From 02c4eea17a67b29c7e693d00b98628b63ce89d40 Mon Sep 17 00:00:00 2001 From: Joachim Schoeberl Date: Tue, 5 Mar 2024 15:14:51 +0100 Subject: [PATCH] Update documentation --- FEM/nonconforming.html | 293 ++++++++++++--------- _sources/FEM/nonconforming.ipynb | 155 ++++++----- _sources/sobolevspaces/SobolevSpaces.ipynb | 2 + _sources/sobolevspaces/Traces.ipynb | 57 ++-- genindex.html | 2 +- intro.html | 2 +- mixedelasticity/dynamics.html | 2 +- plates/tdnnsplate.html | 2 +- search.html | 2 +- searchindex.js | 2 +- sobolevspaces/SobolevSpaces.html | 20 +- sobolevspaces/Traces.html | 168 ++++++------ 12 files changed, 404 insertions(+), 303 deletions(-) diff --git a/FEM/nonconforming.html b/FEM/nonconforming.html index 1c38b885..c4bf9c11 100644 --- a/FEM/nonconforming.html +++ b/FEM/nonconforming.html @@ -32,7 +32,7 @@ - + @@ -441,7 +441,9 @@ `); - + @@ -457,6 +459,18 @@

Non-conforming Finite Element Methods

@@ -468,63 +482,64 @@

Non-conforming Finite Element Methods

19. Non-conforming Finite Element Methods#

-

\label{sec_nonconforming} -In a conforming finite element method, one chooses a sub-space \(V_h \subset V\), and defines the finite element approximation as -$\( +

In a conforming finite element method, one chooses a sub-space \(V_h \subset V\), and defines the finite element approximation as

+
+\[ \mbox{Find } u_h \in V_h: \qquad A(u_h, v_h) = f(v_h) \qquad \forall \, v_h \in V_{h} -\)\( -For reasons of simpler implementation, or even of higher accuracy, the -conforming framework is often violated. Examples are: -\begin{itemize} -\item -The finite element space \)V_h\( is not a sub-space of \)V = H^m\(. -Examples are the non-conforming \)P^1\( triangle, and the Morley element for -approximation of \)H^2$. -\item -The Dirichlet boundary conditions are interpolated in the boundary vertices. -\item -The curved domain is approximated by straight sided elements -\item -The bilinear-form and the linear-form are approximated by -inexact numerical integration -\end{itemize}

-

\noindent -The lemmas by Strang are the extension of Cea’s lemma to the +\]

+

For reasons of simpler implementation, or even of higher accuracy, the +conforming framework is often violated. Examples are:

+ +

The lemmas by Strang are the extension of Cea’s lemma to the non-conforming setting.

-

\subsubsection{The First Lemma of Strang} -In the first step, let \(V_h \subset V\), but the bilinear-form and the linear-form are replaced by mesh-dependent forms -$\( +

+

19.1. The First Lemma of Strang#

+

In the first step, let \(V_h \subset V\), but the bilinear-form and the linear-form are replaced by mesh-dependent forms

+
+\[ A_h(.,.): V_h \times V_h \rightarrow {\mathbb R} -\)\( -and -\)\( +\]
+

and

+
+\[ f_h(.) : V_h \rightarrow {\mathbb R}. -\)\( -We do not assume that \)A_h\( and \)f_h\( are defined on \)V\(. -We assume that the bilinear-forms \)A_h\( are uniformly coercive, i.e., -there exists an \)\alpha_1\( independent of the mesh-size such that -\)\( +\]
+

We do not assume that \(A_h\) and \(f_h\) are defined on \(V\). +We assume that the bilinear-forms \(A_h\) are uniformly coercive, i.e., +there exists an \(\alpha_1\) independent of the mesh-size such that

+
+\[ A_h (v_h, v_h) \geq \alpha_1 \, \| v_h \|_V^2 \qquad \forall \, v_h \in V_h -\)\( -The finite element problem is defined as -\)\( +\]
+

