diff --git a/_sources/aposteriori/aposteriori.ipynb b/_sources/aposteriori/aposteriori.ipynb index 3efc6014..80292b86 100644 --- a/_sources/aposteriori/aposteriori.ipynb +++ b/_sources/aposteriori/aposteriori.ipynb @@ -8,7 +8,7 @@ "# A posteriori error estimates\n", "\n", "We will derive methods to estimate the error of the computed finite\n", - "element approximation. Such {\\em a posteriori} error estimates may use the finite element solution $u_h$, and input data such as the source term $f$. \n", + "element approximation. Such **a posteriori** error estimates may use the finite element solution $u_h$, and input data such as the source term $f$. \n", "\n", "$$\n", "\\eta(u_h, f)\n", @@ -93,7 +93,7 @@ "$$\n", "\\| p - \\tilde p_h \\|_{L_2} \\leq \\alpha \\, \\| p - p_h \\|_{L_2}\n", "$$\n", - "holds with a small constant $\\alpha \\ll 1$. This property is known as *em super-convergence*.It is indeed true on (locally) uniform meshes, and smoothness assumptions onto the source term $f$. \n", + "holds with a small constant $\\alpha \\ll 1$. This property is known as *super-convergence*.It is indeed true on (locally) uniform meshes, and smoothness assumptions onto the source term $f$. \n", "\n", "The ZZ error estimator replaces the true gradient in the error $p-p_h$ by the good approximation $\\tilde p_h$:\n", "\n", @@ -117,7 +117,7 @@ "It is also efficient, a similar short application of the triangle inequality.\n", "\n", "There is a rigorous analysis of the ZZ error estimator, e.g., by showing equivalence\n", - "to the following residual error estimator." + "to the residual error estimator below." ] }, { @@ -348,7 +348,7 @@ "id": "a5b25c26-17d4-49a9-aa5e-cb6b14bda367", "metadata": {}, "source": [ - "We observe a very strong refinement along the material interface. This is unnecessary, since the solution is smooth on both sides, we only expect singularities at the corners. The problem is that the flux-averaging did an averaging of the full flux vector. The tangential component of it is not supposed to be continuous, and this is wrongly measured as error:" + "We observe a very strong refinement along the material interface. This is unnecessary, since the solution is smooth on both sides, we only expect singularities at the corners. The problem is that the flux-averaging did an averaging of the full flux vector. The tangential component of it is not supposed to be continuous, and this it highly overestimates the error:" ] }, { @@ -392,7 +392,7 @@ "source": [ "### Flux recovery in $H(\\operatorname{div})$\n", "\n", - "The over-refinement along the edges can be overcome by averaging only the normal component of the flux, since only this is supposed to be continuous. This we obtain by recovering the flux in an $H(\\operatorname{div})$ finite element spaces " + "The over-refinement along the edges can be overcome by averaging only the normal component of the flux, since only this is supposed to be continuous. This we obtain by recovering the flux in an $H(\\operatorname{div})$ finite element space: " ] }, { diff --git a/_sources/aposteriori/residualEE.ipynb b/_sources/aposteriori/residualEE.ipynb index ff6ca128..e42460d3 100644 --- a/_sources/aposteriori/residualEE.ipynb +++ b/_sources/aposteriori/residualEE.ipynb @@ -109,20 +109,22 @@ "*Proof:* First, choose a reference patch $\\widehat \\omega_T$ of \n", "dimension \n", "$\\simeq 1$. The quasi-interpolation operator is bounded on $H^1(\\omega_T)$:\n", - "\\begin{equation}\n", + "\n", + "$$\n", "% \\label{equ_clement_bh}\n", "\\| v - \\Pi_h v \\|_{L_2(\\widehat T)} + \\| \\nabla (v - \\Pi_h v) \\|_{L_2(\\widehat T)} \\preceq \\| v \\|_{H^1(\\widehat \\omega_T)}\n", - "\\end{equation}\n", + "$$\n", "If $v$ is constant on $\\omega_T$, then the mean values in the vertices\n", "take the same values, and also $(\\Pi_h v)_{|T}$ is the same constant.\n", "The constant function (on $\\omega_T$) is in the kernel of \n", "$\\| v - \\Pi_h v \\|_{H^1(T)}$. Due to\n", "the Bramble-Hilbert lemma, we can replace the norm on the right hand side\n", - "of (\\ref{equ_clement_bh}) by the semi-norm:\n", - "\\begin{equation}\n", + " by the semi-norm:\n", + "\n", + "$$ \n", "% \\label{equ_clement_bh2}\n", "\\| v - \\Pi_h v \\|_{L_2(\\widehat T)} + \\| \\nabla (v - \\Pi_h v) \\|_{L_2(\\widehat T)} \\preceq \\| \\nabla v \\|_{L_2(\\widehat \\omega_T)}\n", - "\\end{equation}\n", + "$$\n", "\n", "The rest follows from scaling. Let $F : x \\rightarrow h x$ scale the reference patch $\\widehat \\omega_T$ to the actual patch $\\omega_T$. Then\n", "\n", diff --git a/aposteriori/aposteriori.html b/aposteriori/aposteriori.html index 77d01cad..ac4e00dd 100644 --- a/aposteriori/aposteriori.html +++ b/aposteriori/aposteriori.html @@ -59,7 +59,7 @@ - + @@ -504,7 +504,7 @@

Contents

26. A posteriori error estimates#

We will derive methods to estimate the error of the computed finite -element approximation. Such {\em a posteriori} error estimates may use the finite element solution \(u_h\), and input data such as the source term \(f\).

