meshes and Spaces": [[9, "distributed-meshes-and-spaces"]], "Distributed finite element spaces": [[9, "distributed-finite-element-spaces"]], "The ParallelDofs class": [[9, "the-paralleldofs-class"]], "Introduction to MPI with mpi4py": [[14, "introduction-to-mpi-with-mpi4py"]], "The MPI library": [[14, "the-mpi-library"]], "Iteration methods in parallel": [[15, "iteration-methods-in-parallel"]], "Richardson iteration": [[15, "richardson-iteration"]], "Solving Stokes in parallel": [[16, "solving-stokes-in-parallel"]], "Using PETSc": [[17, "using-petsc"]], "Consistent and Distributed Vectors": [[18, "consistent-and-distributed-vectors"]], "Consistent vectors:": [[18, "consistent-vectors"]], "Distributed vectors and matrices": [[18, "distributed-vectors-and-matrices"]], "Inner products:": [[18, "inner-products"]], "Matrix vector multiplication:": [[18, "matrix-vector-multiplication"]], "Vector operations:": [[18, "vector-operations"]], "This repository contains an interactive introduction to the Finite Element Method": [[19, "this-repository-contains-an-interactive-introduction-to-the-finite-element-method"]], "Basic properties": [[20, "basic-properties"]], "Coercive variational problems and their approximation": [[21, "coercive-variational-problems-and-their-approximation"]], "Approximation of coercive variational problems": [[21, "approximation-of-coercive-variational-problems"]], "Riesz representation theorem and symmetric variational problems": [[22, "riesz-representation-theorem-and-symmetric-variational-problems"]], "Symmetric variational problems": [[22, "symmetric-variational-problems"]], "Projection onto subspaces": [[24, "projection-onto-subspaces"]], "BDDC - Preconditioner": [[25, "bddc-preconditioner"]], "FETI methods": [[26, "feti-methods"]], "Preconditioner for S": [[26, "preconditioner-for-s"]], "FETI-DP": [[27, "feti-dp"]], "Introduction to Non-overlapping Domain Decomposition": [[28, "introduction-to-non-overlapping-domain-decomposition"]], "Domain Decomposition with Lagrange parameters": [[28, "domain-decomposition-with-lagrange-parameters"]], "Traces spaces": [[29, "traces-spaces"]], "Natural trace space": [[29, "natural-trace-space"]], "Interpolation space H^s": [[29, "interpolation-space-h-s"]], "Trace norm on bottom edge": [[29, "trace-norm-on-bottom-edge"]], "Trace norm on boundary sub-domains": [[29, "trace-norm-on-boundary-sub-domains"]], "Implement a parallel BDDC preconditioner": [[30, "implement-a-parallel-bddc-preconditioner"]], "Helmholtz Equation": [[31, "helmholtz-equation"]], "Grating": [[32, "grating"]], "An Interactive Introduction to the Finite Element Method": [[33, "an-interactive-introduction-to-the-finite-element-method"], [34, "an-interactive-introduction-to-the-finite-element-method"]], "The Galerkin Method": [[33, "the-galerkin-method"], [34, null]], "Abstract Theory": [[33, "abstract-theory"], [41, "abstract-theory"], [34, null]], "Mixed Finite Element Methods": [[33, "mixed-finite-element-methods"], [34, null]], "Discontinuous Galerkin Methods": [[33, "discontinuous-galerkin-methods"], [34, null]], "Mixed Methods for Second Order Equations": [[33, "mixed-methods-for-second-order-equations"], [34, null]], "Mixed Methods for Elasticity": [[33, "mixed-methods-for-elasticity"]], "Mixed Methods for Plates and Shells": [[33, "mixed-methods-for-plates-and-shells"]], "The Helmholtz equation": [[33, "the-helmholtz-equation"]], "Iterative Solvers": [[33, "iterative-solvers"], [65, "iterative-solvers"]], "Iteration Methods": [[33, "iteration-methods"], [34, null]], "Sub-space Correction Methods": [[33, "sub-space-correction-methods"], [34, null]], "Multigrid Methods": [[33, "multigrid-methods"], [34, null]], "Saddle-point Problems": [[33, "saddle-point-problems"], [34, null]], "Non-overlapping Domain Decomposition Methods": [[33, "non-overlapping-domain-decomposition-methods"], [34, null]], "Parallel Solvers": [[33, "parallel-solvers"], [34, null]], "Time-dependent Problems": [[33, "time-dependent-problems"]], "A practical introduction": [[33, "a-practical-introduction"]], "Ordinary differential equations": [[33, "ordinary-differential-equations"]], "Numerical analysis of Parabolic Equations": [[33, "numerical-analysis-of-parabolic-equations"]], "Numerical analysis of Wave Equations": [[33, "numerical-analysis-of-wave-equations"]], "Additional methods": [[33, "additional-methods"]], "Appendix": [[33, "appendix"]], "Some inequalities in Sobolev spaces": [[33, "some-inequalities-in-sobolev-spaces"]], "The Chebyshev Method": [[35, "the-chebyshev-method"]], "Chebyshev polynomials": [[35, "chebyshev-polynomials"]], "The Chebyshev iteration": [[35, "the-chebyshev-iteration"]], "The Richardson Iteration": [[36, "the-richardson-iteration"]], "Optimizing the relaxation parameter \\alpha": [[36, "optimizing-the-relaxation-parameter-alpha"]], "Experiments with the Richardson iteration": [[36, "experiments-with-the-richardson-iteration"]], "Conjugate Gradients": [[37, "conjugate-gradients"]], "Solving the minimization problem": [[37, "solving-the-minimization-problem"]], "Expanding the Krylov-space": [[37, "expanding-the-krylov-space"]], "The Gradient Method": [[38, "the-gradient-method"]], "Preconditioning": [[39, "preconditioning"]], "The preconditioned Richardson iteration": [[39, "the-preconditioned-richardson-iteration"]], "The preconditioned gradient