\newcommand{\Matlab}{{\sc Matlab}}
\newcommand{\Cpp}{{\sc C++}}
+\newcommand{\llangle}{\langle\!\langle}
+\newcommand{\rrangle}{\rangle\!\rangle}
+
\newcommand{\showMesh}[2][]{\begin{figure}[ht]
\caption{#1}
\label{#2}
In der schwachen Formulierung lautet die Laplace-Gleichung
\begin{align*} % \label{math:slp:lapGLS:weak}
- \varDelta u &= 0 \quad \text{ in } H^{-1}(\Omega), \\
-\gamma_0 u &= g \quad \text{ auf }H^{1/2}(\Gamma),
+\gamma_0 u &= g \quad \text{ auf }H^{1/2}(\Gamma).
\end{align*}
Gemäß \todo{cite Steinbach} kann $\tilde V$ auch als Operator $\tilde V \in L(H^{-1/2}(\Gamma),H^{1}(\Omega))$ aufgefasst werden, und es gilt
\begin{align*}
\begin{align}\label{math:slp:gls}
V \phi = g
\end{align}
-mit $V := \gamma_0\tilde V$ gilt. Ziel ist es nun, aus \eqref{math:slp:gls} eine Funktion $\phi$ zu bestimmen, die die obige Gleichung erfüllt, denn dann ist $\tilde V\phi$ die Lösung des Problems \eqref{math:slp:lapGLS}.
+mit $V := \gamma_0\tilde V$ gilt. Ziel ist es nun, aus \eqref{math:slp:gls} eine Funktion $\phi$ zu bestimmen, die die obige Gleichung erfüllt, denn dann ist mit festen Dirichlet-Daten $g$ $\tilde V\phi$ die eindeutige Lösung des Problems \eqref{math:slp:lapGLS}.
+
+\noindent
+Da wir das Problem \eqref{math:slp:gls} im allgemeinen nicht Lösen können, werden wir es mithilfe des Galerkin-Verfahrens näherungsweise lösen. Die Idee dabei ist $H^{-1/2}(\Gamma)$ durch einen endlich dimensionalen Unterraum zu ersetzen.
+
+\noindent
+Bezeichne nun $\langle \cdot, \cdot \rangle$ das erweiterte $L^2$-Skalarprodukt, so existiert, da $V$ ein symmetrischer und elliptischer Isomorphismus ist, auf $\widetilde H^{-1/2}$ ein äquivalentes Skalarprodukt $\llangle \cdot, \cdot \rrangle$ mit $\llangle \phi, \psi \rrangle := \langle V\phi,\psi\rangle$ und der induzierten Norm $\enorm{\cdot}$.
+Sei nun $\phi$ die eindeutige Lösung und bezeichne $g$ die Dirichlet-Daten am Rand. Dann gilt für die schwache Formulierung \eqref{math:slp:gls}
+\begin{align}
+ \llangle \phi,\psi\rrangle = \langle g,\psi\rangle \quad \text{für alle }\psi \in \widetilde H^{-1/2}.
+\end{align}
+
+\noindent
+Sei nun $\T_{\ell} = \{T_1,T_2,\ldots,T_N\}$ eine Partition des Randes $\Gamma$ in $N$ Randstücke $T_1,T_2,\ldots,T_N$ mit der charakteristischen Funktion
+\begin{align}
+ \chi_i(\bs x) &=
+ \begin{cases}
+ 1&\text{für alle }\bs x \in T_i\\
+ 0&\text{sonst}
+ \end{cases}
+\end{align}
+für alle $i\in\{1,2,\ldots,N\}$. Dann kann die Funktion $\chi_i$ als Basis vom Raum $\P^0(\T_{\ell})$ verstanden werden. Wir suchen also die Lösung $\psi_{\ell} \in P^0(\T_{\ell})$ für
+\begin{align}
+ \llangle \phi_{\ell},\psi_{\ell} \rrangle & = \langle f,\psi_{\ell} \rangle\quad\text{für alle } \psi_{\ell} \in P^0(\T_{\ell}),
+\end{align}
+was aufgrund der Basiseigenschaft von $\chi_i$ eindeutig und äquivalent ist zu
+\begin{align}
+ \llangle \phi_{\ell},\chi_j \rrangle & = \langle f,\chi_j \rangle\quad\text{für alle } j\in\{1,2,\ldots,N\}.