+element approximation. Such a posteriori error estimates may use the finite element solution \(u_h\), and input data such as the source term \(f\).

\[ \eta(u_h, f) @@ -567,7 +567,7 @@

26.1. The Zienkiewicz Zhu error estimato \[ \| p - \tilde p_h \|_{L_2} \leq \alpha \, \| p - p_h \|_{L_2} \]

-

holds with a small constant \(\alpha \ll 1\). This property is known as em super-convergence.It is indeed true on (locally) uniform meshes, and smoothness assumptions onto the source term \(f\).

+

holds with a small constant \(\alpha \ll 1\). This property is known as super-convergence.It is indeed true on (locally) uniform meshes, and smoothness assumptions onto the source term \(f\).

The ZZ error estimator replaces the true gradient in the error \(p-p_h\) by the good approximation \(\tilde p_h\):

\[ @@ -588,7 +588,7 @@

26.1. The Zienkiewicz Zhu error estimato \]

It is also efficient, a similar short application of the triangle inequality.

There is a rigorous analysis of the ZZ error estimator, e.g., by showing equivalence -to the following residual error estimator.

+to the residual error estimator below.

26.2. Mesh refinement algorithms#

@@ -688,7 +688,7 @@

26.3. Flux-recovery error estimates with
-
+
@@ -743,7 +743,7 @@

26.3.1. Variable coefficients -

+
@@ -757,7 +757,7 @@

26.3.1. Variable coefficients

-

We observe a very strong refinement along the material interface. This is unnecessary, since the solution is smooth on both sides, we only expect singularities at the corners. The problem is that the flux-averaging did an averaging of the full flux vector. The tangential component of it is not supposed to be continuous, and this is wrongly measured as error:

+

We observe a very strong refinement along the material interface. This is unnecessary, since the solution is smooth on both sides, we only expect singularities at the corners. The problem is that the flux-averaging did an averaging of the full flux vector. The tangential component of it is not supposed to be continuous, and this it highly overestimates the error:

Draw (lam*grad(gfu)[0], mesh, order=2, min=-0.5, max=0.5);
@@ -765,7 +765,7 @@ 
26.3.1. Variable coefficients
-
+
@@ -774,7 +774,7 @@

26.3.1. Variable coefficients -

+
@@ -783,12 +783,12 @@

26.3.1. Variable coefficients -

+

26.3.2. Flux recovery in \(H(\operatorname{div})\)#

-

The over-refinement along the edges can be overcome by averaging only the normal component of the flux, since only this is supposed to be continuous. This we obtain by recovering the flux in an \(H(\operatorname{div})\) finite element spaces

+

The over-refinement along the edges can be overcome by averaging only the normal component of the flux, since only this is supposed to be continuous. This we obtain by recovering the flux in an \(H(\operatorname{div})\) finite element space:

mesh = MakeTwoMaterialMesh()
@@ -812,7 +812,7 @@ 
26.3.2. Flux recovery in 
-
+
diff --git a/aposteriori/residualEE.html b/aposteriori/residualEE.html index caefa426..ffb9df3c 100644 --- a/aposteriori/residualEE.html +++ b/aposteriori/residualEE.html @@ -583,22 +583,22 @@

27.1. The Clément- operator\(\widehat \omega_T\) of dimension \(\simeq 1\). The quasi-interpolation operator is bounded on \(H^1(\omega_T)\):

-
-(27.1)#\[\begin{equation} +
+\[ % \label{equ_clement_bh} \| v - \Pi_h v \|_{L_2(\widehat T)} + \| \nabla (v - \Pi_h v) \|_{L_2(\widehat T)} \preceq \| v \|_{H^1(\widehat \omega_T)} -\end{equation}\]
+\]

If \(v\) is constant on \(\omega_T\), then the mean values in the vertices take the same values, and also \((\Pi_h v)_{|T}\) is the same constant. The constant function (on \(\omega_T\)) is in the kernel of \(\| v - \Pi_h v \|_{H^1(T)}\). Due to the Bramble-Hilbert lemma, we can replace the norm on the right hand side -of (\ref{equ_clement_bh}) by the semi-norm:

-
-(27.2)#\[\begin{equation} +by the semi-norm:

+
+\[ % \label{equ_clement_bh2} \| v - \Pi_h v \|_{L_2(\widehat T)} + \| \nabla (v - \Pi_h v) \|_{L_2(\widehat T)} \preceq \| \nabla v \|_{L_2(\widehat \omega_T)} -\end{equation}\]
+\]