method": [[39, "the-preconditioned-gradient-method"]], "Jacobi and Gauss Seidel Preconditioners": [[39, "jacobi-and-gauss-seidel-preconditioners"]], "Basic Iterative Methods": [[40, "basic-iterative-methods"]], "Constrained minimization problem": [[41, "constrained-minimization-problem"]], "Stokes equation within the abstract theory": [[41, "stokes-equation-within-the-abstract-theory"]], "Dirichlet boundary conditions as mixed system": [[41, "dirichlet-boundary-conditions-as-mixed-system"]], "Mixed method for second order equation": [[41, "mixed-method-for-second-order-equation"]], "Abstract theory for mixed finite element methods": [[42, "abstract-theory-for-mixed-finite-element-methods"]], "Finite element error estimates": [[42, "finite-element-error-estimates"]], "Proving discrete LBB by the Fortin operator": [[42, "proving-discrete-lbb-by-the-fortin-operator"]], "Example: Finite elements for Stokes": [[42, "example-finite-elements-for-stokes"]], "Boundary Conditions": [[43, "boundary-conditions"], [61, "boundary-conditions"]], "Parameter Dependent Problems": [[44, "parameter-dependent-problems"], [69, "parameter-dependent-problems"]], "Example: Dirichlet boundary condition by penalty": [[44, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Nearly incompressible materials": [[44, "example-nearly-incompressible-materials"]], "Mixed Methods for second order equations": [[45, "mixed-methods-for-second-order-equations"]], "Stokes Equation": [[46, "stokes-equation"]], "Variational Formulation": [[46, "variational-formulation"]], "Finite Element Spaces": [[46, "finite-element-spaces"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[48, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "Primal mixed method:": [[48, "primal-mixed-method"]], "Dual mixed method": [[48, "dual-mixed-method"]], "\\DeclareMathOperator{\\opdiv}{div}\n\\DeclareMathOperator{\\opcurl}{curl}\n\\DeclareMathOperator{\\eps}{\\varepsilon}\nReduced symmetry methods": [[49, "declaremathoperator-opdiv-div-declaremathoperator-opcurl-curl-declaremathoperator-eps-varepsilon-reduced-symmetry-methods"]], "Tangential displacement normal normal stress continuous finite elements": [[50, "tangential-displacement-normal-normal-stress-continuous-finite-elements"]], "The divergence of nn-continuous piece-wise smooth functions:": [[50, "the-divergence-of-nn-continuous-piece-wise-smooth-functions"]], "TDNNS Variational formulation:": [[50, "tdnns-variational-formulation"]], "Error estimates:": [[50, "error-estimates"]], "Multigrid and Multilevel Methods": [[51, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[51, "multilevel-preconditioner"]], "Multigrid Preconditioning": [[51, "multigrid-preconditioning"]], "Projection matrices from the finest level": [[51, "projection-matrices-from-the-finest-level"]], "Analysis of the Multigrid Iteration": [[52, "analysis-of-the-multigrid-iteration"]], "The Algorithm": [[52, "the-algorithm"]], "The Approximation Property": [[52, "the-approximation-property"]], "The Smothing Property": [[52, "the-smothing-property"]], "Optimal convergence of the V-cycle": [[52, "optimal-convergence-of-the-v-cycle"]], "Analysis of the multi-level preconditioner": [[53, "analysis-of-the-multi-level-preconditioner"]], "Nearly optimal analysis of the ML - preconditioner": [[53, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Optimal analysis of the multi-level preconditioner": [[53, "optimal-analysis-of-the-multi-level-preconditioner"]], "Multi-level Extension": [[54, "multi-level-extension"]], "Efficiently computable multi-level decomposition": [[54, "efficiently-computable-multi-level-decomposition"]], "Algorithm": [[54, "algorithm"]], "Extending boundary data": [[54, "extending-boundary-data"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[55, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "Kirchhoff Plate equation": [[55, "kirchhoff-plate-equation"]], "Hellan-Herrmann-Johnson method": [[55, "hellan-herrmann-johnson-method"]], "Relationship between HHJ and TDNNS": [[56, "relationship-between-hhj-and-tdnns"]], "Commuting diagram for H^1 - H(\\opcurl)": [[56, "commuting-diagram-for-h-1-h-opcurl"]], "Reissner Mindlin Plates": [[57, "reissner-mindlin-plates"]], "Nonlinear Shells": [[58, "nonlinear-shells"]], "Preamble": [[60, "preamble"]], "Natural boundary conditions": [[61, "natural-boundary-conditions"]], "Essential boundary conditions": [[61, "essential-boundary-conditions"]], "3D Solid Mechanics": [[62, "d-solid-mechanics"]], "Linear elasticity": [[62, "linear-elasticity"]], "Variational formulation:": [[62, "variational-formulation"]], "Exercises": [[63, "exercises"]], "Get started with Netgen-Opencascade geometric modeling": [[63, "get-started-with-netgen-opencascade-geometric-modeling"]], "Experiment with CoefficientFunctions": [[63, "experiment-with-coefficientfunctions"]], "Work with GridFunctions": [[63, "work-with-gridfunctions"]], "Linear Algebra": [[63, "linear-algebra"]], "Experiments with BilienarForms and LinearForms": [[63, "experiments-with-bilienarforms-and-linearforms"]], "Computing dual norms": [[63, "computing-dual-norms"]], "Variable Coefficients": [[66, "variable-coefficients"]], "The Bramble-Pasciak Transformation": [[67, "the-bramble-pasciak-transformation"]], "A Small Number of Constraints": [[68, "a-small-number-of-constraints"]], "Projected