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b/doc/doc.tex @@ -1,4 +1,4 @@ -\documentclass[a4paper,11pt,fleqn]{article} +\documentclass[a4paper,12pt,fleqn]{article} \usepackage{fullpage} %Seiten etwas Größer \usepackage[ngerman]{babel} %Sprachpacket für Überschriften @@ -47,6 +47,9 @@ \def\T{\mathcal{T}} \def\K{\mathcal{K}} \def\S{\mathcal{S}} +\def\P{\mathcal{P}} +\def\C{\mathcal{C}} +\def\I{\mathcal{I}} \def\oder{\vee} \def\und{\wedge} @@ -236,11 +239,20 @@ Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. \begin{defi}[Rechteck] Sei $\bs x \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $a,b \in \R$ mit $a,b > 0$. Dann heißt \begin{align} - T := \{\bs x + \lambda a {\bs a} + \gamma b {\bs b} ~|~ \lambda,\gamma \in[0,1]\} + T := \{\bs x + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\} \end{align} achsenorientiertes Rechteck. \end{defi} +\begin{defi} + Sei $T$ ein Rechteck, Dann heißt + \begin{align} + \gamma := [0,1]^2 \to T: \lambda_1,\lambda_2 \mapsto \bs x + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} + \end{align} +die zu $T$ zugehörige Parametrisierung. +\end{defi} + + \begin{defi}[hängender Knoten] Ist $c \in T_j \cap T_k$ ein Eckpunkt von $T_j$ aber nicht von $T_k$, so nennen wir $c$ einen hängenden Knoten. \end{defi} @@ -306,7 +318,7 @@ Ein Element $T \in \T$ wird isotrop in vier Elemente $T_1,\ldots,T_4$ geteilt, w \item \label{alg:refine:first} Definiere $\S_{\ell}^{(i+1)} := \{ e_T, \tilde e_T ~|~ T \in \T_{\ell}$ mit $\exists$ hängender-Knoten auf $T$ und $\exists T' \in\T_{\ell}$ sodass $T \cap T' = e_{T'},\in \S_{\ell}^{(i)} \}$ \item - Falls $\S_{ell}^{(i)} \subsetneq \S_{ell}^{(i+1)}$, $i = i+1$ und gehe zu Schritt \ref{alg:refine:first} + Falls $\S_{\ell}^{(i)} \subsetneq \S_{\ell}^{(i+1)}$, $i = i+1$ und gehe zu Schritt \ref{alg:refine:first} \item Teile alle Elemente aus $\T_{\ell}$ bezüglich der markierten Kanten $\S_{\ell}^{(0,1,\ldots,i)}$, wie in der lokalen Verfeinerung vorgegeben \end{enumerate} \end{alg} @@ -338,17 +350,78 @@ beziehungsweise als Spezialfall davon die Berechnung des Integrals A_{jk} &=\frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x}. \end{align} unter bestimmten Vorraussetzungen an die affinen Randstücke $T_j,T_k$ und den asymptotisch glatten Integranden $\kappa : \R^3 \times \R^3 \to \R$. - - +\subsection{Interpolation} +Zunächst wollen wir die Chebyshev-Interpolation auf dem Intervall [0,1] definieren. Hierzu sei $\P_p$ die Menge aller Polynome vom Grad $p$. +\begin{defi} + Seien $x_0,\ldots,x_p \in [0,1]$ paarweise verschiedene Knoten und $y_0,\ldots,y_p$ die zugehörigen Funktionswerte. Die Lagrange'sche Interpolationsaufgabe lautet dann folgendermaßen: Finde ein Polynom $q$ vom Grad $p-1$, sodass + \begin{align*} + q(x_i) &=y_i \quad \text{für alle} i=0,\ldots,p. + \end{align*} +\end{defi} +Ferner definieren wir die zu den Knoten $x_0,\ldots,x_p$ zugehörigen Lagrange-Polynome durch +\begin{align} + L_j(x) &= \prod_{i=1,i\neq j}^p \frac{x-x_i}{x_j-xi}. +\end{align} +\todo{cite} zeigt, dass die