The rest follows from scaling. Let \(F : x \rightarrow h x\) scale the reference patch \(\widehat \omega_T\) to the actual patch \(\omega_T\). Then

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Introduction to the Finite Element Method": [[45, "an-interactive-introduction-to-the-finite-element-method"], [46, "an-interactive-introduction-to-the-finite-element-method"]], "Analysis of the 2-level method:": [[109, "analysis-of-the-2-level-method"]], "Analysis of the DD preconditioner": [[109, "analysis-of-the-dd-preconditioner"]], "Analysis of the Multigrid Iteration": [[64, "analysis-of-the-multigrid-iteration"]], "Analysis of the method": [[108, "analysis-of-the-method"]], "Analysis of the multi-level preconditioner": [[65, "analysis-of-the-multi-level-preconditioner"]], "Appendix": [[45, "appendix"]], "Application of the abstract theory": [[89, "application-of-the-abstract-theory"]], "Approximation of Dirichlet boundary conditions": [[8, "approximation-of-dirichlet-boundary-conditions"]], "Approximation of coercive variational problems": [[28, "approximation-of-coercive-variational-problems"]], "Approximation of inf-sup stable variational problems": [[31, "approximation-of-inf-sup-stable-variational-problems"]], "BDDC - Preconditioner": [[37, "bddc-preconditioner"]], "Basic Iterative Methods": [[52, "basic-iterative-methods"]], "Basic properties": [[27, "basic-properties"]], "Block-Jacobi and general additive Schwarz preconditioners": [[105, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Block-preconditioning": [[84, "block-preconditioning"]], "Boundary Conditions": [[55, "boundary-conditions"], [73, "boundary-conditions"]], "Bramble-Hilbert Lemma": [[102, "bramble-hilbert-lemma"]], "Building systems from building-blocks": [[30, "building-systems-from-building-blocks"]], "Butcher tableaus of simple methods": [[111, "butcher-tableaus-of-simple-methods"]], "Chebyshev polynomials": [[47, "chebyshev-polynomials"]], "Coercive examples": [[30, "coercive-examples"]], "Coercive variational problems and their approximation": [[28, "coercive-variational-problems-and-their-approximation"]], "Commuting diagram for H^1 - H(\\opcurl)": [[68, "commuting-diagram-for-h-1-h-opcurl"]], "Comparison to DD with minimal overlap": [[109, "comparison-to-dd-with-minimal-overlap"]], "Computation of element-vectors and element-matrices": [[13, "computation-of-element-vectors-and-element-matrices"]], "Computation of the lifting \\| \\sigma^\\Delta \\|": [[34, "computation-of-the-lifting-sigma-delta"]], "Computing dual norms": [[75, "computing-dual-norms"]], "Conjugate Gradients": [[49, "conjugate-gradients"]], "Consistent and Distributed Vectors": [[24, "consistent-and-distributed-vectors"]], "Consistent vectors:": [[24, "consistent-vectors"]], "Constrained minimization problem": [[53, "constrained-minimization-problem"]], "Constrained minimization problems": [[84, "constrained-minimization-problems"]], "Convergence of Runge Kutta methods": [[111, "convergence-of-runge-kutta-methods"]], "DG - Methods for elliptic problems": [[2, "dg-methods-for-elliptic-problems"]], "Diagonal preconditioner for L_2-norm": [[106, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[106, "diagonal-preconditioner-for-the-h-1-norm"]], "Diagonally implicit Runge-Kutta methods:": [[111, "diagonally-implicit-runge-kutta-methods"]], "Dirichlet boundary conditions as mixed system": [[53, "dirichlet-boundary-conditions-as-mixed-system"]], "Dirichlet boundary conditions by penalty:": [[83, "dirichlet-boundary-conditions-by-penalty"]], "Discontinuous Galerkin Methods": [[45, "discontinuous-galerkin-methods"], [46, null]], "Discontinuous Galerkin for the Wave Equation": [[121, "discontinuous-galerkin-for-the-wave-equation"]], "Discontinuous Galerkin method": [[6, "discontinuous-galerkin-method"]], "Distributed Meshes and Spaces": [[15, "distributed-meshes-and-spaces"]], "Distributed finite element spaces": [[15, "distributed-finite-element-spaces"]], "Distributed vectors and matrices": [[24, "distributed-vectors-and-matrices"]], "Domain Decomposition with Lagrange parameters": [[40, "domain-decomposition-with-lagrange-parameters"]], "Domain Decomposition with minimal overlap": [[108, "domain-decomposition-with-minimal-overlap"]], "Dual mixed formulation": [[89, "dual-mixed-formulation"]], "Dual mixed method": [[60, "dual-mixed-method"]], "Efficiency of the residual error estimator": [[36, "efficiency-of-the-residual-error-estimator"]], "Efficient implementation:": [[121, "efficient-implementation"]], "Efficiently computable multi-level decomposition": [[66, "efficiently-computable-multi-level-decomposition"]], "Eigenvalues of the discretized