preconditioner": [[68, "projected-preconditioner"]], "Dirichlet boundary conditions by penalty:": [[69, "dirichlet-boundary-conditions-by-penalty"]], "Penalty formulation for the flux:": [[69, "penalty-formulation-for-the-flux"]], "Maxwell equations:": [[69, "maxwell-equations"]], "Penalty formulation for the Stokes equation:": [[69, "penalty-formulation-for-the-stokes-equation"]], "Robust two-level methods for parameter dependent problems": [[69, "robust-two-level-methods-for-parameter-dependent-problems"]], "Robust smoothers": [[69, "robust-smoothers"]], "Robust coarse-grid correction": [[69, "robust-coarse-grid-correction"]], "Two-level analysis for Maxwell equations": [[69, "two-level-analysis-for-maxwell-equations"]], "Structure of Saddle-point Problems": [[70, "structure-of-saddle-point-problems"]], "Constrained minimization problems": [[70, "constrained-minimization-problems"]], "Examples": [[70, "examples"], [93, "examples"]], "Schur complement iteration": [[70, "schur-complement-iteration"]], "Block-preconditioning": [[70, "block-preconditioning"]], "Finite Element Error Analysis": [[72, "finite-element-error-analysis"]], "Local post-processing": [[72, "local-post-processing"]], "Error Analysis in L_2 \\times H^1": [[73, "error-analysis-in-l-2-times-h-1"]], "Finite Elements in H(\\operatorname{div})": [[74, "finite-elements-in-h-operatorname-div"]], "Piola Transformation": [[74, "piola-transformation"]], "Application of the abstract theory": [[75, "application-of-the-abstract-theory"]], "Dual mixed formulation": [[75, "dual-mixed-formulation"]], "Primal mixed formulation": [[75, "primal-mixed-formulation"]], "The function space H(\\operatorname{div})": [[76, "the-function-space-h-operatorname-div"]], "Normal-trace of functions in H(\\opdiv)": [[76, "normal-trace-of-functions-in-h-opdiv"]], "H(\\opdiv) on sub-domains": [[76, "h-opdiv-on-sub-domains"]], "Hybridization Techniques": [[77, "hybridization-techniques"]], "Friedrichs\u2019 Inequality": [[78, "friedrichs-inequality"], [81, "friedrichs-inequality"]], "The Poincar\u00e9 inequality": [[82, "the-poincare-inequality"]], "The Trace Inequality": [[83, "the-trace-inequality"]], "Numerical verification": [[83, "numerical-verification"]], "Proof of the trace inequality:": [[83, "proof-of-the-trace-inequality"]], "Generalized derivatives": [[84, "generalized-derivatives"]], "Sobolev spaces": [[85, "sobolev-spaces"]], "Trace theorems and their applications": [[86, "trace-theorems-and-their-applications"]], "Integration by parts": [[86, "integration-by-parts"]], "Sobolev spaces over sub-domains}": [[86, "sobolev-spaces-over-sub-domains"]], "Extension operators": [[86, "extension-operators"]], "The trace space H^{1/2}": [[86, "the-trace-space-h-1-2"]], "Equivalent norms on H^1 and on sub-spaces": [[87, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Additive Schwarz Methods": [[88, "additive-schwarz-methods"]], "Block-Jacobi and general additive Schwarz preconditioners": [[88, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[88, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "The Additive Schwarz Lemma": [[88, "the-additive-schwarz-lemma"]], "The upper-bound by the overlap": [[88, "the-upper-bound-by-the-overlap"]], "Same Examples of ASM preconditioners": [[89, "same-examples-of-asm-preconditioners"]], "Diagonal preconditioner for L_2-norm": [[89, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[89, "diagonal-preconditioner-for-the-h-1-norm"]], "H^1-norm with small L_2-term": [[89, "h-1-norm-with-small-l-2-term"]], "Exercise: Robust preconditioners": [[90, "exercise-robust-preconditioners"]], "Exercise A:": [[90, "exercise-a"]], "Exercise B:": [[90, "exercise-b"]], "Overlapping Domain Decomposition Methods": [[92, "overlapping-domain-decomposition-methods"]], "Experiments with overlapping DD": [[92, "experiments-with-overlapping-dd"]], "Analysis of the DD preconditioner": [[92, "analysis-of-the-dd-preconditioner"]], "Overlapping DD Methods with coarse grid": [[92, "overlapping-dd-methods-with-coarse-grid"]], "Analysis of the 2-level method:": [[92, "analysis-of-the-2-level-method"]], "Comparison to DD with minimal overlap": [[92, "comparison-to-dd-with-minimal-overlap"]], "Runge Kutta Methods": [[93, "runge-kutta-methods"]], "Butcher tableaus of simple methods": [[93, "butcher-tableaus-of-simple-methods"]], "Convergence of Runge Kutta methods": [[93, "convergence-of-runge-kutta-methods"]], "Explicit methods:": [[93, "explicit-methods"]], "Diagonally implicit Runge-Kutta methods:": [[93, "diagonally-implicit-runge-kutta-methods"]], "Exercise:": [[93, "exercise"], [104, "exercise"]], "Single-step methods": [[94, "single-step-methods"]], "Simple methods based on the integral equation": [[94, "simple-methods-based-on-the-integral-equation"]], "Explicit Euler method (EE)": [[94, "explicit-euler-method-ee"]], "Implicit Euler method (IE)": [[94, "implicit-euler-method-ie"]], "Trapezoidal method": [[94, "trapezoidal-method"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[94, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Linear stability classification": [[94, "linear-stability-classification"]], "Stability function": [[94, "stability-function"]], "Stability conditions": [[94, "stability-conditions"]], "Single step methods and parabolic equations": [[94, "single-step-methods-and-parabolic-equations"]], "Single