The finite element problem is defined as

+
+\[ \mbox{Find } u_h \in V_h: \qquad A_h (u_h, v_h) = f_h (v_h) \qquad \forall \, v_h \in V_h -\)$

-

\begin{lemma}[First Lemma of Strang] Assume that -\begin{itemize} -\item \(A(.,.)\) is continuous on \(V\) -\item \(A_h(.,.)\) is uniformly coercive -\end{itemize} -Then there holds

+\]
+
+

First Lemma of Strang: Assume that

+
    +

  • \(A(.,.)\) is continuous on \(V\)

    • +

  • \(A_h(.,.)\) is uniformly coercive +Then there holds

    • +
\[\begin{eqnarray*} -\| u - u_h \| & \preceq & \inf_{v_h \in V_h} \left\{ - \| u - v_h \| + \sup_{w_h \in V_h} \frac{|A(v_h, w_h) - A_h (v_h, w_h)|}{\| w_h \|} \right\} \\ - & & - + \sup_{w_h \in V_h} \frac{f(w_h) - f_h (w_h)}{\| w_h \|} -\end{eqnarray*}\]
-

\end{lemma}
-{\em Proof:} + \| u - u_h \| & \preceq & \inf_{v_h \in V_h} \left\{ + \| u - v_h \| + \sup_{w_h \in V_h} \frac{|A(v_h, w_h) - A_h (v_h, w_h)|}{\| w_h \|} \right\} \\ + & & + + \sup_{w_h \in V_h} \frac{f(w_h) - f_h (w_h)}{\| w_h \|} + \end{eqnarray*}\]

+
+

Proof: Choose an arbitrary \(v_h \in V_h\), and set \(w_h := u_h - v_h\). We use the uniform coercivity, and the definitions of \(u\) and \(u_h\):

@@ -532,69 +547,75 @@

19. Non-conforming Finite Element Method \alpha_1 \| u_h - v_h \|_V^2 & \leq & A_h (u_h - v_h, u_h - v_h) = A_h (u_h - v_h, w_h) \\ & = & A(u-v_h, w_h) + [ A(v_h, w_h) - A_h(v_h, w_h) ] + [ A_h (u_h, w_h) - A(u, w_h)] \\ & = & A(u-v_h, w_h) + [ A(v_h, w_h) - A_h(v_h, w_h) ] + [ f_h(w_h) - f(w_h)] + \end{eqnarray*}\]

Divide by \(\| u_h - v_h \| = \| w_h \|\), and use the continuity of \(A(.,.)\):

-
-(19.1)#\[\begin{equation} +
+()#\[\begin{equation} \label{equ_strang1a} \| u_h - v_h \| \preceq \| u - v_h \| + \frac{|A(v_h, w_h) - A_h(v_h, w_h)|}{\| w_h \|} + \frac{ | f(w_h) - f_h(w_h) | } { \| w_h \| } \end{equation}\]
-

Using the triangle inequality, the error \(\| u - u_h \|\) is bounded by -$\( +

Using the triangle inequality, the error \(\| u - u_h \|\) is bounded by

+
+\[ \| u - u_h \| \leq \inf_{v_h \in V_h} \| u - v_h \| + \| v_h - u_h \| -\)\( -The combination with (\ref{equ_strang1a}) proves the result. -\hfill \)\Box$

-

\bigskip

-

{\bf Example:} Lumping of the \(L_2\) bilinear-form: \newline -Define the \(H^1\) - bilinear-form -$\( +\]

+

The combination with (\ref{equ_strang1a}) proves the result. +\hfill \(\Box\)

+

Example: Lumping of the \(L_2\) bilinear-form: \newline +Define the \(H^1\) - bilinear-form

+
+\[ A(u,v) = \int_\Omega \nabla u \cdot \nabla v + \int_\Omega u v \, dx, -\)\( -and perform Galerkin discretization with \)P^1\( triangles. -The second term leads to a non-diagonal matrix. -The vertex integration rule -\)\( +\]
+

and perform Galerkin discretization with \(P^1\) triangles. +The second term leads to a non-diagonal matrix. +The vertex integration rule