Laplace-operator": [[121, "eigenvalues-of-the-discretized-laplace-operator"]], "Eqivalent versions of the Poincar\u00e9 inequality": [[102, "eqivalent-versions-of-the-poincare-inequality"]], "Equilibrated Residual Error Estimates": [[34, "equilibrated-residual-error-estimates"]], "Equivalent norms on H^1 and on sub-spaces": [[101, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Error Analysis in L_2 \\times H^1": [[87, "error-analysis-in-l-2-times-h-1"]], "Error estimate of the L_2 projection": [[79, "error-estimate-of-the-l-2-projection"]], "Error estimates in L_2-norm": [[8, "error-estimates-in-l-2-norm"]], "Error estimates:": [[62, "error-estimates"]], "Essential boundary conditions": [[73, "essential-boundary-conditions"]], "Example": [[35, "example"]], "Example: Dirichlet boundary condition by penalty": [[56, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Finite elements for Stokes": [[54, "example-finite-elements-for-stokes"]], "Example: Nearly incompressible materials": [[56, "example-nearly-incompressible-materials"]], "Examples": [[84, "examples"], [111, "examples"]], "Exercise A:": [[107, "exercise-a"]], "Exercise B:": [[107, "exercise-b"]], "Exercise:": [[11, "exercise"], [111, "exercise"], [121, "exercise"]], "Exercise: Robust preconditioners": [[107, "exercise-robust-preconditioners"]], "Exercises": [[30, "exercises"], [75, "exercises"], [102, "exercises"]], "Exercises:": [[112, "exercises"]], "Expanding the Krylov-space": [[49, "expanding-the-krylov-space"]], "Experiment with CoefficientFunctions": [[75, "experiment-with-coefficientfunctions"]], "Experiments with BilienarForms and LinearForms": [[75, "experiments-with-bilienarforms-and-linearforms"]], "Experiments with norms": [[103, "experiments-with-norms"]], "Experiments with overlapping DD": [[109, "experiments-with-overlapping-dd"]], "Experiments with the Richardson iteration": [[48, "experiments-with-the-richardson-iteration"]], "Explicit Euler method (EE)": [[112, "explicit-euler-method-ee"]], "Explicit methods:": [[111, "explicit-methods"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[112, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Exponential Integrators for Parabolic Equations": [[115, "exponential-integrators-for-parabolic-equations"]], "Extending boundary data": [[66, "extending-boundary-data"]], "Extension operators": [[100, "extension-operators"]], "FETI methods": [[38, "feti-methods"]], "FETI-DP": [[39, "feti-dp"]], "Finite Element Error Analysis": [[86, "finite-element-error-analysis"]], "Finite Element Method": [[9, "finite-element-method"], [46, null]], "Finite Element Spaces": [[58, "finite-element-spaces"]], "Finite Elements in H(\\operatorname{div})": [[88, "finite-elements-in-h-operatorname-div"]], "Finite element error analysis": [[8, "finite-element-error-analysis"]], "Finite element error estimates": [[54, "finite-element-error-estimates"]], "Finite element system assembling": [[13, "finite-element-system-assembling"]], "Flux recovery in H(\\operatorname{div})": [[33, "flux-recovery-in-h-operatorname-div"]], "Flux-recovery error estimates with NGSolve": [[33, "flux-recovery-error-estimates-with-ngsolve"]], "Fourth Order Equation": [[3, "fourth-order-equation"]], "Fractional Sobolev spaces": [[102, "fractional-sobolev-spaces"]], "Friedrichs\u2019 Inequality": [[92, "friedrichs-inequality"], [95, 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"graph-based-mesh-partitioning"]], "Grating": [[44, "grating"]], "H(\\opdiv) on sub-domains": [[90, "h-opdiv-on-sub-domains"]], "H(div)-conforming Stokes": [[7, "h-div-conforming-stokes"]], "H^1-norm with small L_2-term": [[106, "h-1-norm-with-small-l-2-term"]], "Heat Equation": [[113, "heat-equation"]], "Hellan-Herrmann-Johnson method": [[67, "hellan-herrmann-johnson-method"]], "Helmholtz Equation": [[43, "helmholtz-equation"]], "High Order Finite Elements": [[46, null]], "High order elements": [[8, "high-order-elements"]], "Hybrid DG for elliptic equations": [[1, "hybrid-dg-for-elliptic-equations"]], "Hybrid Interfaces": [[0, "hybrid-interfaces"]], "Hybridization Techniques": [[91, "hybridization-techniques"]], "Hybridized C^0-continuous interior penalty method:": [[3, "hybridized-c-0-continuous-interior-penalty-method"]], "Implement a parallel BDDC preconditioner": [[42, "implement-a-parallel-bddc-preconditioner"]], "Implement our own system assembling": [[11, "implement-our-own-system-assembling"]], "Implementation of Finite Elements": [[10, "implementation-of-finite-elements"]], "Implementation of High Order Finite Elements": [[80, "implementation-of-high-order-finite-elements"]], "Implicit Euler method (IE)": [[112, "implicit-euler-method-ie"]], "Implicit Euler