+\end{align}
+Dadurch können wir das Problem als Summe anschreiben
+\begin{align}
+ \sum_{j=1}^N x_{\ell,j} \llangle \chi_j,\chi_i\rrangle = \langle g, \chi_i\rangle,
+\end{align}
+wobei $\bs x_{\ell} = \{x_{\ell,1},x_{\ell,2},\ldots,x_{\ell,N}\}$ der Koordinatenvektor zur Basis $\chi_1,\ldots,\chi_N$ ist.
+So schreiben wir
+\begin{align}
+ \bs V \bs x_{\ell} = \bs g_{\ell},
+\end{align}
+mit Bezeichnungen $\bs V = \llangle \chi_j,\chi_i\rrangle$ und sei $\bs g_{\ell}\in\R^N$ mit den Koeffizienten $g_{\ell,i} = \langle g, \chi_j\rangle$.
+In Integralschreibweise können wir $\bs V$ anschreiben als
+\begin{align}
+ V := \frac 1 {4\pi} \int_{T_j} \int_{T_k} \frac {1}{\abs{\bs x - \bs y}} ds_{\bs y} ds_{\bs x}
+\end{align}
-\subsection{old Galerkin}
+
+% \subsection{old Galerkin}
% Wir wissen, dass die Laplace-Gleichung erfüllt wird durch:
% \begin{align}
% V\phi &= g \label{Formel}
% \begin{align}
% V : H^{-1/2+s}(\Gamma) &\rightarrow H^{1/2+s}(\Gamma)& \text{mit } s\in [-1/2,1/2]
% \end{align}
-\begin{lem}[Lax-Milgram]
-Sei eine Abbildung $a: X\times X \rightarrow \R$ wobei $X$ ein reflexiver Banachraum. Und gilt:
-\begin{itemize}
- \item $a$ stetig, d.h. : $\abs{a(x,y)} \leq C \cdot \norm{x} \cdot \norm{y}$
- \item $a$ eliptisch, d.h. : $a(x,x) \geq X \cdot \norm{x}^2$
-\end{itemize}
-So folgt daraus
-$\forall f \in X'$ $\exists $ eindeutiges $x \in X$ mit $a(x,\cdot) = f$.
-\end{lem}
-\begin{defi}
-Sei $\langle \cdot, \cdot \rangle$ das erweiterte $L_2$ - Skalarprodukt
-\end{defi}
-Wendet man nun das Lax-Milgram Lemma auf die schwache Formulierung an,
-\begin{align}
- \langle V \phi, \psi\rangle &= \langle f,\psi\rangle & \psi \in H^{-1/2}\\
-a(\phi,\psi) &:= \langle V\phi,\psi\rangle&:H^{-1/2}(\Gamma)\times H^{-1/2}(\Gamma) \rightarrow \R
-\end{align}
-zeigen wir noch, dass:
-\begin{itemize}
- \item $H^{-1/2}(\Gamma)$ ist reflexiver Banachraum, welches aus der Definition von $H^{-1/2}$ folgt
- \item $ \abs{\langle V\phi,\psi \rangle} \leq C \cdot \norm{\phi}_{H^{-1/2}} \cdot \norm{\psi}_{H^{-1/2}}$
- \item $ \langle V\phi,\phi \rangle \geq C \cdot \norm{\phi}_{H^{-1/2}}^2$
-\end{itemize}
-Daraus folgt nun dass, $\forall f \in H^{-1/2}$ $\exists$ eindeutige Lösung $\phi \in H^{-1/2}(\Gamma)$ von
-\begin{align}
- \langle V \phi, \psi \rangle &=\langle f, \psi \rangle & \forall \psi \in H^{-1/2}(\Gamma)
-\end{align}
-Wollen wir nun das Galerkin-Verfahren anwenden benötigen wir die schwache Formulierung:
-\begin{align}
- \int_{\Gamma} V \phi(x) \cdot \psi(x) dx &= \int_{\Gamma} f(x)\cdot\psi(x) dx
-\end{align}
-Nun wählen wir einen endlich-dimensionalen Teilraum $P^0(\T_n) \subseteq H^{-1/2}$ und betrachten
-
-\begin{defi}\label{1}
-\begin{align}
- \langle V\phi_{\ell},\psi_{\ell} \rangle & = \langle f,\psi_{\ell} \rangle& \forall \psi_{\ell} \in P^0(\T_{\ell})
-\end{align}
-\end{defi}
-Gesucht ist jetzt also $\phi_{\ell} \in P^0(\T_{\ell})$
-
-\noindent
-Aus dem Max-Milgram Lemma und $X = P^0(\T_{\ell})$ folgt wiederum, es $\exists$ eindeutige Lösung $\phi_{\ell} \in P^0(\T_{\ell})$, da $\psi_{\ell} \in P^0(\T_{\ell}),\phi_{\ell} \in P^0(\T_{\ell})$.
