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zP)VJ@RQRbOWbkm49dANVMSfDm6%G{NSfF{x53JBUgXXjHe4tKHpV0iS6y!BhzQ2RT8`}+^JSW5$`d-qG zJ48LlIn(aoUfe;B?!*AZ#f?T>q5tB7v$Tt|8-WaPwEzEN1Zg=sYR!llKRqUHdPt}i Rv}NFwP|M70fVaEW{{c4Ud}IIs diff --git a/doc/doc.tex b/doc/doc.tex index f684b39..dc5e71a 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -47,7 +47,8 @@ \newcommand{\dif}[2]{\frac{\partial#1}{\partial#2}} -\newcommand{\enorm}[1]{\left| \! \left| \! \left|#1\right| \! \right| \! \right|} +% \newcommand{\enorm}[1]{\left| \! \left| \! \left|#1\right| \! \right| \! \right|} +\newcommand{\enorm}[1]{| \! | \! |#1| \! | \! |} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\abs}[1]{\lvert#1\rvert} @@ -85,117 +86,123 @@ \psfrag{T4}{\scriptsize $T_4$} \author{Peter Schaefer} +\title{Stabile Berechnung der Galerkin-Matrix für das Einfachschichtpotential auf anisotrop verfeinerten Gittern} \begin{document} \tableofcontents \clearpage \section{Einleitung} - -\subsection{Allgemein} -In dieser Arbeit beschäftigen wir uns mit der Randelementmethode für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen -\begin{align} - - \varDelta u &= 0 &\text{ in } \Omega& \subset \R^3,\\ - u &= f &\text{ auf } \Gamma&:=\partial\Omega, -\end{align} -wobei $\varDelta u := \partial_x^2u+\partial_y^2u$ den Laplace-Operator bezeichnet und $\Omega \subset \R^3$ eine beschränkte Teilmenge von $\R^3$ mit Lipschitz-Rand $\Gamma := \partial \Omega$ ist.\\ - -\noindent -Wir wissen, dass die Laplace-Gleichung erfüllt wird durch: -\begin{align} -V\phi &= f -\end{align} -% $\abs{u(x)} = O(\abs{x}^{-1}) \nonumber$ -Sei nun die $G$ Fundamentallösung, -\begin{align} - G(\bs z) := \frac 1 {4 \pi} \frac 1 {\abs{\bs z}} -\end{align} -dann können wir $\tilde V$ anschreiben durch -\begin{align} -\tilde V \phi (\bs x) &:= \int G (\bs x- \bs y) \phi(\bs y) d\bs y & \bs x\in \R^3\backslash \Gamma -\end{align} -Daraus folgt nun mit $\gamma_0 = \rm{spur}$ : -\begin{align} - V \phi &: \gamma_0 \tilde V \phi_0 -\end{align} -Weiterhin kann man nun Zeigen, dass: -\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: +\todo{ \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$ + \item kurz + praegnant, worum es geht (vgl. Mayr-Bakk, wo das exakt +eine Seite ist) \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} +In dieser Arbeit beschäftigen wir uns mit der Randelementmethode für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen \begin{align} - \langle V\phi_{\ell},\psi_{\ell} \rangle & = \langle f,\psi_{\ell} \rangle& \forall \psi_{\ell} \in P^0(\T_{\ell}) + - \varDelta u &= 0 \text{ in } \Omega \subset \R^3,\\ + u &= f \text{ auf } \Gamma, \end{align} -\end{defi} -Gesucht ist jetzt also $\phi_{\ell} \in P^0(\T_{\ell})$ +wobei $\varDelta u := \partial_x^2u+\partial_y^2u$ den Laplace-Operator bezeichnet und $\Omega \subset \R^3$ eine beschränkte Teilmenge von $\R^3$ mit Lipschitz-Rand $\Gamma := \partial \Omega$ ist.