step methods and hyperbolic equations": [[94, "single-step-methods-and-hyperbolic-equations"]], "Exercises:": [[94, "exercises"]], "Heat Equation": [[96, "heat-equation"]], "Variational formulation in space": [[96, "variational-formulation-in-space"]], "Galerkin method in space": [[96, "galerkin-method-in-space"]], "Implicit Euler time-stepping": [[96, "implicit-euler-time-stepping"]], "Wave Equation": [[97, "wave-equation"]], "Newmark time-stepping method": [[97, "newmark-time-stepping-method"]], "Exponential Integrators for Parabolic Equations": [[98, "exponential-integrators-for-parabolic-equations"]], "Mass-lumping and Local time-stepping": [[101, "mass-lumping-and-local-time-stepping"]], "The Verlet method:": [[101, "the-verlet-method"]], "Geometry with local details:": [[101, "geometry-with-local-details"]], "Nano-optics: A ring-resonator": [[102, "nano-optics-a-ring-resonator"], [103, "nano-optics-a-ring-resonator"]], "Discontinuous Galerkin for the Wave Equation": [[104, "discontinuous-galerkin-for-the-wave-equation"]], "Testing the differential operators": [[104, "testing-the-differential-operators"]], "Efficient implementation:": [[104, "efficient-implementation"]], "Solving the wave equation:": [[104, "solving-the-wave-equation"]], "Eigenvalues of the discretized Laplace-operator": [[104, "eigenvalues-of-the-discretized-laplace-operator"]], "Literature": [[34, "literature"]], "Installing NGSolve": [[34, "installing-ngsolve"]], "Sobolev Spaces": [[34, null]], "Inf-sup stable variational problems": [[23, "inf-sup-stable-variational-problems"]], "Approximation of inf-sup stable variational problems": [[23, "approximation-of-inf-sup-stable-variational-problems"]], "Non-linear dynamics": [[47, "non-linear-dynamics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[59, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]], "Solving the Poisson Equation": [[64, "solving-the-poisson-equation"]], "Weak formulation": [[64, "weak-formulation"]], "The Sobolev space H^1, linear and bilinear forms": [[64, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Finite Element Method": [[64, "the-finite-element-method"]], "Poisson equation in NGSolve:": [[64, "poisson-equation-in-ngsolve"]], "Visualizing the basis functions": [[64, "visualizing-the-basis-functions"]], "Domain Decomposition with minimal overlap": [[91, "domain-decomposition-with-minimal-overlap"]], "Analysis of the method": [[91, "analysis-of-the-method"]], "Adding a coarse grid space": [[91, "adding-a-coarse-grid-space"]], "Graph-based mesh partitioning": [[91, "graph-based-mesh-partitioning"]]}, "indexentries": {}})
\ No newline at end of file
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"sobolev/Untitled", "sobolev/Untitled1", "sobolev/Untitled2", "sobolev/poincare", "sobolev/trace", "sobolevspaces/GeneralizedDerivatives", "sobolevspaces/SobolevSpaces", "sobolevspaces/Traces", "sobolevspaces/equivalentnorms", "subspacecorrection/asm", "subspacecorrection/examples", "subspacecorrection/exercise", "subspacecorrection/minimaldd", "subspacecorrection/overlapping", "timedependent/ODEs/runge_kutta", "timedependent/ODEs/singlestep", "timedependent/intro/Untitled", "timedependent/intro/heatequation", "timedependent/intro/waveequation", "timedependent/parabolic/exponential", "timedependent/waves/Maxwell-DG", "timedependent/waves/Untitled", "timedependent/waves/lts", "timedependent/waves/ringresonator", "timedependent/waves/testpml/ringresonator", "timedependent/waves/wave-leapfrogDG"], "filenames": ["DG/Nitsche.ipynb", "DG/elliptic.ipynb", "DG/elliptic_stdDG.ipynb", "DG/fourthorder.ipynb", "DG/instationary.ipynb", "DG/splitting.ipynb", "DG/stationary.ipynb", "DG/stokes.ipynb", "MPIparallel/PETSc_interface.ipynb", "MPIparallel/distmesh.ipynb", "MPIparallel/hdivnstsimple/NavierStokesMT.ipynb", "MPIparallel/hdivnstsimple/TestConvert.ipynb", "MPIparallel/hdivnstsimple/veclaplace.ipynb", "MPIparallel/hdivnstsimple/veclaplace-auxiliary.ipynb", "MPIparallel/intro.ipynb", "MPIparallel/paralleliteration.ipynb", "MPIparallel/parallelstokes.ipynb", "MPIparallel/petsc.ipynb", "MPIparallel/vectypes.ipynb", "README.md", "abstracttheory/BasicProperties.ipynb", "abstracttheory/Coercive.ipynb", "abstracttheory/RieszRepresentation.ipynb", "abstracttheory/infsup.ipynb", "abstracttheory/subspaceprojection.ipynb", "domaindecomposition/BDDC.ipynb", "domaindecomposition/FETI.ipynb", "domaindecomposition/FETI-DP.ipynb", "domaindecomposition/introduction.ipynb", "domaindecomposition/traces.ipynb", "ex20210609.ipynb", "helmholtz/absorbing.ipynb", "helmholtz/grating.ipynb", "iFEM.ipynb", "intro.md", "iterative/Chebyshev.ipynb", "iterative/Richardson.ipynb", "iterative/conjugategradients.ipynb", "iterative/gradientmethod.ipynb", "iterative/preconditioning.ipynb", "iterative/simple.ipynb", "mixed/abstract.ipynb", "mixed/abstractfem.ipynb", "mixed/boundary.ipynb", "mixed/pardep.ipynb", "mixed/secondorder.ipynb", "mixed/stokes.ipynb", "mixedelasticity/dynamics.ipynb", "mixedelasticity/hellingerreissner.ipynb", "mixedelasticity/reducedsymmetry.ipynb", "mixedelasticity/tdnns.ipynb", "multigrid/algorithms.ipynb", "multigrid/analysisMG.ipynb", "multigrid/analysisML.ipynb", "multigrid/multilevel-extension.ipynb", "plates/hhj.ipynb", "plates/hhj-tdnns.ipynb", "plates/reissnermindlin.ipynb", "plates/shells.ipynb", "plates/tdnnsplate.ipynb", "preamble