-\begin{defi}
- Sei nun die Basis von $P^0(\T_{\ell})$ die charakteristischen Funktionen
-\begin{align}
- \{\chi_T | T\in\T_{\ell}\} &= \{\chi_{T_1},\chi_{T_2},\dots\}
-\end{align}
-\end{defi}
-So können wir mit $N = \dim P^0(\T_{\ell})$ und $\psi_{\ell},\phi_{\ell}\in\R$ wobei $l\in \{1\dots N\}$ schreiben
-\begin{align}
- \psi_{\ell} &= \sum_{l=1}^N \psi_{\ell} \cdot \chi_{T_{\ell}} \\
- \phi_{\ell} &= \sum_{l=1}^N \phi_{\ell} \cdot \chi_{T_{\ell}}
-\end{align}
-Dadurch können wir Definition \ref{1} nun einfacher Lösen durch:
-\begin{align}
- \langle V \phi_{\ell},\chi_k\rangle & = \langle f, \chi_k \rangle & k = 1\dots N
-\end{align}
-Aufgrund der Linearität von $V$ und dem Skalarprodukt schreiben wir:
-\begin{align}
-\sum_{l=1}^N\langle V\phi_{\ell}\chi_{\ell},\chi_k\rangle & = \langle f,\chi_k\rangle
-\end{align}
-welches sich wiederum so schreiben lässt
-\begin{defi}[Galerkinapproximation]
-\begin{align}
-\ul{\ul{V}} \cdot \ul{\phi} = \ul{f}
-\end{align}
-wobei $\ul{\ul{V}}\in R^{N \times N},\ul{\phi}\in\R^{N \times 1},\ul{f}\in\R^{N \times 1}$
-\begin{align}
- \ul{\ul{V}}_{\ell,k} &= \langle V \chi_{\ell}, \chi_k \rangle\\
- \ul{\phi}_{\ell} &= \phi_{\ell} \nonumber\\
- \ul{f}_k &= \langle f, \chi_k\rangle \nonumber
-\end{align}
-Damit ist $\phi_{\ell}$ die Galerkinapproximation an $\phi$
-\end{defi}
+% \begin{lem}[Lax-Milgram]
+% Sei eine Abbildung $a: X\times X \rightarrow \R$ wobei $X$ ein reflexiver Banachraum. Und gilt:
+% \begin{itemize}
+% \item $a$ stetig, d.h. : $\abs{a(x,y)} \leq C \cdot \norm{x} \cdot \norm{y}$
+% \item $a$ eliptisch, d.h. : $a(x,x) \geq X \cdot \norm{x}^2$
+% \end{itemize}
+% So folgt daraus
+% $\forall f \in X'$ $\exists $ eindeutiges $x \in X$ mit $a(x,\cdot) = f$.