time-stepping": [[113, "implicit-euler-time-stepping"]], "Inf-sup stable variational problems": [[31, "inf-sup-stable-variational-problems"]], "Inner products:": [[24, "inner-products"]], "Installing NGSolve": [[46, "installing-ngsolve"]], "Instationary Transport Equation": [[4, "instationary-transport-equation"]], "Integration by parts": [[100, "integration-by-parts"]], "Interfaces": [[0, "interfaces"]], "Interpolation space H^s": [[41, "interpolation-space-h-s"]], "Interpolation spaces": [[100, "interpolation-spaces"]], "Introduction to MPI with mpi4py": [[20, "introduction-to-mpi-with-mpi4py"]], "Introduction to Non-overlapping Domain Decomposition": [[40, 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[46, null]], "Multigrid Preconditioning": [[63, "multigrid-preconditioning"]], "Multigrid and Multilevel Methods": [[63, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[63, "multilevel-preconditioner"]], "NGSolve - PETSc interface": [[14, "ngsolve-petsc-interface"]], "Nano-optics: A ring-resonator": [[119, "nano-optics-a-ring-resonator"], [120, "nano-optics-a-ring-resonator"]], "Natural boundary conditions": [[73, "natural-boundary-conditions"]], "Natural trace space": [[41, "natural-trace-space"]], "Nearly optimal analysis of the ML - preconditioner": [[65, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Newmark time-stepping method": [[114, "newmark-time-stepping-method"]], "Nitsche\u2019s Method for boundary and interface conditions": [[0, "nitsche-s-method-for-boundary-and-interface-conditions"]], "Nitsche\u2019s method:": [[0, "nitsche-s-method"]], "Non-conforming Finite Element Methods": [[12, "non-conforming-finite-element-methods"]], "Non-linear 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97, 100, 103], "transform": [81, 88], "transport": [4, 6], "trapezoid": 112, "triangl": [12, 79], "two": 83, "upper": 105, "us": [11, 23], "v": 64, "valu": [30, 102], "varepsilon": 61, "variabl": [33, 78], "variat": [28, 29, 31, 58, 62, 74, 113], "vector": [13, 24], "verif": 97, "verlet": 118, "version": 102, "vertex": 8, "visual": 76, "wave": [45, 114, 121], "we": 11, "weak": [76, 104], "wise": 62, "within": 53, "work": 75, "zhu": 33, "zienkiewicz": 33}}) \ No newline at end of file +Search.setIndex({"alltitles": {"3D Solid Mechanics": [[74, "d-solid-mechanics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[71, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]], "A Small Number of Constraints": [[82, "a-small-number-of-constraints"]], "A posteriori error estimates": [[33, "a-posteriori-error-estimates"], [46, null]], "A practical introduction": [[45, "a-practical-introduction"]], "Abstract Theory": [[45, "abstract-theory"], [46, null], [53, "abstract-theory"]], "Abstract theory for mixed finite element methods": [[54, "abstract-theory-for-mixed-finite-element-methods"]], "Adding a coarse grid space": [[108, "adding-a-coarse-grid-space"]], "Additional methods": [[45, "additional-methods"]], "Additive Schwarz Methods": [[105, "additive-schwarz-methods"]], "Algorithm": [[66, "algorithm"]], "An Interactive Introduction to the Finite Element Method": [[45, "an-interactive-introduction-to-the-finite-element-method"], [46, "an-interactive-introduction-to-the-finite-element-method"]], "Analysis of the 2-level method:": [[109, "analysis-of-the-2-level-method"]], "Analysis of the DD preconditioner": [[109, "analysis-of-the-dd-preconditioner"]], "Analysis of the Multigrid Iteration": [[64, "analysis-of-the-multigrid-iteration"]], "Analysis of the method": [[108, "analysis-of-the-method"]], "Analysis of the multi-level preconditioner": [[65, "analysis-of-the-multi-level-preconditioner"]], "Appendix": [[45, "appendix"]], "Application of the abstract theory": [[89, "application-of-the-abstract-theory"]], "Approximation of Dirichlet boundary conditions": [[8, "approximation-of-dirichlet-boundary-conditions"]], "Approximation of coercive variational problems": [[28, "approximation-of-coercive-variational-problems"]], "Approximation of inf-sup stable variational problems": [[31, "approximation-of-inf-sup-stable-variational-problems"]], "BDDC - Preconditioner": [[37, "bddc-preconditioner"]], "Basic Iterative Methods": [[52, "basic-iterative-methods"]], "Basic properties": [[27, "basic-properties"]], "Block-Jacobi and general additive Schwarz preconditioners": [[105, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Block-preconditioning": [[84, "block-preconditioning"]], "Boundary Conditions": [[55, "boundary-conditions"], [73, "boundary-conditions"]], "Bramble-Hilbert Lemma": [[102, "bramble-hilbert-lemma"]], "Building systems from building-blocks": [[30, "building-systems-from-building-blocks"]], "Butcher tableaus of simple