+
+\[ \int_T v \, dx \approx \frac{|T|}{3} \sum_{\alpha = 1}^3 v(x_{T,\alpha}) -\)\( -is exact for \)v \in P^1\(. We apply this integration rule for the term -\)\int u v , dx\(: -\)\( +\]
+

is exact for \(v \in P^1\). We apply this integration rule for the term +\(\int u v \, dx\):

+
+\[ A_h(u,v) = \int \nabla u \cdot \nabla v + \sum_{T \in {\cal T}} \frac{|T|}{3} \sum_{\alpha = 1}^3 u(x_{T,\alpha}) v(x_{T,\alpha}) -\)\( -The bilinear form is now defined only for \)u, v \in V_h\(. -The integration is not exact, since \)u v \in P^2$ on each triangle.

+\]
+

The bilinear form is now defined only for \(u, v \in V_h\). +The integration is not exact, since \(u v \in P^2\) on each triangle.

Inserting the nodal basis \(\varphi_i\), we obtain a diagonal matrix for -the second term: -$\( +the second term:

+
+\[\begin{split} \varphi_i (x_{T,\alpha}) \varphi_j (x_{T,\alpha}) = \left\{ \begin{array}{cl} 1 & \mbox{for } x_i = x_j = x_{T,\alpha} \\ 0 & \mbox{else} \end{array} \right. -\)$

+\end{split}\]

To apply the first lemma of Strang, we have to verify the uniform coercivity

-
-(19.2)#\[\begin{equation} +
+()#\[\begin{equation} \label{equ_uniformell} \sum_T \frac{|T|}{3} \sum_{\alpha = 1}^3 |v_h(x_{T,\alpha})|^2 \geq \alpha_1 \sum_T \int_T | v_h |^2 \, dx \qquad \forall \, v_h \in V_h, \end{equation}\]

which is done by transformation to the reference element. The consistency error can be estimated by

-
-(19.3)#\[\begin{equation} +
+()#\[\begin{equation} \label{equ_consist} | \int_T u_h v_h \, dx - \frac{|T|}{3} \sum_{\alpha=1}^3 u_h(x_\alpha) v_h(x_\alpha) | \preceq h_T^2 \, \| \nabla u_h \|_{L_2(T)} \, \| \nabla v_h \|_{L_2(T)} \end{equation}\]
-

Summation over the elements give -$\( +

Summation over the elements give

+
+\[ A(u_h, v_h) - A_h (u_h, v_h) \preceq h^2 \| u_h \|_{H^1(\Omega)} \, \| v_h \|_{H^1(\Omega)} -\)$ -The first lemma of Strang proves that this modification of the bilinear-form +\]
+

The first lemma of Strang proves that this modification of the bilinear-form preserves the order of the discretization error:

\[\begin{eqnarray*} @@ -613,33 +634,39 @@

19. Non-conforming Finite Element Method problems, one has to solve linear equations with the \(L_2\)-matrix. This becomes cheap for diagonal matrices. \end{itemize}

-

\subsubsection{The Second Lemma of Strang}

+

+
+

19.2. The Second Lemma of Strang#

In the following, we will also skip the requirement \(V_h \subset V\). Thus, the norm \(\|.\|_V\) cannot be used on \(V_h\), and it will be replaced by mesh-dependent norms \(\|.\|_h\). These norms must be defined for \(V + V_h\). As well, the mesh-dependent forms \(A_h(.,.)\) and \(f_h(.)\) are defined -on \(V + V_h\). We assume -\begin{itemize} -\item uniform coercivity: -$\( -A_h (v_h, v_h) \geq \alpha_1 \| v_h \|_h^2 \qquad \forall \, v_h \in V_h -\)\( -\item continuity: -\)\( -A_h (u, v_h) \leq \alpha_2 \| u \|_h \| v_h \|_h \qquad \forall \, u \in V + V_h, \; \forall \, v_h \in V_h -\)$ -\end{itemize}