+% \end{lem}
+% \begin{defi}
+% Sei $\langle \cdot, \cdot \rangle$ das erweiterte $L_2$ - Skalarprodukt
+% \end{defi}
+% Wendet man nun das Lax-Milgram Lemma auf die schwache Formulierung an,
+% \begin{align}
+% \langle V \phi, \psi\rangle &= \langle f,\psi\rangle & \psi \in H^{-1/2}\\
+% a(\phi,\psi) &:= \langle V\phi,\psi\rangle&:H^{-1/2}(\Gamma)\times H^{-1/2}(\Gamma) \rightarrow \R
+% \end{align}
+% zeigen wir noch, dass:
+% \begin{itemize}
+% \item $H^{-1/2}(\Gamma)$ ist reflexiver Banachraum, welches aus der Definition von $H^{-1/2}$ folgt
+% \item $ \abs{\langle V\phi,\psi \rangle} \leq C \cdot \norm{\phi}_{H^{-1/2}} \cdot \norm{\psi}_{H^{-1/2}}$
+% \item $ \langle V\phi,\phi \rangle \geq C \cdot \norm{\phi}_{H^{-1/2}}^2$
+% \end{itemize}
+% Daraus folgt nun dass, $\forall f \in H^{-1/2}$ $\exists$ eindeutige Lösung $\phi \in H^{-1/2}(\Gamma)$ von
+% \begin{align}
+% \langle V \phi, \psi \rangle &=\langle f, \psi \rangle & \forall \psi \in H^{-1/2}(\Gamma)
+% \end{align}
+% Wollen wir nun das Galerkin-Verfahren anwenden benötigen wir die schwache Formulierung:
+% \begin{align}
+% \int_{\Gamma} V \phi(x) \cdot \psi(x) dx &= \int_{\Gamma} f(x)\cdot\psi(x) dx
+% \end{align}
+% Nun wählen wir einen endlich-dimensionalen Teilraum $P^0(\T_n) \subseteq H^{-1/2}$ und betrachten
+%
+% \begin{defi}\label{1}
+% \begin{align}
+% \llangle \phi_{\ell},\psi_{\ell} \rrangle & = \langle f,\psi_{\ell} \rangle\quad\text{für alle } \psi_{\ell} \in P^0(\T_{\ell})
+% \end{align}
+% \end{defi}
+% Gesucht ist jetzt also $\phi_{\ell} \in P^0(\T_{\ell})$
+%
+% \noindent
+% Aus dem Max-Milgram Lemma und $X = P^0(\T_{\ell})$ folgt wiederum, es $\exists$ eindeutige Lösung $\phi_{\ell} \in P^0(\T_{\ell})$, da $\psi_{\ell} \in P^0(\T_{\ell}),\phi_{\ell} \in P^0(\T_{\ell})$.
+% \begin{defi}
+% Sei nun die Basis von $P^0(\T_{\ell})$ die charakteristischen Funktionen
+% \begin{align}
+% \{\chi_T | T\in\T_{\ell}\} &= \{\chi_{T_1},\chi_{T_2},\dots\}
+% \end{align}
+% \end{defi}
+% So können wir mit $N = \dim P^0(\T_{\ell})$ und $\psi_{\ell},\phi_{\ell}\in\R$ wobei $l\in \{1\dots N\}$ schreiben
+% \begin{align}
+% \psi_{\ell} &= \sum_{l=1}^N \psi_{\ell} \cdot \chi_{T_{\ell}} \\
+% \phi_{\ell} &= \sum_{l=1}^N \phi_{\ell} \cdot \chi_{T_{\ell}}
+% \end{align}
+% Dadurch können wir Definition \ref{1} nun einfacher Lösen durch:
+% \begin{align}
+% \langle V \phi_{\ell},\chi_k\rangle & = \langle f, \chi_k \rangle & k = 1\dots N
+% \end{align}
+% Aufgrund der Linearität von $V$ und dem Skalarprodukt schreiben wir:
+% \begin{align}
+% \sum_{l=1}^N\langle V\phi_{\ell}\chi_{\ell},\chi_k\rangle & = \langle f,\chi_k\rangle
+% \end{align}
+% welches sich wiederum so schreiben lässt
+% \begin{defi}[Galerkinapproximation]
+% \begin{align}
+% \ul{\ul{V}} \cdot \ul{\phi} = \ul{f}
+% \end{align}
+% wobei $\ul{\ul{V}}\in R^{N \times N},\ul{\phi}\in\R^{N \times 1},\ul{f}\in\R^{N \times 1}$
+% \begin{align}
+% \ul{\ul{V}}_{\ell,k} &= \langle V \chi_{\ell}, \chi_k \rangle\\
+% \ul{\phi}_{\ell} &= \phi_{\ell} \nonumber\\
+% \ul{f}_k &= \langle f, \chi_k\rangle \nonumber
+% \end{align}
+% Damit ist $\phi_{\ell}$ die Galerkinapproximation an $\phi$
+% \end{defi}
% \subsection{Vorkonditionieren}
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
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gr
/c10 { 0.600000 0.000000 0.600000 sr} bdef
c10
-103 25 100 71 89 71 94 71 85 66 69 63 78 63 62 60
-79 57 55 50 74 47 70 50 73 61 97 64 86 59 114 65
-90 58 165 58 106 53 184 58 176 61 396 55 977 969 23 MP stroke
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0 j
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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-DP
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DP
gr
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-gs 926 1239 2548 1361 MR c np
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0 j
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
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0 j
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
DD
/c11 { 0.900000 0.600000 0.000000 sr} bdef
c11
-103 40 100 40 89 35 94 37 85 34 69 27 78 31 62 24
-79 31 55 22 74 29 70 28 73 28 97 39 86 34 114 45
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DO