methods": [[111, "butcher-tableaus-of-simple-methods"]], "Chebyshev polynomials": [[47, "chebyshev-polynomials"]], "Coercive examples": [[30, "coercive-examples"]], "Coercive variational problems and their approximation": [[28, "coercive-variational-problems-and-their-approximation"]], "Commuting diagram for H^1 - H(\\opcurl)": [[68, "commuting-diagram-for-h-1-h-opcurl"]], "Comparison to DD with minimal overlap": [[109, "comparison-to-dd-with-minimal-overlap"]], "Computation of element-vectors and element-matrices": [[13, "computation-of-element-vectors-and-element-matrices"]], "Computation of the lifting \\| \\sigma^\\Delta \\|": [[34, "computation-of-the-lifting-sigma-delta"]], "Computing dual norms": [[75, "computing-dual-norms"]], "Conjugate Gradients": [[49, "conjugate-gradients"]], "Consistent and Distributed Vectors": [[24, "consistent-and-distributed-vectors"]], "Consistent vectors:": [[24, "consistent-vectors"]], "Constrained minimization problem": [[53, "constrained-minimization-problem"]], "Constrained minimization problems": [[84, "constrained-minimization-problems"]], "Convergence of Runge Kutta methods": [[111, "convergence-of-runge-kutta-methods"]], "DG - Methods for elliptic problems": [[2, "dg-methods-for-elliptic-problems"]], "Diagonal preconditioner for L_2-norm": [[106, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[106, "diagonal-preconditioner-for-the-h-1-norm"]], "Diagonally implicit Runge-Kutta methods:": [[111, "diagonally-implicit-runge-kutta-methods"]], "Dirichlet boundary conditions as mixed system": [[53, "dirichlet-boundary-conditions-as-mixed-system"]], "Dirichlet boundary conditions by penalty:": [[83, "dirichlet-boundary-conditions-by-penalty"]], "Discontinuous Galerkin Methods": [[45, "discontinuous-galerkin-methods"], [46, null]], "Discontinuous Galerkin for the Wave Equation": [[121, "discontinuous-galerkin-for-the-wave-equation"]], "Discontinuous Galerkin method": [[6, "discontinuous-galerkin-method"]], "Distributed Meshes and Spaces": [[15, "distributed-meshes-and-spaces"]], "Distributed finite element spaces": [[15, "distributed-finite-element-spaces"]], "Distributed vectors and matrices": [[24, "distributed-vectors-and-matrices"]], "Domain Decomposition with Lagrange parameters": [[40, "domain-decomposition-with-lagrange-parameters"]], "Domain Decomposition with minimal overlap": [[108, "domain-decomposition-with-minimal-overlap"]], "Dual mixed formulation": [[89, "dual-mixed-formulation"]], "Dual mixed method": [[60, "dual-mixed-method"]], "Efficiency of the residual error estimator": [[36, "efficiency-of-the-residual-error-estimator"]], "Efficient implementation:": [[121, "efficient-implementation"]], "Efficiently computable multi-level decomposition": [[66, "efficiently-computable-multi-level-decomposition"]], "Eigenvalues of the discretized Laplace-operator": [[121, "eigenvalues-of-the-discretized-laplace-operator"]], "Eqivalent versions of the Poincar\u00e9 inequality": [[102, "eqivalent-versions-of-the-poincare-inequality"]], "Equilibrated Residual Error Estimates": [[34, "equilibrated-residual-error-estimates"]], "Equivalent norms on H^1 and on sub-spaces": [[101, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Error Analysis in L_2 \\times H^1": [[87, "error-analysis-in-l-2-times-h-1"]], "Error estimate of the L_2 projection": [[79, "error-estimate-of-the-l-2-projection"]], "Error estimates in L_2-norm": [[8, "error-estimates-in-l-2-norm"]], "Error estimates:": [[62, "error-estimates"]], "Essential boundary conditions": [[73, "essential-boundary-conditions"]], "Example": [[35, "example"]], "Example: Dirichlet boundary condition by penalty": [[56, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Finite elements for Stokes": [[54, "example-finite-elements-for-stokes"]], "Example: Nearly incompressible materials": [[56, "example-nearly-incompressible-materials"]], "Examples": [[84, "examples"], [111, "examples"]], "Exercise A:": [[107, "exercise-a"]], "Exercise B:": [[107, "exercise-b"]], "Exercise:": [[11, "exercise"], [111, "exercise"], [121, "exercise"]], "Exercise: Robust preconditioners": [[107, "exercise-robust-preconditioners"]], "Exercises": [[30, "exercises"], [75, "exercises"], [102, "exercises"]], "Exercises:": [[112, "exercises"]], "Expanding the Krylov-space": [[49, "expanding-the-krylov-space"]], "Experiment with CoefficientFunctions": [[75, "experiment-with-coefficientfunctions"]], "Experiments with BilienarForms and LinearForms": [[75, "experiments-with-bilienarforms-and-linearforms"]], "Experiments with norms": [[103, "experiments-with-norms"]], "Experiments with overlapping DD": [[109, "experiments-with-overlapping-dd"]], "Experiments with the