+on \(V + V_h\). We assume

+
    +

  • uniform coercivity:

    +
    +\[ + A_h (v_h, v_h) \geq \alpha_1 \| v_h \|_h^2 \qquad \forall \, v_h \in V_h + \]
    +
    • +

  • continuity:

    +
    +\[ + A_h (u, v_h) \leq \alpha_2 \| u \|_h \| v_h \|_h \qquad \forall \, u \in V + V_h, \; \forall \, v_h \in V_h + \]
    +
    • +

The error can now be measured only in the discrete norm \(\| u - u_h \|_{V_h}\). \begin{lemma} Under the above assumptions there holds

-
-(19.4)#\[\begin{equation} +
+()#\[\begin{equation} \label{equ_strang2} \| u - u_h \|_h \preceq \inf_{v_h \in V_h} \| u - v_h \|_h + \sup_{w_h \in V_h} \frac{| A_h(u,w_h) - f_h(w_h) |}{\| w_h \|_h} \end{equation}\]
-

\end{lemma} -{\em Remark}: The first term in (\ref{equ_strang2}) is the approximation +

\end{lemma}

+

Remark: The first term in (\ref{equ_strang2}) is the approximation error, the second one is called consistency error. \ {\em Proof:} Let \(v_h \in V_h\). Again, set \(w_h = u_h - v_h\), and use the \(V_h\)-coercivity:

@@ -653,24 +680,28 @@

19. Non-conforming Finite Element Method \| u_h - v_h \|_h \preceq \| u - v_h \|_h + \frac{A_h(u,w_h) - f_h(w_h)}{\| w_h \|_h} \)\( The rest follows from the triangle inequality. \hfill \)\Box$

-

\subsubsection{The non-conforming \(P^1\) triangle}

+
+

19.2.1. The non-conforming \(P^1\) triangle}#

The non-conforming \(P^1\) triangle is also called the Crouzeix-Raviart element.

The finite element space generated by the non-conforming \(P^1\) element -is -$\( +is

+
+\[ V_h^{nc} := \{ v \in L_2 : v_{|T} \in P^1(T), \mbox{and }v \mbox{ is continuous in edge mid-points} \} -\)$

+\]

The functions in \(V_h^{nc}\) are not continuous across edges, and thus, \(V_h^{nc}\) is not a sub-space of \(H^1\). We have to extend the bilinear-form and -the norm in the following way: -$\( +the norm in the following way:

+
+\[ A_h (u,v) = \sum_{T \in {\cal T}} \int_T \nabla u \nabla v \, dx \qquad \forall \, u, v \in V + V_h^{nc} -\)\( -and -\)\( +\]
+

and

+
+\[ \| v \|_h^2 := \sum_{T \in {\cal T}} \| \nabla v \|_{L_2(T)}^2 \qquad \forall \, v \in V + V_h^{nc} -\)$

+\]

We consider the Dirichlet-problem with \(u = 0\) on \(\Gamma_D\).

We will apply the second lemma of Strang.

The continuous \(P^1\) finite element space \(V_h^c\) is a sub-space of @@ -750,18 +781,17 @@

19. Non-conforming Finite Element Method \| u - u_h \| \preceq h \, \| u \|_{H^2} \)$

There are several applications where the non-conforming \(P^1\) triangle -is of advantage: -\begin{itemize} -\item -The \(L_2\) matrix is diagonal (exercises) -\item -It can be used for the approximation of problems in fluid dynamics -described by the Navier Stokes equations (see later). -\item -The finite element matrix has exactly 5 non-zero entries in each row +is of advantage:

+
    +

  • The \(L_2\) matrix is diagonal (exercises)

    • +

  • It can be used for the approximation of problems in fluid dynamics +described by the Navier Stokes equations (see later).

    • +

  • The finite element matrix has exactly 5 non-zero entries in each row associated with inner edges. That allows simplifications in the matrix -generation code. -\end{itemize}

    +generation code.

    • +
+

+