\\ -\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} +% \noindent +% Wir wissen, dass die Laplace-Gleichung erfüllt wird durch: +% \begin{align} +% V\phi &= f \label{Formel} +% \end{align} +% % $\abs{u(x)} = O(\abs{x}^{-1}) \nonumber$ +% Sei nun die $G$ Fundamentallösung, +% \begin{align} +% G(\bs z) := \frac 1 {4 \pi} \frac 1 {\abs{\bs z}} +% \end{align} +% dann können wir $\tilde V$ anschreiben durch +% \begin{align} +% \tilde V \phi (\bs x) &:= \int G (\bs x- \bs y) \phi(\bs y) d\bs y & \bs x\in \R^3\backslash \Gamma +% \end{align} +% Daraus folgt nun mit $\gamma_0 = \rm{spur}$ : +% \begin{align} +% V \phi &: \gamma_0 \tilde V \phi_0 +% \end{align} +% Weiterhin kann man nun Zeigen, dass: +% \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} % \subsection{Vorkonditionieren} @@ -208,6 +215,15 @@ Damit ist $\phi_{\ell}$ die Galerkinapproximation an $\phi$ % \hat \phi_{\ell} &= D \cdot y\\ % \end{align} + + +\section{Randelementmethode} +\todo{ +\begin{itemize} + \item kuerzer als in Ferraz-DA + \item insbesondere fastregulaere Triangulierung + K-Gitter (vgl. Ferraz-DA) +\end{itemize} +} \subsection{Netze} Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. @@ -273,78 +289,68 @@ anliegen können. Weiterhin wollen wir auch verlangen, dass der auf der Kante li \end{enumerate} \end{alg} +\clearpage - -\subsection{Fehlerschätzer} -In diesem Abschnitt definieren wir die a-posteriori Fehlerschätzer, die wir im Folgenden zur Steuerung des adaptiven Algorithmus einsetzen werden. Wir verwenden dazu die $h-h/2$ Strategie aus Ferraz-Leite (Provet. 108),wo die folgende Aussage bewiesen wird. - -\begin{defi}[A-Posteriori Fehlerschätzer $\mu$] -sollte - \begin{enumerate} - \item Berechenbar sein, also keine unbekannten enthalten - \item Zuverlässig sein, -\begin{align} - \norm{\phi-\phi_{\ell}} &\leq C_{ref} \cdot \mu -\end{align} - \item Effizient sein, -\begin{align} - \mu &\leq C_{eff} \cdot \norm{\phi - \phi_{\ell}} -\end{align} - \end{enumerate} -\end{defi} -\begin{defi}[Saturationsannahme] -\begin{align} - \norm{\phi -\hat \phi_{\ell}} &\leq C_{sat} \cdot \norm{\phi - \phi_{\ell}} & 0 < C_{sat} < 1 -\end{align} -\end{defi} - -\begin{defi}[$\ell-\ell/2$ - Schätzer] -\begin{align} - \eta_{\ell} &:= \norm{\hat \phi_{\ell} - \phi_{\ell}}_{H^{-1/2}} -\end{align} -Der Fehlerschätzer ist aber nur unter der Saturationsannahme zuverlässig und ist effizient mit $C_{\tt eff} = 1$ -\end{defi} -Da aber die $\norm{\cdot}_{H^{1/2}}$ schlecht zu berechnen ist, wenden wir die $L_2$-Projektion an. -\begin{defi}[ersetzen von $\norm{\cdot}_{H^{1/2}}$] - \begin{align} - \mu_{\ell} &:= \norm{ \varrho^{1/2} (\hat \phi_{\ell} - {\phi_{\ell}})}_{L_2(\Gamma)} - \end{align} -$\mu_{\ell}$ ist da $\mu_{\ell} \approx \eta_{\ell}$ gilt noch immer zuverlässig und effizient. -\end{defi} -Um sich unnötige Berechnungen zu sparen ist es sinnvoll $\phi_{\ell}$ zu ersetzen. -\begin{defi}[ersetzen von $\phi_{\ell}$] - \begin{align} - \tilde \mu_{\ell} &:= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell})}_{L_2(\Gamma)} - \end{align} -wobei $\Pi_{\ell}$ die $L_2$ Projektion auf $P^0(\T_{\ell})$ ist. -\end{defi} -\begin{sat}[A-posteriori Fehlerschätzer] Seien also: -\begin{align} - \eta_{\ell} &= \enorm{\hat \phi_{\ell} - \phi_{\ell}}\\ - \tilde\eta_{\ell} &= \enorm{\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell}}\\ - \mu_{\ell} &= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \phi_{\ell})}_{L^2(\Gamma)}\\ - \tilde\mu_{\ell} &= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell})}_{L^2(\Gamma)} -\end{align} -Dann gilt auf isotropen Netzen: +\section{Analytische und Semi-analytische Berechnung} +\todo{ \begin{itemize} - \item Schätzer sind equivalent\\ $\tilde \mu_{\ell} \leq \mu_{\ell} \leq \sqrt{2}C_3 \eta_{\ell}$ - \item sie sind effizient - \item sie sind unter Saturationsannahme auch zuverlässig + \item Modellproblem $A_{kj} = \int_{Tj}\int_{Tk} \kappa(x,y) ds_y ds_x$ +mit $\kappa(.,.)$ asymptotisch glatt und $Tj, Tk$ achsenorientierte +Rechtecke. +\item Das ist das Kernstueck der Arbeit (Uebertragung der Mayr-Bakk auf 3D). +\item Warum erwarten man Instabilitaet bei analytischer Berechnung? +\item Beweis von exponentieller Konvergenz bei semidiskreter/volldiskreter +Berechnung von gewissen Matrix-Eintraegen +\item Was fuer Faelle sollte man betrachten? Wann rechnet man +analytisch/semidiskret/volldiskret \end{itemize} -\end{sat} -\begin{bew} - Siehe S.F. Paper $\mapsto$ THM 3.2 \& 3.4 -\end{bew} +} +\subsection{Gauss-Quadratur} +\begin{displaymath} + \int_0^l f(x) dx \approx \sum_{k=0}^n w_kf(x_k) =: \Q(f) +\end{displaymath} +\subsection{Quadratur über ein Element} +\begin{displaymath} + \int_{T} \int_{\tilde T} \approx \q_{T} \int_{\tilde T} +\end{displaymath} +Das Integral über \Tb~ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über \Ta~ wird nun mittels einer doppelten Gauss-Quadratur berechnet. +\subsubsection{Zulässigkeitsbedingung} +\begin{displaymath} + \D (T, \tilde T) \geq \mu \min\{ \G(T) , \G(\tilde T)\} +\end{displaymath} +Nur wenn die Zulässigkeitsbedingung gültig ist wird die Quadratur verwendet. Sonst wird Analytisch gerechnet. -\clearpage +\subsection{Quadratur über eine Achse} +\begin{displaymath} + \int_{T} \int_{\tilde T} \approx \q_{I_1} \q_{J_1} \int_{I_2}\int_{J_2} +\end{displaymath} +\subsubsection{Zulässigkeitsbedingung} +\begin{displaymath} + \D (I_1, J_1) \geq \mu \max\{ \L(I_2) , \L(J_2)\} +\end{displaymath} +\subsection{Quadratur über eine Seite} +\begin{displaymath} + \int_{T} \int_{\tilde T} \approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2} +\end{displaymath} + +$\enorm{a}-\norm{b}$ -\section{Analytische Berechnung der Integrale} + +\section{Analytische Berechnung von Integral im Fall des +Einfachschichtpotentials} +\todo{ +\begin{itemize} + \item Zusammenfassung des Maischak-Papers + \item Ergebnisse ohne Beweise + \item Betonen, wo Sie im Maischak-Paper einen Fehler gefunden haben +\end{itemize} +} Es seien $T_j,T_k \subseteq\R^3$ zwei beschränkte, achsenorientierte Rechtecke in $\R^3$. Die Berechnung der Matrix für das Galerkin-Verfahren benötigt die Auswertung von \begin{eqnarray}\label{math:V} @@ -363,11 +369,11 @@ Für beliebige $x,p,y, \lambda \in \R$ mit $\lambda \neq 0$ gilt die Rekursionsf \end{eqnarray*} Für $p \in \{1/2 , 0 , -1/2 , -1, -3/2\}$ gilt explizit: \begin{eqnarray*} - g(1/2;y;x;\lambda) &:=& \frac{y-x}{2}\{(y-x)^2+\lambda^2\}^{1/2} + \frac{\lambda^2}{2} \text{arsinh} \frac{y-x}{|\lambda|}\\ - g(0;y;x;\lambda) &:=& y-x\\ - g(-1/2;y;x;\lambda) &:=& \text{arsinh} \frac{y-x}{|\lambda|}\\ - g(-1;y;x;\lambda) &:=& \frac{\text{arctan} \frac{y-x}{|\lambda|}}{ \todo{\lambda}}\\ - g(-3/2;y;x;\lambda) &:=& \frac{y-x}{\lambda^2}\{(y-x)^2+\lambda^2\}^{-1/2} + g(1/2;y;x;\lambda) &=& \frac{y-x}{2}\{(y-x)^2+\lambda^2\}^{1/2} + \frac{\lambda^2}{2} \text{arsinh} \frac{y-x}{|\lambda|}\\ + g(0;y;x;\lambda) &=& y-x\\ + g(-1/2;y;x;\lambda) &=& \text{arsinh} \frac{y-x}{|\lambda|}\\ + g(-1;y;x;\lambda) &=& \frac{\text{arctan} \frac{y-x}{|\lambda|}}{ \todo{\lambda}}\\ + g(-3/2;y;x;\lambda) &=& \frac{y-x}{\lambda^2}\{(y-x)^2+\lambda^2\}^{-1/2} \end{eqnarray*} \subsection{doppel Integral} @@ -441,144 +447,107 @@ F_{ort}(x_1,x_2,y_1,y_2,\delta_1,\delta_2,\delta_3) &:=& \int \int \int \int \le dy_3 dy_2 dx_2 dx_1 \end{eqnarray*} -\subsection{Bestimmtes Integral} -\begin{eqnarray*} -&&\frac{1}{4\pi} \int_T \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ -&\approx& \frac{1}{4\pi} \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1}\int_0^{\tilde k_2} - \dif{}{y_2} \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,y_1,y_2) -dy_2 dy_1 dx_2 dx_1\\ -% -&=& \frac{1}{4\pi}\big( \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1} -\dif{}{y_1} \dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,y_1,\tilde k_2) -- -\dif{}{y_1} \dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,y_1,0) \big) - dy_1 dx_2 dx_1\\ -% -&=& \frac{1}{4\pi}\big( \int_0^{k_1}\int_0^{k_2} - \dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,\tilde k_1,\tilde k_2) -- - \dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,\tilde k_1,0) \\ -&&- -\dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,0,\tilde k_2) -+ - \dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,0,0)\big) - dx_2 dx_1\\ -% -&=& \frac{1}{4\pi}\big( \int_0^{k_1} -\dif{}{x_1} - F_{par/ort}(x_1,k_2,\tilde k_1,\tilde k_2) -- -\dif{}{x_1} - F_{par/ort}(x_1,k_2,\tilde k_1,0) \\ -&&- -\dif{}{x_1} - F_{par/ort}(x_1,k_2,0,\tilde k_2) -+ -\dif{}{x_1} - F_{par/ort}(x_1,k_2,0,0)\\ -&&- %% -\dif{}{x_1} - F_{par/ort}(x_1,0,\tilde k_1,\tilde k_2) -+ -\dif{}{x_1} - F_{par/ort}(x_1,0,\tilde k_1,0) \\ -&&+ -\dif{}{x_1} - F_{par/ort}(x_1,0,0,\tilde k_2) -- -\dif{}{x_1} - F_{par/ort}(x_1,0,0,0)\big) - dx_1\\ -% -&=& \frac{1}{4\pi}\big( - F_{par/ort}(k_1,k_2,\tilde k_1,\tilde k_2) -- - F_{par/ort}(k_1,k_2,\tilde k_1,0) -- - F_{par/ort}(k_1,k_2,0,\tilde k_2) -+ - F_{par/ort}(k_1,k_2,0,0)\\ -&&- - F_{par/ort}(k_1,0,\tilde k_1,\tilde k_2) -+ - F_{par/ort}(k_1,0,\tilde k_1,0) -+ - F_{par/ort}(k_1,0,0,\tilde k_2) -- - F_{par/ort}(k_1,0,0,0)\\ -&&- %% - F_{par/ort}(0,k_2,\tilde k_1,\tilde k_2) -- - F_{par/ort}(0,k_2,\tilde k_1,0) -- - F_{par/ort}(0,k_2,0,\tilde k_2) -+ - F_{par/ort}(0,k_2,0,0)\\ -&&- - F_{par/ort}(0,0,\tilde k_1,\tilde k_2) -+ - F_{par/ort}(0,0,\tilde k_1,0) -+ - F_{par/ort}(0,0,0,\tilde k_2) -- - F_{par/ort}(0,0,0,0)\big) -\end{eqnarray*} - - - -\section{Semi-analytische Berechnung bestimmter