Richardson iteration": [[48, "experiments-with-the-richardson-iteration"]], "Explicit Euler method (EE)": [[112, "explicit-euler-method-ee"]], "Explicit methods:": [[111, "explicit-methods"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[112, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Exponential Integrators for Parabolic Equations": [[115, "exponential-integrators-for-parabolic-equations"]], "Extending boundary data": [[66, "extending-boundary-data"]], "Extension operators": [[100, "extension-operators"]], "FETI methods": [[38, "feti-methods"]], "FETI-DP": [[39, "feti-dp"]], "Finite Element Error Analysis": [[86, "finite-element-error-analysis"]], "Finite Element Method": [[9, "finite-element-method"], [46, null]], "Finite Element Spaces": [[58, "finite-element-spaces"]], "Finite Elements in H(\\operatorname{div})": [[88, "finite-elements-in-h-operatorname-div"]], "Finite element error analysis": [[8, "finite-element-error-analysis"]], "Finite element error estimates": [[54, "finite-element-error-estimates"]], "Finite element system assembling": [[13, "finite-element-system-assembling"]], "Flux recovery in H(\\operatorname{div})": [[33, "flux-recovery-in-h-operatorname-div"]], "Flux-recovery error estimates with NGSolve": [[33, "flux-recovery-error-estimates-with-ngsolve"]], "Fourth Order Equation": [[3, "fourth-order-equation"]], "Fractional Sobolev spaces": [[102, "fractional-sobolev-spaces"]], "Friedrichs\u2019 Inequality": [[92, "friedrichs-inequality"], [95, "friedrichs-inequality"]], "Friedrichs\u2019 inequality": [[102, "friedrichs-inequality"], [103, "friedrichs-inequality"]], "Friedrichs\u2019 inequality in 1D": [[102, "friedrichs-inequality-in-1d"]], "Friedrichs\u2019 inequality on the square": [[102, "friedrichs-inequality-on-the-square"]], "Galerkin method in space": [[113, "galerkin-method-in-space"]], "General definition:": [[100, "general-definition"]], "General framework": [[34, "general-framework"]], "Generalized derivatives": [[98, "generalized-derivatives"]], "Geometric mesh refinement": [[8, "geometric-mesh-refinement"]], "Geometry with local details:": [[118, "geometry-with-local-details"]], "Get started with Netgen-Opencascade geometric modeling": [[75, "get-started-with-netgen-opencascade-geometric-modeling"]], "Goal driven error estimates": [[35, "goal-driven-error-estimates"]], "Graded meshes around vertex singularities": [[8, "graded-meshes-around-vertex-singularities"]], "Graph-based mesh partitioning": [[108, "graph-based-mesh-partitioning"]], "Grating": [[44, "grating"]], "H(\\opdiv) on sub-domains": [[90, "h-opdiv-on-sub-domains"]], "H(div)-conforming Stokes": [[7, "h-div-conforming-stokes"]], "H^1-norm with small L_2-term": [[106, "h-1-norm-with-small-l-2-term"]], "Heat Equation": [[113, "heat-equation"]], "Hellan-Herrmann-Johnson method": [[67, "hellan-herrmann-johnson-method"]], "Helmholtz Equation": [[43, "helmholtz-equation"]], "High Order Finite Elements": [[46, null]], "High order elements": [[8, "high-order-elements"]], "Hybrid DG for elliptic equations": [[1, "hybrid-dg-for-elliptic-equations"]], "Hybrid Interfaces": [[0, "hybrid-interfaces"]], "Hybridization Techniques": [[91, "hybridization-techniques"]], "Hybridized C^0-continuous interior penalty method:": [[3, "hybridized-c-0-continuous-interior-penalty-method"]], "Implement a parallel BDDC preconditioner": [[42, "implement-a-parallel-bddc-preconditioner"]], "Implement our own system assembling": [[11, "implement-our-own-system-assembling"]], "Implementation of Finite Elements": [[10, "implementation-of-finite-elements"]], "Implementation of High Order Finite Elements": [[80, "implementation-of-high-order-finite-elements"]], "Implicit Euler method (IE)": [[112, "implicit-euler-method-ie"]], "Implicit Euler time-stepping": [[113, "implicit-euler-time-stepping"]], "Inf-sup stable variational problems": [[31, "inf-sup-stable-variational-problems"]], "Inner products:": [[24, "inner-products"]], "Installing NGSolve": [[46, "installing-ngsolve"]], "Instationary Transport Equation": [[4, "instationary-transport-equation"]], "Integration by parts": [[100, "integration-by-parts"]], "Interfaces": [[0, "interfaces"]], "Interpolation space H^s": [[41, "interpolation-space-h-s"]], "Interpolation spaces": [[100, "interpolation-spaces"]], "Introduction to MPI with mpi4py": [[20, "introduction-to-mpi-with-mpi4py"]], "Introduction to Non-overlapping Domain Decomposition": [[40, "introduction-to-non-overlapping-domain-decomposition"]], "Inverse estimates": [[103, "inverse-estimates"]], "Iteration Methods": [[45, "iteration-methods"], [46, null]], "Iteration methods in parallel": [[21, "iteration-methods-in-parallel"]], "Iterative