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a/doc/doc.tex +++ b/doc/doc.tex @@ -111,7 +111,7 @@ \clearpage \section{Einleitung} -\todo{ +\todo{ \scriptsize \begin{itemize} \item kurz + praegnant, worum es geht (vgl. Mayr-Bakk, wo das exakt eine Seite ist) \end{itemize} @@ -119,11 +119,30 @@ \noindent 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 &= g &&\text{ auf }\Gamma,&& -\end{align} +\begin{align*} +- \varDelta u &= 0 \quad\text{ in } \Omega \subset \R^3,\\ +u &= g \quad \text{ auf }\Gamma, +\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.\\ +In Abschnitt 2 stellen wir zunächst die Randelementmethode für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen vor. Dabei verwenden wir den indirekten Ansatz um anschließend mithilfe des Galerkin-Verfahrens die Gleichung zu lösen. An dieser Stelle werden wir auch kurz die Parametrisierung des Randes vorstellen. Denn wir werden im Folgenden den Rand in affine achsenorientierte Rechtecke $T$ zerlegen, das heißt die Punkte in einem Rechteck liegen in einer zu den Achsen des Koordinatensystem parallelen Ebene.\\ +In Abschnitt 3 werden wir uns mit der approximativen Berechnung des Doppelintegrals +\begin{align}\label{math:intro:int} + \int_{T_j} \int_{T_k} \kappa(\bs x, \bs y) ds_{\bs y} ds_{\bs x} +\end{align} +beschäftigen. $\kappa$ sei hierbei ein asymptotisch glatte Kernfunktion. Speziell interessieren wir uns hierbei für die asymptotisch glatte Funktion $\abs{\bs x - \bs y}^{-1}$. Ziel wird es sein das äußere Integral durch eine Gauss-Quadratur zu erstetzen. Unter bestimmten Zulässigkeits\-beding\-ungen werden wir zeigen, dass die Gauss-Quadratur bei steigendem Quadraturgrad exponentiell schnell gegen den exakten Wert konvergiert. Weiterhin betrachten wir die in Abschnitt 2 auftretende Matrix $A \in R^{n\times n}$ deren Einträge $A_{jk}$ die Gestalt \eqref{math:intro:int} hat. Denn wir werden eine approximative Matrix $A_p$ aufstellen, welche das durch Quadratur approximierte Integral für die zulässigen Einträge und für alle anderen das exakte Integral verwendet. Hiermit können wir dann Zeigen, dass die approximative Matrix unter der Frobeninusnorm exponentiell schnell gegen $A$ konvergiert.\\ +In Abschnitt 4 fassen wir kurz zusammen, wie wir das Doppelintegral in einfache Integrale zerlegen und andschließend voll analytisch berechnen können. Hierzu werden wir uns weitgehend an \cite{mai:3dbem} orientieren.\\ +Abschließend werden wir kurz die numerische Umsetzung der Techniken vorstellen und anhand von numerischen Beispielen vergleiche. Hierbei wird uns die voll analytische und approximative Berechnung, sowie die adaptive und uniforme Netzverfeinerung besonders Interessieren. + +\clearpage + +\section{Randelementmethode} +\todo{ \scriptsize +\begin{itemize} +\item kuerzer als in Ferraz-DA \cite{fer:errbem} +\item insbesondere fastregulaere Triangulierung + K-Gitter (vgl. Ferraz-DA \cite{fer:errbem}) +\end{itemize} +} + % \noindent % Wir wissen, dass die Laplace-Gleichung erfüllt wird durch: @@ -233,15 +252,7 @@ wobei $\varDelta u := \partial_x^2u+\partial_y^2u$ den Laplace-Operator bezeichn % \hat \phi_{\ell} &= D \cdot y\\ % \end{align} -\clearpage -\section{Randelementmethode} -\todo{ -\begin{itemize} -\item kuerzer als in Ferraz-DA \cite{fer:errbem} -\item insbesondere fastregulaere Triangulierung + K-Gitter (vgl. Ferraz-DA \cite{fer:errbem}) -\end{itemize} -} \subsection{Netze} Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. @@ -335,7 +346,7 @@ Für eine Partition $\T_{\ell}$ bezeichnen wir mit $\widehat \T_{\ell}$ jene Par \clearpage \section{Analytische und Semi-analytische Berechnung} -\todo{ +\todo{ \scriptsize \begin{itemize} \item Modellproblem $A_{kj} = \int_{Tj}\int_{Tk} \kappa(x,y) ds_y ds_x$ mit $\kappa(.,.)$ asymptotisch glatt und $Tj, Tk$ achsenorientiertes @@ -573,7 +584,7 @@ wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ in $x_{\b \clearpage \section{Analytische Berechnung} -\todo{ +\todo{ \scriptsize \begin{itemize} \item Zusammenfassung des Maischak-Papers \cite{mai:3dbem} \item Ergebnisse ohne Beweise @@ -582,11 +593,11 @@ wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ in $x_{\b } \noindent In diesem Abschnitt wollen wir uns mit der analytischen Berechnung der Galerkin-Matrix -\begin{align*}\label{math:V} -A_{jk} &= \frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x} \in \R^3. -\end{align*} +\begin{align}\label{math:analy:int} +\frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x} \in \R^3. +\end{align} mit zwei beschränkte, achsenorientierte Rechtecke $T_j,T_k \subseteq\R^3$ beschäftigen. -Dazu wollen wir angelehnt an \cite{mai:3dbem} zwei Integrale anschauen, welche durch das Aufspalten des Integrals $A_{jk}$ auftreten. +Dazu wollen wir angelehnt an \cite{mai:3dbem} zwei Integrale anschauen, welche durch das Aufspalten des Integrals \eqref{math:analy:int} auftreten. \\\noindent \begin{lem} Für das Integral @@ -646,7 +657,7 @@ Für alle weiteren relevante Fälle $p = -3/2 +k : k \in \N\backslash0$ können \end{lem} \subsection{Integral über zwei Elemente} -Bei der Berechnung von \eqref{math:V} werden wir geometrisch zwischen zwei Fällen zu unterscheiden. Entweder die beiden Elemente liegen in parallelen Ebenen oder in orthogonalen Ebenen. +Bei der Berechnung von \eqref{math:analy:int} werden wir geometrisch zwischen zwei Fällen zu unterscheiden. Entweder die beiden Elemente liegen in parallelen Ebenen oder in orthogonalen Ebenen. \subsubsection{Parallele Elemente} @@ -815,7 +826,7 @@ für orthogonal liegende Elemente schreiben: \clearpage \section{fasträguläre Partionierung in \Matlab} -\todo{ +\todo{ \scriptsize \begin{itemize} \item Datenstruktur \item Netzverfeinerung @@ -964,7 +975,7 @@ Zum Plotten (Abb.\ref{fig:exmplAA_2DQuad})werden noch folgende Schritte ausgefü \clearpage \section{Numerische Experimente} -\todo{ +\todo{ \scriptsize \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 -- 2.47.3