From 62937562b31f68024250681a14785c05d747433f Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Thu, 11 Apr 2013 11:49:33 +0200 Subject: [PATCH] [doc] besseres Quad Plot [doc] Kapitel 2 Fertig? --- doc/code.pdf | Bin 242883 -> 241348 bytes doc/doc.pdf | Bin 322699 -> 318247 bytes doc/doc.tex | 200 +++--- doc/fig/132t05n05_3DFichCube_cond.eps | 369 +++++------ doc/fig/132t05n05_3DFichCube_error.eps | 754 +++++++++++------------ doc/fig/132t05n05_3DFichCube_hminmax.eps | 736 +++++++++++----------- doc/fig/132t05n05_3DFichCube_time.eps | 361 +++++------ src/export_exmpl.m | 2 +- 8 files changed, 1151 insertions(+), 1271 deletions(-) diff --git a/doc/code.pdf b/doc/code.pdf index 1c527fedb2db112ff54bb0c88e89520bed5a8f77..829ab3b461d882740d7376666afdbbc1fe835bff 100644 GIT binary patch delta 33125 zcmY(~Wmp_d(Is{%k}qy7lM^Mnx=D+m#Rk(p$8uAPpLm2YsE3e*GHZvbPV`i+f4eucY3d7jqV1& zQ@{JP3BK+LCEPrXc6yRL)$sooLIJ)gQ&fjd?GnEs?qiWm4!R}`X1|RC#XdgIG#&mV 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\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*} @@ -231,9 +234,52 @@ Wir machen nun den sogenannten indirekten Ansatz $u = \tilde V\phi$, wodurch \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} @@ -255,80 +301,80 @@ mit $V := \gamma_0\tilde V$ gilt. Ziel ist es nun, aus \eqref{math:slp:gls} eine % \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} diff --git a/doc/fig/132t05n05_3DFichCube_cond.eps b/doc/fig/132t05n05_3DFichCube_cond.eps index 3fd178d..42d23c5 100644 --- a/doc/fig/132t05n05_3DFichCube_cond.eps +++ b/doc/fig/132t05n05_3DFichCube_cond.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 -%%Creator: MATLAB, The MathWorks, Inc. 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