Solvers": [[45, "iterative-solvers"], [77, "iterative-solvers"]], "Jacobi and Gauss Seidel Preconditioners": [[51, "jacobi-and-gauss-seidel-preconditioners"]], "Kirchhoff Plate equation": [[67, "kirchhoff-plate-equation"]], "Korn\u2019s inequality": [[103, "korn-s-inequality"]], "Legendre Polynomials": [[79, "legendre-polynomials"]], "Linear Algebra": [[75, "linear-algebra"]], "Linear elasticity": [[74, "linear-elasticity"]], "Linear stability classification": [[112, "linear-stability-classification"]], "Literature": [[46, "literature"]], "Local post-processing": [[86, "local-post-processing"]], "Marked edge bisection": [[33, "marked-edge-bisection"]], "Mass-lumping and Local time-stepping": [[118, "mass-lumping-and-local-time-stepping"]], "Matrix vector multiplication:": [[24, "matrix-vector-multiplication"]], "Maxwell equations:": [[83, "maxwell-equations"]], "Mean-value interpolation": [[102, "mean-value-interpolation"]], "Mesh refinement algorithms": [[33, "mesh-refinement-algorithms"]], "Minimization problem": [[30, "minimization-problem"]], "Mixed Finite Element Methods": [[45, "mixed-finite-element-methods"], [46, null]], "Mixed Methods for Elasticity": [[45, "mixed-methods-for-elasticity"]], "Mixed Methods for Plates and Shells": [[45, "mixed-methods-for-plates-and-shells"]], "Mixed Methods for Second Order Equations": [[45, "mixed-methods-for-second-order-equations"], [46, null]], "Mixed Methods for second order equations": [[57, "mixed-methods-for-second-order-equations"]], "Mixed method for second order equation": [[53, "mixed-method-for-second-order-equation"]], "Multi-level Extension": [[66, "multi-level-extension"]], "Multigrid Methods": [[45, "multigrid-methods"], [46, null]], "Multigrid Preconditioning": [[63, "multigrid-preconditioning"]], "Multigrid and Multilevel Methods": [[63, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[63, "multilevel-preconditioner"]], "NGSolve - PETSc interface": [[14, "ngsolve-petsc-interface"]], "Nano-optics: A ring-resonator": [[119, "nano-optics-a-ring-resonator"], [120, "nano-optics-a-ring-resonator"]], "Natural boundary conditions": [[73, "natural-boundary-conditions"]], "Natural trace space": [[41, "natural-trace-space"]], "Nearly optimal analysis of the ML - preconditioner": [[65, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Newmark time-stepping method": [[114, "newmark-time-stepping-method"]], "Nitsche\u2019s Method 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Boundary Conditions", "37. Parameter Dependent Problems", "34. Mixed Methods for second order equations", "32. Stokes Equation", "Non-linear dynamics", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\nHellinger Reissner mixed formulation", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\n\\(\\DeclareMathOperator{\\opcurl}{curl}\\)\n\\(\\DeclareMathOperator{\\eps}{\\varepsilon}\\)\nReduced symmetry methods", "Tangential displacement normal normal stress continuous finite elements", "67. Multigrid and Multilevel Methods", "69. Analysis of the Multigrid Iteration", "68. Analysis of the multi-level preconditioner", "70. 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", "2. Boundary Conditions", "5. 3D Solid Mechanics", "6. Exercises", "1. Solving the Poisson Equation", "4. Iterative Solvers", "3. Variable Coefficients", "30. hp - Finite Elements", "31. Implementation of High Order Finite Elements", "72. The Bramble-Pasciak Transformation", "73. A Small Number of Constraints", "74. Parameter Dependent Problems", "71. Structure of Saddle-point Problems", "<no title>", "49. Finite Element Error Analysis", "50. Error Analysis in \\(L_2 \\times H^1\\)", "48. Finite Elements in \\(H(\\operatorname{div})\\)", "46. Application of the abstract theory", "47. The function space \\(H(\\operatorname{div})\\)", "51. Hybridization Techniques", "Friedrichs\u2019 Inequality", "<no title>", "<no title>", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "13. Generalized derivatives", "14. Sobolev spaces", "15. Trace theorems and their applications", "16. Equivalent norms on \\(H^1\\) and on sub-spaces", "19. Exercises", "18. Experiments with norms", "17. The weak formulation of the Poisson equation", "61. Additive Schwarz Methods", "62. Some Examples of ASM preconditioners", "66. Exercise: Robust preconditioners", "64. Domain Decomposition with minimal overlap", "65. Overlapping Domain Decomposition Methods", "63. Schwarz preconditioners for high order finite elements", "Runge Kutta Methods", "Single-step methods", "52. Heat Equation", "53. Wave Equation", "Exponential Integrators for Parabolic Equations", "<no title>", "<no title>", "54. 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