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+132 20 119 18 126 18 113 17 92 14 105 15 82 12 106 16
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DA
-103 62 100 58 89 53 94 56 85 50 69 41 78 46 62 37
-79 47 55 32 74 44 70 41 73 44 97 57 86 51 114 68
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c2
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gr
c2
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624 3206 mt 642 3206 L
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gs 624 270 3721 2937 MR c np
/c8 { 0.000000 0.300000 0.300000 sr} bdef
c8
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gr
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DP
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DP
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DP
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DP
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DP
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DP
gr
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gr
/c9 { 0.000000 0.600000 0.600000 sr} bdef
c9
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16 W
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16 W
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16 W
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16 W
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16 W
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16 W
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16 W
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16 W
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16 W
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16 W
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16 W
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16 W
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gr
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
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gr
/c10 { 0.600000 0.000000 0.600000 sr} bdef
c10
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0 j
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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-DP
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DP
gr
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
DA
c2
-0 -2227 3177 2930 2 MP stroke
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gr
c2
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+%%Creator: MATLAB, The MathWorks, Inc. Version 8.1.0.604 (R2013a). Operating System: Linux 3.5.0-27-generic #46-Ubuntu SMP Mon Mar 25 19:58:17 UTC 2013 x86_64.
%%Title: ../doc/fig/132t05n05_3DFichCube_time.eps
-%%CreationDate: 04/08/2013 13:19:07
+%%CreationDate: 04/11/2013 10:22:37
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gs 624 269 3721 2938 MR c np
/c8 { 0.000000 0.300000 0.300000 sr} bdef
c8
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-79 -95 55 -72 74 -87 70 -85 73 -91 97 -141 86 -113 114 -132
-90 -96 165 -155 106 -77 184 -144 176 -194 396 173 977 2500 23 MP stroke
-gs 926 282 2548 2443 MR c np
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gr
/c9 { 0.000000 0.600000 0.600000 sr} bdef
c9
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-79 -98 55 -66 74 -90 70 -87 73 -92 97 -127 86 -122 114 -131
-90 -101 165 -146 106 -79 184 -187 176 -128 396 -277 977 2915 23 MP stroke
-gs 926 281 2548 2686 MR c np
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0 j
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-DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
/c10 { 0.600000 0.000000 0.600000 sr} bdef
c10
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-79 -100 55 -66 74 -92 70 -90 73 -99 97 -127 86 -134 114 -160
-90 -84 165 -119 106 -138 184 -124 176 -196 396 -237 977 2969 23 MP stroke
-gs 926 283 2548 2738 MR c np
- 960 2952 mt 994 2986 L
- 994 2952 mt 960 2986 L
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gr
gr
%% Semianalytisch
A_plots({'meshSave/132t05n05_2DQuad_30'},'../doc/fig/132t05n05_2DQuad')
-A_plots({'meshSave/132t05n05_3DFichCube_23'},'../doc/fig/132t05n05_3DFichCube')
+A_plots({'meshSave/132t05n05_3DFichCube_22'},'../doc/fig/132t05n05_3DFichCube')
close all
\ No newline at end of file
projects / bacc.git / commitdiff