Integrale} -\subsection{Gauss-Quadratur} -\begin{displaymath} - \int_0^l f(x) dx \approx \sum_{k=0}^n w_kf(x_k) =: \Q(f) -\end{displaymath} - -\subsection{Quadratur über ein Element} -\begin{displaymath} - \int_{T} \int_{\tilde T} \approx \q_{T} \int_{\tilde T} -\end{displaymath} -Das Integral über \Tb~ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über \Ta~ wird nun mittels einer doppelten Gauss-Quadratur berechnet. -\subsubsection{Zulässigkeitsbedingung} -\begin{displaymath} - \D (T, \tilde T) \geq \mu \min\{ \G(T) , \G(\tilde T)\} -\end{displaymath} -Nur wenn die Zulässigkeitsbedingung gültig ist wird die Quadratur verwendet. Sonst wird Analytisch gerechnet. - - -\subsection{Quadratur über eine Achse} -\begin{displaymath} - \int_{T} \int_{\tilde T} \approx \q_{I_1} \q_{J_1} \int_{I_2}\int_{J_2} -\end{displaymath} -\subsubsection{Zulässigkeitsbedingung} -\begin{displaymath} - \D (I_1, J_1) \geq \mu \max\{ \L(I_2) , \L(J_2)\} -\end{displaymath} - - -\subsection{Quadratur über eine Seite} -\begin{displaymath} - \int_{T} \int_{\tilde T} \approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2} -\end{displaymath} - -$\enorm{a}-\norm{b}$ - - -\section{Zulässigkeitsbedingungen} -\subsection{Lagen der Elemente} -so so oder so... -\subsection{Fehlerschätzer} -\subsection{Möglichkeiten für Kriterien} - -\section{V berechnen} - - -\section{Implementierung} +% \subsection{Bestimmtes Integral} +% \begin{eqnarray*} +% &&\frac{1}{4\pi} \int_T \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ +% &\approx& \frac{1}{4\pi} \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1}\int_0^{\tilde k_2} +% \dif{}{y_2} \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} +% F_{par/ort}(x_1,x_2,y_1,y_2) +% dy_2 dy_1 dx_2 dx_1\\ +% % +% &=& \frac{1}{4\pi}\big( \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1} +% \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} +% F_{par/ort}(x_1,x_2,y_1,\tilde k_2) +% - +% \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} +% F_{par/ort}(x_1,x_2,y_1,0) \big) +% dy_1 dx_2 dx_1\\ +% % +% &=& \frac{1}{4\pi}\big( \int_0^{k_1}\int_0^{k_2} +% \dif{}{x_2} \dif{}{x_1} +% F_{par/ort}(x_1,x_2,\tilde k_1,\tilde k_2) +% - +% \dif{}{x_2} \dif{}{x_1} +% F_{par/ort}(x_1,x_2,\tilde k_1,0) \\ +% &&- +% \dif{}{x_2} \dif{}{x_1} +% F_{par/ort}(x_1,x_2,0,\tilde k_2) +% + +% \dif{}{x_2} \dif{}{x_1} +% F_{par/ort}(x_1,x_2,0,0)\big) +% dx_2 dx_1\\ +% % +% &=& \frac{1}{4\pi}\big( \int_0^{k_1} +% \dif{}{x_1} +% F_{par/ort}(x_1,k_2,\tilde k_1,\tilde k_2) +% - +% \dif{}{x_1} +% F_{par/ort}(x_1,k_2,\tilde k_1,0) \\ +% &&- +% \dif{}{x_1} +% F_{par/ort}(x_1,k_2,0,\tilde k_2) +% + +% \dif{}{x_1} +% F_{par/ort}(x_1,k_2,0,0)\\ +% &&- %% +% \dif{}{x_1} +% F_{par/ort}(x_1,0,\tilde k_1,\tilde k_2) +% + +% \dif{}{x_1} +% F_{par/ort}(x_1,0,\tilde k_1,0) \\ +% &&+ +% \dif{}{x_1} +% F_{par/ort}(x_1,0,0,\tilde k_2) +% - +% \dif{}{x_1} +% F_{par/ort}(x_1,0,0,0)\big) +% dx_1\\ +% % +% &=& \frac{1}{4\pi}\big( +% F_{par/ort}(k_1,k_2,\tilde k_1,\tilde k_2) +% - +% F_{par/ort}(k_1,k_2,\tilde k_1,0) +% - +% F_{par/ort}(k_1,k_2,0,\tilde k_2) +% + +% F_{par/ort}(k_1,k_2,0,0)\\ +% &&- +% F_{par/ort}(k_1,0,\tilde k_1,\tilde k_2) +% + +% F_{par/ort}(k_1,0,\tilde k_1,0) +% + +% F_{par/ort}(k_1,0,0,\tilde k_2) +% - +% F_{par/ort}(k_1,0,0,0)\\ +% &&- %% +% F_{par/ort}(0,k_2,\tilde k_1,\tilde k_2) +% - +% F_{par/ort}(0,k_2,\tilde k_1,0) +% - +% F_{par/ort}(0,k_2,0,\tilde k_2) +% + +% F_{par/ort}(0,k_2,0,0)\\ +% &&- +% F_{par/ort}(0,0,\tilde k_1,\tilde k_2) +% + +% F_{par/ort}(0,0,\tilde k_1,0) +% + +% F_{par/ort}(0,0,0,\tilde k_2) +% - +% F_{par/ort}(0,0,0,0)\big) +% \end{eqnarray*} + + + + +\section{fasträguläre Partionierung in \Matlab} +\todo{ +\begin{itemize} + \item Datenstruktur + \item Netzverfeinerung + \item Code + kurze Dokumentation des Codes (was passiert wie/wo) +\end{itemize} +} \subsection{Datenstruktur} Für die Implementierung in \Matlab~und C++ wollen wollen wir eine einheitliche Datenstruktur einführen. @@ -626,14 +595,14 @@ Da wir im weiteren Verlauf sowohl adaptive also auch anisotrope Netzverfeinerung \end{enumerate} Zusätzlich wurde auch Typ 5. belegt, welcher als Ergebnis eine volle Teilung 2. ausführt, diese aber schrittweise durch eine 3. Teilung und zwei 4. Teilungen. Aus Sicherheitsgründen wird auch jede vierte volle 2. Teilung durch eine 5. Teilung ausgeführt, da sonst kurzzeitig Seiten mit mehr als zwei Elementen auftreten könnten. -Damit jedem Element $T_i$ eine Teilungsart zugeordnet werden kann, haben wir einen Mar\-kierungsvektor $marked \in {1,2,3,4,5}^M$ eingeführt. Dabei entspricht $marked_i$ der Art der Teilung für das Element $T_i$. Um isotrope und auch uniforme Teilungen zu erleichtern kann statt dem Vektor $marked$ auch nur ein Skalar übergeben werden $marked \in {1,2\dots5}$, wodurch jedes Element mit der gewählten Art verfeinert wird. +Damit jedem Element $T_i$ eine Teilungsart zugeordnet werden kann, haben wir einen Mar\-kierungs\-vektor $marked \in {1,2,3,4,5}^M$ eingeführt. Dabei entspricht $marked_i$ der Art der Teilung für das Element $T_i$. Um isotrope und auch uniforme Teilungen zu erleichtern kann statt dem Vektor $marked$ auch nur ein Skalar übergeben werden $marked \in {1,2\dots5}$, wodurch jedes Element mit der gewählten Art verfeinert wird. -Relevant zum Verfeinern eines Netzes sind also die Koordinaten $COO$, Elemente $ELE$ sowie die Nachbarschaftsrelationen $NTI$ und der Markierungsvektor $marked$. +Relevant zum Verfeinern eines Netzes sind also die Koordinaten $COO$, Elemente $ELE$ sowie die Nachbarschaftsrelationen $NTI$ und der Markierungs\-vektor $marked$. Da wir später einen Fehlerschätzer berechnen wollen, ist es wichtig sich zu jedem Element seine Teilelemente zu merken. Dazu legen wir während der Teilung eine $M \times 4$ Matrix an, in der die maximal vier Elementindizes gespeichert sind. Wenn wir also ein Element in vier gleich große Teile verfeinern, so wird das neue Element links unten das erste sein und alle weiteren folgen gegen den Uhrzeigersinn. Teilen wir ein Element in zwei gleich große Elemente, so werden die doppelt belegten Quadranten auch doppelt eingetragen. Ein gar nicht geteiltes Element wird also vier mal den alten Indizes speichern. Dadurch wird sicher gestellt, dass das