.ipynb", "primal/boundary_conditions.ipynb", "primal/elasticity3D.ipynb", "primal/exercises.ipynb", "primal/first_example.ipynb", "primal/solvers.ipynb", "primal/subdomains.ipynb", "saddlepoint/bramblepasciak.ipynb", "saddlepoint/fewconstraints.ipynb", "saddlepoint/parameterdependent.ipynb", "saddlepoint/structure.ipynb", "saddlepoint/useBP.ipynb", "secondorder/erroranalysis.ipynb", "secondorder/erroranalysisl2h1.ipynb", "secondorder/finiteelements.ipynb", "secondorder/framework.ipynb", "secondorder/hdiv.ipynb", "secondorder/hybridization.ipynb", "sobolev/Friedrichs.ipynb", "sobolev/Untitled.ipynb", "sobolev/Untitled1.ipynb", "sobolev/Untitled2.ipynb", "sobolev/poincare.ipynb", "sobolev/trace.ipynb", "sobolevspaces/GeneralizedDerivatives.ipynb", "sobolevspaces/SobolevSpaces.ipynb", "sobolevspaces/Traces.ipynb", "sobolevspaces/equivalentnorms.ipynb", "subspacecorrection/asm.ipynb", "subspacecorrection/examples.ipynb", "subspacecorrection/exercise.ipynb", "subspacecorrection/minimaldd.ipynb", "subspacecorrection/overlapping.ipynb", "timedependent/ODEs/runge_kutta.ipynb", "timedependent/ODEs/singlestep.ipynb", "timedependent/intro/Untitled.ipynb", "timedependent/intro/heatequation.ipynb", "timedependent/intro/waveequation.ipynb", "timedependent/parabolic/exponential.ipynb", "timedependent/waves/Maxwell-DG.ipynb", "timedependent/waves/Untitled.ipynb", "timedependent/waves/lts.ipynb", "timedependent/waves/ringresonator.ipynb", "timedependent/waves/testpml/ringresonator.ipynb", "timedependent/waves/wave-leapfrogDG.ipynb"], "titles": ["<span class=\"section-number\">20. </span>Nitsche\u2019s Method for boundary and interface conditions", "<span class=\"section-number\">22. </span>Hybrid DG for elliptic equations", "DG - Methods for elliptic problems", "<span class=\"section-number\">24. </span>Fourth Order Equation", "<span class=\"section-number\">19. </span>Instationary Transport Equation", "<span class=\"section-number\">23. </span>Splitting Methods for the time-dependent convection diffusion equation", "<span class=\"section-number\">18. </span>Stationary Transport Equation", "<span class=\"section-number\">25. </span>H(div)-conforming Stokes", "<span class=\"section-number\">66. </span>NGSolve - PETSc interface", "<span class=\"section-number\">61. </span>Distributed Meshes and Spaces", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "&lt;no title&gt;", "<span class=\"section-number\">60. </span>Introduction to MPI with mpi4py", "<span class=\"section-number\">64. </span>Iteration methods in parallel", "<span class=\"section-number\">67. </span>Solving Stokes in parallel", "<span class=\"section-number\">65. </span>Using PETSc", "<span class=\"section-number\">63. </span>Consistent and Distributed Vectors", "This repository contains an interactive introduction to the Finite Element Method", "<span class=\"section-number\">7. </span>Basic properties", "<span class=\"section-number\">10. </span>Coercive variational problems and their approximation", "<span class=\"section-number\">9. </span>Riesz representation theorem and symmetric variational problems", "<span class=\"section-number\">11. </span>Inf-sup stable variational problems", "<span class=\"section-number\">8. </span>Projection onto subspaces", "<span class=\"section-number\">59. </span>BDDC - Preconditioner", "<span class=\"section-number\">57. </span>FETI methods", "<span class=\"section-number\">58. </span>FETI-DP", "<span class=\"section-number\">54. </span>Introduction to Non-overlapping Domain Decomposition", "<span class=\"section-number\">55. </span>Traces spaces", "Implement a parallel BDDC preconditioner", "Helmholtz Equation", "Grating", "An Interactive Introduction to the Finite Element Method", "An Interactive Introduction to the Finite Element Method", "<span class=\"section-number\">37. </span>The Chebyshev Method", "<span class=\"section-number\">33. </span>The Richardson Iteration", "<span class=\"section-number\">38. </span>Conjugate Gradients", "<span class=\"section-number\">34. </span>The Gradient Method", "<span class=\"section-number\">35. </span>Preconditioning", "<span class=\"section-number\">32. </span>Basic Iterative Methods", "<span class=\"section-number\">15. </span>Abstract Theory", "<span class=\"section-number\">16. </span>Abstract theory for mixed finite element methods", "<span class=\"section-number\">13. </span>Boundary Conditions", "<span class=\"section-number\">17. </span>Parameter Dependent Problems", "<span class=\"section-number\">14. </span>Mixed Methods for second order equations", "<span class=\"section-number\">12. </span>Stokes Equation", "Non-linear dynamics", "<span class=\"math notranslate nohighlight\">\\(\\DeclareMathOperator{\\opdiv}{div}\\)</span>\nHellinger Reissner mixed formulation", "<span class=\"math notranslate nohighlight\">\\(\\DeclareMathOperator{\\opdiv}{div}\\)</span>\n<span class=\"math notranslate nohighlight\">\\(\\DeclareMathOperator{\\opcurl}{curl}\\)</span>\n<span class=\"math notranslate nohighlight\">\\(\\DeclareMathOperator{\\eps}{\\varepsilon}\\)</span>\nReduced symmetry methods", "Tangential displacement normal normal stress continuous finite elements", "<span class=\"section-number\">44. </span>Multigrid and Multilevel Methods", "<span class=\"section-number\">46. </span>Analysis of the Multigrid Iteration", "<span