arithmetische Mittel über die Elemente immer gültig auszuführen ist. (Siehe Figur:\ref{exmpl13:f2s}) -$[\\hat T_{\ell}, F2S ] = refineQuad(\T_{\ell}, marked);$\\ +$[\T_{\ell}, F2S ] = refineQuad(\T_{\ell}, marked);$\\ $[COO_{fine}, ELE_{fine}, NEI_{fine}, F2S ] = refineQuad(COO, ELE, NEI, marked);$ \begin{figure}[ht] @@ -644,30 +613,6 @@ $[COO_{fine}, ELE_{fine}, NEI_{fine}, F2S ] = refineQuad(COO, ELE, NEI, marked); \end{figure} -\subsection{Fehlerschätzer} -\begin{eqnarray*} -%\mu_{\ell}^2 & = & \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ -%& = &\sum_{T\in \T_{\ell}}\mu_{\ell}(T)^2\\ -\mu_{\ell}(T)^2 & = & \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ -& = & h_{min}(T)\norm{\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ -&& T_j \in \tau_{\ell}, t_j^{(1)},\dots,t_j^{(4)} \in \tau_{\frac l 2} \\ -\phi_{\frac l 2}|_{T_j} & = &x_j^{(1)}\chi_{t_j^{(1)}}+\dots+x_j^{(4)}\chi_{t_j^{(4)}}\\ -\Pi_{\ell}\phi_{\frac l 2}|_{T_j}&=&\frac 1 {\abs{T_j}}\int_{T_j}\phi_{\frac l 2}d\Gamma\\ -& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\int_{t^{(k)}}1d\Gamma\\ -& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\abs{t_j^{(k)}}\\ -& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\frac {\abs{T_j}} 4 \\ -& =&\frac 1 4 (x_j^{(1)} +\dots+x_j^{(4)} )=: m_j\\ -\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2(T_j)}^2 -&=&\sum_{k=1}^4\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2\left(t_j^{(k)}\right)}^2\\ -&=&\sum_{k=1}^4 \int_{t_j^{(k)}}(\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2})^2d\Gamma\\ -&=&\sum_{k=1}^4 \int_{t_j^{(k)}}(x_j^{(k)}-m_j)^2d\Gamma\\ -&=&\sum_{k=1}^4 \abs{t_j^{(k)}}(x_j^{(k)}-m_j)^2\\ -&=&\frac {\abs{T_j}} 4\sum_{k=1}^4 (x_j^{(k)}-m_j)^2\\ -\mu_{\ell}(T_j)^2 & = & \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} -\end{eqnarray*} - - - \subsection{Markieren} @@ -743,7 +688,93 @@ Zum Plotten (Abb.\ref{exmplAA_2DQuad})werden noch folgende Schritte ausgeführt \end{itemize} -\section{Auswertung} +\section{Numerische Experimente} +\todo{ +\begin{itemize} + \item Zusammenfassung der h-h/2 Strategie (Ergebnisse aus Ferraz-DA und +Paper), interessant sind nur eta und tilde-mu +\item adaptiver Algorithmus als Algorithmus und MATLAB-Code +\item Warum braucht man Adaptivitaet? Warum reicht isotrope Verfeinerung +nicht aus? ggf. anhand numerischem Beispiel +\item Beispiele, bei denen anisotrop gerechnet wird, bis Instabilitaet +auftritt, verschiedene Strategien (voll-analytisch + ...) +\item in Plots: Fehler/eta/tilde-mu +\item genau beschreiben: Was ist das Beispiel? Was sieht man im Plot? +Wie interpretiert man das? +\end{itemize} +} + +\subsection{Fehlerschätzer} +In diesem Abschnitt definieren wir die a-posteriori Fehlerschätzer, die wir im Folgenden zur Steuerung des adaptiven Algorithmus einsetzen werden. Wir verwenden dazu die $h-h/2$ Strategie aus Ferraz-Leite (Provet. 108),wo die folgende Aussage bewiesen wird. + +\begin{defi}Es bezeichne $\phi$ die Lösung von Formel \ref{Formel}, $\phi_{\ell}$ die Galerkinlösung auf dem Gitter $\T_{\ell}$ und $\hat \phi_{\ell}$ die Galerkinlösung auf dem uniform verfeinerten Gitter $\hat \T_{\ell}$. Dann gilt, der Schätzer +\begin{align} + \eta_{\ell} &:= \enorm{\hat \phi_{\ell} - \phi_{\ell}} +\end{align} +ist effizient +\begin{align} + \eta_{ell} &\leq \enorm{\phi - \phi_{\ell}} +\end{align} +und unter der Saturationsannahme +\begin{align} + \norm{\phi -\hat \phi_{\ell}} &\leq C_{sat} \cdot \norm{\phi - \phi_{\ell}} & 0 < C_{sat} < 1 +\end{align} +zuverlässig +\begin{align} + \enorm{\phi - \phi_{\ell}}&\leq \frac 1{\sqrt{1-C_{sat}}}\eta_{\ell}. +\end{align} + +\end{defi} +Der Schätzer ist berechenbar, liefert aber keine lokalen Beiträge und kann daher nicht unmittelbar verwendet werden, um einen adaptiven Algorithmus zu steuern. + +\begin{sat}[A-posteriori Fehlerschätzer] Seien also: +\begin{align} + \eta_{\ell} &= \enorm{\hat \phi_{\ell} - \phi_{\ell}}\\ + \tilde\eta_{\ell} &= \enorm{\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell}}\\ + \mu_{\ell} &= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \phi_{\ell})}_{L^2(\Gamma)}\\ + \tilde\mu_{\ell} &= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell})}_{L^2(\Gamma)} +\end{align} +wobei $\Pi_{\ell}$ die $L_2$ Projektion auf $P^0(\T_{\ell})$ ist. +\end{sat} +$\mu_{\ell}$ ist da $\mu_{\ell} \approx \eta_{\ell}$ gilt noch immer zuverlässig und effizient.\\ +Dann gilt auf isotropen Netzen: +\begin{itemize} + \item Schätzer sind equivalent\\ $\tilde \mu_{\ell} \leq \mu_{\ell} \leq \sqrt{2}C_3 \eta_{\ell}$ + \item sie sind effizient + \item sie sind unter Saturationsannahme auch zuverlässig +\end{itemize} + +\begin{bew} + Siehe S.F. Paper $\mapsto$ THM 3.2 \& 3.4 +\end{bew} + + +\begin{eqnarray*} +%\mu_{\ell}^2 & = & \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ +%& = &\sum_{T\in \T_{\ell}}\mu_{\ell}(T)^2\\ +\mu_{\ell}(T)^2 & = & \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ +& = & h_{min}(T)\norm{\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ +&& T_j \in \tau_{\ell}, t_j^{(1)},\dots,t_j^{(4)} \in \tau_{\frac l 2} \\ +\phi_{\frac l 2}|_{T_j} & = &x_j^{(1)}\chi_{t_j^{(1)}}+\dots+x_j^{(4)}\chi_{t_j^{(4)}}\\ +\Pi_{\ell}\phi_{\frac l 2}|_{T_j}&=&\frac 1 {\abs{T_j}}\int_{T_j}\phi_{\frac l 2}d\Gamma\\ +& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\int_{t^{(k)}}1d\Gamma\\ +& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\abs{t_j^{(k)}}\\ +& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\frac {\abs{T_j}} 4 \\ +& =&\frac 1 4 (x_j^{(1)} +\dots+x_j^{(4)} )=: m_j\\ +\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2(T_j)}^2 +&=&\sum_{k=1}^4\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2\left(t_j^{(k)}\right)}^2\\ +&=&\sum_{k=1}^4 \int_{t_j^{(k)}}(\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2})^2d\Gamma\\ +&=&\sum_{k=1}^4 \int_{t_j^{(k)}}(x_j^{(k)}-m_j)^2d\Gamma\\ +&=&\sum_{k=1}^4 \abs{t_j^{(k)}}(x_j^{(k)}-m_j)^2\\ +&=&\frac {\abs{T_j}} 4\sum_{k=1}^4 (x_j^{(k)}-m_j)^2\\ +\mu_{\ell}(T_j)^2 & = & \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} +\end{eqnarray*} + + + + + + \todo{Viele Bunte Bilder und Testergebnisse} \begin{itemize} \item gerechnet wurde $V\phi = 1$ -- 2.47.3