class=\"section-number\">45. </span>Analysis of the multi-level preconditioner", "<span class=\"section-number\">47. </span>Multi-level Extension", "The Hellan Herrmann Johnson Method for Kirchhoff plates", "Relationship between HHJ and TDNNS", "Reissner Mindlin Plates", "Nonlinear Shells", "3D-TDNNS Elasticity and Reissner Mindlin Plate elements", "Preamble", "<span class=\"section-number\">2. </span>Boundary Conditions", "<span class=\"section-number\">5. </span>3D Solid Mechanics", "<span class=\"section-number\">6. </span>Exercises", "<span class=\"section-number\">1. </span>Solving the Poisson Equation", "<span class=\"section-number\">4. </span>Iterative Solvers", "<span class=\"section-number\">3. </span>Variable Coefficients", "<span class=\"section-number\">49. </span>The Bramble-Pasciak Transformation", "<span class=\"section-number\">50. </span>A Small Number of Constraints", "<span class=\"section-number\">52. </span>Parameter Dependent Problems", "<span class=\"section-number\">48. </span>Structure of Saddle-point Problems", "&lt;no title&gt;", "<span class=\"section-number\">29. </span>Finite Element Error Analysis", "<span class=\"section-number\">30. </span>Error Analysis in <span class=\"math notranslate nohighlight\">\\(L_2 \\times H^1\\)</span>", "<span class=\"section-number\">28. </span>Finite Elements in <span class=\"math notranslate nohighlight\">\\(H(\\operatorname{div})\\)</span>", "<span class=\"section-number\">26. </span>Application of the abstract theory", "<span class=\"section-number\">27. </span>The function space <span class=\"math notranslate nohighlight\">\\(H(\\operatorname{div})\\)</span>", "<span class=\"section-number\">31. </span>Hybridization Techniques", "Friedrichs\u2019 Inequality", "&lt;no title&gt;", "&lt;no title&gt;", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "Generalized derivatives", "Sobolev spaces", "Trace theorems and their applications", "Equivalent norms on <span class=\"math notranslate nohighlight\">\\(H^1\\)</span> and on sub-spaces", "<span class=\"section-number\">39. </span>Additive Schwarz Methods", "<span class=\"section-number\">40. </span>Same Examples of ASM preconditioners", "<span class=\"section-number\">43. </span>Exercise: Robust preconditioners", "<span class=\"section-number\">41. </span>Domain Decomposition with minimal overlap", "<span class=\"section-number\">42. </span>Overlapping Domain Decomposition Methods", "Runge Kutta Methods", "Single-step methods", "&lt;no title&gt;", "Heat Equation", "Wave Equation", "Exponential Integrators for Parabolic Equations", "&lt;no title&gt;", "&lt;no title&gt;", "Mass-lumping and Local time-stepping", "Nano-optics: A ring-resonator", "Nano-optics: A ring-resonator", "Discontinuous Galerkin for the Wave Equation"], "terms": {"from": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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Methods": [[40, "basic-iterative-methods"]], "Constrained minimization problem": [[41, "constrained-minimization-problem"]], "Stokes equation within the abstract theory": [[41, "stokes-equation-within-the-abstract-theory"]], "Dirichlet boundary conditions as mixed system": [[41, "dirichlet-boundary-conditions-as-mixed-system"]], "Mixed method for second order equation": [[41, "mixed-method-for-second-order-equation"]], "Abstract theory for mixed finite element methods": [[42, "abstract-theory-for-mixed-finite-element-methods"]], "Finite element error estimates": [[42, "finite-element-error-estimates"]], "Proving discrete LBB by the Fortin operator": [[42, "proving-discrete-lbb-by-the-fortin-operator"]], "Example: Finite elements for Stokes": [[42, "example-finite-elements-for-stokes"]], "Boundary Conditions": [[43, "boundary-conditions"], [61, "boundary-conditions"]], "Parameter Dependent Problems": [[44, "parameter-dependent-problems"], [69, "parameter-dependent-problems"]], "Example: Dirichlet boundary condition by penalty": [[44, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Nearly incompressible materials": [[44, "example-nearly-incompressible-materials"]], "Mixed Methods for second order equations": [[45, "mixed-methods-for-second-order-equations"]], "Stokes Equation": [[46, "stokes-equation"]], "Variational Formulation": [[46, "variational-formulation"]], "Finite Element Spaces": [[46, "finite-element-spaces"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[48, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "Primal mixed method:": [[48, "primal-mixed-method"]], "Dual mixed method": [[48, "dual-mixed-method"]], "\\DeclareMathOperator{\\opdiv}{div}\n\\DeclareMathOperator{\\opcurl}{curl}\n\\DeclareMathOperator{\\eps}{\\varepsilon}\nReduced symmetry methods": [[49, "declaremathoperator-opdiv-div-declaremathoperator-opcurl-curl-declaremathoperator-eps-varepsilon-reduced-symmetry-methods"]], "Tangential displacement normal normal stress continuous finite elements": [[50, "tangential-displacement-normal-normal-stress-continuous-finite-elements"]], "The divergence of nn-continuous piece-wise smooth functions:": [[50, "the-divergence-of-nn-continuous-piece-wise-smooth-functions"]], "TDNNS Variational formulation:": [[50, "tdnns-variational-formulation"]], "Error estimates:": [[50, "error-estimates"]], "Multigrid and Multilevel Methods": [[51, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[51, "multilevel-preconditioner"]], "Multigrid Preconditioning": [[51, "multigrid-preconditioning"]], "Projection matrices from the finest level": [[51, "projection-matrices-from-the-finest-level"]], "Analysis of the Multigrid Iteration": [[52, "analysis-of-the-multigrid-iteration"]], "The Algorithm": [[52, "the-algorithm"]], "The Approximation Property": [[52, "the-approximation-property"]], "The Smothing Property": [[52, "the-smothing-property"]], "Optimal convergence of the V-cycle": [[52, "optimal-convergence-of-the-v-cycle"]], "Analysis of the multi-level preconditioner": [[53, "analysis-of-the-multi-level-preconditioner"]], "Nearly optimal analysis of the ML - preconditioner": [[53, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Optimal analysis of the multi-level preconditioner": [[53, "optimal-analysis-of-the-multi-level-preconditioner"]], "Multi-level Extension": [[54, "multi-level-extension"]], "Efficiently computable multi-level decomposition": [[54, "efficiently-computable-multi-level-decomposition"]], "Algorithm": [[54, "algorithm"]], "Extending boundary data": [[54, "extending-boundary-data"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[55, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "Kirchhoff Plate equation": [[55, "kirchhoff-plate-equation"]], "Hellan-Herrmann-Johnson method": [[55, "hellan-herrmann-johnson-method"]], "Relationship between HHJ and TDNNS": [[56, "relationship-between-hhj-and-tdnns"]], "Commuting diagram for H^1 - H(\\opcurl)": [[56, "commuting-diagram-for-h-1-h-opcurl"]], "Reissner Mindlin Plates": [[57, "reissner-mindlin-plates"]], "Nonlinear Shells": [[58, "nonlinear-shells"]], "Preamble": [[60, "preamble"]], "Natural boundary conditions": [[61, "natural-boundary-conditions"]], "Essential boundary conditions": [[61, "essential-boundary-conditions"]], "3D Solid Mechanics": [[62, "d-solid-mechanics"]], "Linear elasticity": [[62, "linear-elasticity"]], "Variational formulation:": [[62, "variational-formulation"]], "Exercises": [[63, "exercises"]], "Get started with Netgen-Opencascade geometric modeling": [[63, "get-started-with-netgen-opencascade-geometric-modeling"]], "Experiment with CoefficientFunctions": [[63, "experiment-with-coefficientfunctions"]], "Work with GridFunctions": [[63, "work-with-gridfunctions"]], "Linear Algebra": [[63, "linear-algebra"]], "Experiments with BilienarForms and LinearForms": [[63, "experiments-with-bilienarforms-and-linearforms"]], "Computing dual norms": [[63, "computing-dual-norms"]], "Variable Coefficients": [[66, "variable-coefficients"]], "The Bramble-Pasciak Transformation": [[67, "the-bramble-pasciak-transformation"]], "A Small Number of Constraints": [[68, "a-small-number-of-constraints"]], "Projected preconditioner": [[68, "projected-preconditioner"]], "Dirichlet boundary conditions by penalty:": [[69, "dirichlet-boundary-conditions-by-penalty"]], "Penalty formulation for the flux:": [[69, "penalty-formulation-for-the-flux"]], "Maxwell equations:": [[69, "maxwell-equations"]], "Penalty formulation for the Stokes equation:": [[69, "penalty-formulation-for-the-stokes-equation"]], "Robust two-level methods for parameter dependent problems": [[69, "robust-two-level-methods-for-parameter-dependent-problems"]], "Robust smoothers": [[69, "robust-smoothers"]], "Robust coarse-grid correction": [[69, "robust-coarse-grid-correction"]], "Two-level analysis for Maxwell equations": [[69, "two-level-analysis-for-maxwell-equations"]], "Structure of Saddle-point Problems": [[70, "structure-of-saddle-point-problems"]], "Constrained minimization problems": [[70, "constrained-minimization-problems"]], "Examples": [[70, "examples"], [93, "examples"]], "Schur complement iteration": [[70, "schur-complement-iteration"]], "Block-preconditioning": [[70, "block-preconditioning"]], "Finite Element Error Analysis": [[72, "finite-element-error-analysis"]], "Local post-processing": [[72, "local-post-processing"]], "Error Analysis in L_2 \\times H^1": [[73, "error-analysis-in-l-2-times-h-1"]], "Finite Elements in H(\\operatorname{div})": [[74, "finite-elements-in-h-operatorname-div"]], "Piola Transformation": [[74, "piola-transformation"]], "Application of the abstract theory": [[75, "application-of-the-abstract-theory"]], "Dual mixed formulation": [[75, "dual-mixed-formulation"]], "Primal mixed formulation": [[75, "primal-mixed-formulation"]], "The function space H(\\operatorname{div})": [[76, "the-function-space-h-operatorname-div"]], "Normal-trace of functions in H(\\opdiv)": [[76, "normal-trace-of-functions-in-h-opdiv"]], "H(\\opdiv) on sub-domains": [[76, "h-opdiv-on-sub-domains"]], "Hybridization Techniques": [[77, "hybridization-techniques"]], "Friedrichs\u2019 Inequality": [[78, "friedrichs-inequality"], [81, "friedrichs-inequality"]], "The Poincar\u00e9 inequality": [[82, "the-poincare-inequality"]], "The Trace Inequality": [[83, "the-trace-inequality"]], "Numerical verification": [[83, "numerical-verification"]], "Proof of the trace inequality:": [[83, "proof-of-the-trace-inequality"]], "Generalized derivatives": [[84, "generalized-derivatives"]], "Sobolev spaces": [[85, "sobolev-spaces"]], "Trace theorems and their applications": [[86, "trace-theorems-and-their-applications"]], "Integration by parts": [[86, "integration-by-parts"]], "Sobolev spaces over sub-domains}": [[86, "sobolev-spaces-over-sub-domains"]], "Extension operators": [[86, "extension-operators"]], "The trace space H^{1/2}": [[86, "the-trace-space-h-1-2"]], "Equivalent norms on H^1 and on sub-spaces": [[87, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Additive Schwarz Methods": [[88, "additive-schwarz-methods"]], "Block-Jacobi and general additive Schwarz preconditioners": [[88, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[88, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "The Additive Schwarz Lemma": [[88, "the-additive-schwarz-lemma"]], "The upper-bound by the overlap": [[88, "the-upper-bound-by-the-overlap"]], "Same Examples of ASM preconditioners": [[89, "same-examples-of-asm-preconditioners"]], "Diagonal preconditioner for L_2-norm": [[89, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[89, "diagonal-preconditioner-for-the-h-1-norm"]], "H^1-norm with small L_2-term": [[89, "h-1-norm-with-small-l-2-term"]], "Exercise: Robust preconditioners": [[90, "exercise-robust-preconditioners"]], "Exercise A:": [[90, "exercise-a"]], "Exercise B:": [[90, "exercise-b"]], "Overlapping Domain Decomposition Methods": [[92, "overlapping-domain-decomposition-methods"]], "Experiments with overlapping DD": [[92, "experiments-with-overlapping-dd"]], "Analysis of the DD preconditioner": [[92, "analysis-of-the-dd-preconditioner"]], "Overlapping DD Methods with coarse grid": [[92, "overlapping-dd-methods-with-coarse-grid"]], "Analysis of the 2-level method:": [[92, "analysis-of-the-2-level-method"]], "Comparison to DD with minimal overlap": [[92, "comparison-to-dd-with-minimal-overlap"]], "Runge Kutta Methods": [[93, "runge-kutta-methods"]], "Butcher tableaus of simple methods": [[93, "butcher-tableaus-of-simple-methods"]], "Convergence of Runge Kutta methods": [[93, "convergence-of-runge-kutta-methods"]], "Explicit methods:": [[93, "explicit-methods"]], "Diagonally implicit Runge-Kutta methods:": [[93, "diagonally-implicit-runge-kutta-methods"]], "Exercise:": [[93, "exercise"], [104, "exercise"]], "Single-step methods": [[94, "single-step-methods"]], "Simple methods based on the integral equation": [[94, "simple-methods-based-on-the-integral-equation"]], "Explicit Euler method (EE)": [[94, "explicit-euler-method-ee"]], "Implicit Euler method (IE)": [[94, "implicit-euler-method-ie"]], "Trapezoidal method": [[94, "trapezoidal-method"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[94, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Linear stability classification": [[94, "linear-stability-classification"]], "Stability function": [[94, "stability-function"]], "Stability conditions": [[94, "stability-conditions"]], "Single step methods and parabolic equations": [[94, "single-step-methods-and-parabolic-equations"]], "Single step methods and hyperbolic equations": [[94, "single-step-methods-and-hyperbolic-equations"]], "Exercises:": [[94, "exercises"]], "Heat Equation": [[96, "heat-equation"]], "Variational formulation in space": [[96, "variational-formulation-in-space"]], "Galerkin method in space": [[96, "galerkin-method-in-space"]], "Implicit Euler time-stepping": [[96, "implicit-euler-time-stepping"]], "Wave Equation": [[97, "wave-equation"]], "Newmark time-stepping method": [[97, "newmark-time-stepping-method"]], "Exponential Integrators for Parabolic Equations": [[98, "exponential-integrators-for-parabolic-equations"]], "Mass-lumping and Local time-stepping": [[101, "mass-lumping-and-local-time-stepping"]], "The Verlet method:": [[101, "the-verlet-method"]], "Geometry with local details:": [[101, "geometry-with-local-details"]], "Nano-optics: A ring-resonator": [[102, "nano-optics-a-ring-resonator"], [103, "nano-optics-a-ring-resonator"]], "Discontinuous Galerkin for the Wave Equation": [[104, "discontinuous-galerkin-for-the-wave-equation"]], "Testing the differential operators": [[104, "testing-the-differential-operators"]], "Efficient implementation:": [[104, "efficient-implementation"]], "Solving the wave equation:": [[104, "solving-the-wave-equation"]], "Eigenvalues of the discretized Laplace-operator": [[104, "eigenvalues-of-the-discretized-laplace-operator"]], "Inf-sup stable variational problems": [[23, "inf-sup-stable-variational-problems"]], "Approximation of inf-sup stable variational problems": [[23, "approximation-of-inf-sup-stable-variational-problems"]], "Solving the Poisson Equation": [[64, "solving-the-poisson-equation"]], "Weak formulation": [[64, "weak-formulation"]], "The Sobolev space H^1, linear and bilinear forms": [[64, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Finite Element Method": [[64, "the-finite-element-method"]], "Poisson equation in NGSolve:": [[64, "poisson-equation-in-ngsolve"]], "Visualizing the basis functions": [[64, "visualizing-the-basis-functions"]], "Domain Decomposition with minimal overlap": [[91, "domain-decomposition-with-minimal-overlap"]], "Analysis of the method": [[91, "analysis-of-the-method"]], "Adding a coarse grid space": [[91, "adding-a-coarse-grid-space"]], "Graph-based mesh partitioning": [[91, "graph-based-mesh-partitioning"]], "Literature": [[34, "literature"]], "Installing NGSolve": [[34, "installing-ngsolve"]], "Sobolev Spaces": [[34, null]], "Non-linear dynamics": [[47, "non-linear-dynamics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[59, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]]}, "indexentries": {}})
\ No newline at end of file