Lagrange'sche Interpolationsaufgabe eine eindeutige Lösung besitzt, die gegeben ist durch +\begin{align} + q&=\sum_{j=0}^py_jL_j. +\end{align} +Wir definieren den Interpolationsoperator $\I_pu$ durch +\begin{align} + \I_pu &:= \sum_{j=0}^p u(x_j)L_j. +\end{align} +Außerdem definieren wir die Lebesgue-Konstante durch +\begin{align} + \Lambda_p &:= \max_{x\in[0,1]} \sum_{j=0}^p\abs{L_j(x)}. +\end{align} +\todo{cite} stellt einen Bezug zwischen der Bestapproximation und der Polynominterpolation her und ist später von Bedeutung. +\begin{sat} + Sei $\Lambda_p$ die Lebesgue-Konstante zu $p$ paarweise verschiedener Knoten $x_1,\ldots,x_p$, die alle im Intervall $[0,1]$ liegen. Dann gilt für jedes $u \in \C[0,1]$ + \begin{align} + \norm{\I_pu-u}_{\infty} &\leq (1 + \Lambda_p) \min_{v\in \P_p} \norm{u-v}_{\infty}, + \end{align} +wobei $\P_p$ die Menge aller Polynome vom Grad $p$ bezeichnet. +\hfill $\square$ +\end{sat} +Eine einfache Fehlerabschätzung für den Interpolationsfehler liefert \todo{cite}. Unter der Voraussetzung $u \in \C^{p+1}([0,1])$ gilt: +\begin{align} +\norm{u-\I_pu}_{\infty,[0,1]} &\leq \max_{x\in[0,1]} \left| \prod_{j=0}^n (x-x_j) \right| \frac{1}{(p+1)!} \norm{u^{(p+1)}}_{\infty,[0,1]} +\end{align} +Bei der Chebyshev-Interpolation wählt man die Knoten in Bezug auf diese Abschätzung optimal, indem man den ausschließlich durch die Knotenwahl bestimmten Term +\begin{align*} + \max_{x\in[0,1]} \left| \prod_{j=0}^n (x-x_j) \right| +\end{align*} +minimiert. Die Läsung dieses Minimierungsproblrems sind die NUllstellen der Chebyshev-Polynome. Diese sind gegeben durch +\begin{align} + x_j &= \frac12 \cos \left( \frac{2j-1}{p} \frac12 \right) + \frac12 \quad \text{für} j=1,\ldots,p. +\end{align} +Sieh dazu auch \todo{cite}.\\ Im folgenden bezeichne $\I_p$ immer den Chebyshev-Interpolationsoperator. \subsection{Gauss-Quadratur} +Im Folgenden wollen wir die klassische Gauss-Quadratur definieren.\\ +Unter einer Quadratur verstehen wir die approximative Berechnung eines Integrals der Form: +\begin{align} + \int_0^1 f(x) dx +\end{align} +durch eine Summe +\begin{align} + \Q(f) &:= \sum_{k=0}^n w_kf(x_k). +\end{align} +Die $x_k$ sind hierbei die Knoten, die $w_k$ Gewichte. +Eine Quadratur heißt exakt für eine Funktion $f$, falls $\Q(f) = \int_0^s f(x) dx$ ist.\\ +Eine Quadratur hat den Exaktheitsgrad $m$, wenn sie für Polynome bis zum Grad $m$ exakt ist.\\ +Eine Quadratur hießt interpolatorisch (vom Grad $n$), falls für jede integrierbare, auf $(0,1)$ stetige Funktion $f$ +\begin{align} + \Q_n(f) &:= \sum_{k=0}^n w_kp(x_k) +\end{align} +gilt, wobei $p$ das Interpolationspolynom von $f$ vom Grad $n$ zu den Knoten $x_0,\ldots,x_n$ ist. \begin{align} \Q(f) := \sum_{k=0}^n w_kf(x_k) \approx \int_0^s f(x) dx \end{align} \subsection{Bedingung} \begin{align} - dist (T, \tilde T)&\geq \mu \min\{ diam(T) , diam(\tilde T)\}\\ - dist (I_1, J_1) &\geq \mu \max\{ len(I_2) , len(J_2)\} + dist (T, \tilde T)&\geq \zeta \min\{ diam(T) , diam(\tilde T)\}\\ + dist (I_1, J_1) &\geq \zeta \max\{ len(I_2) , len(J_2)\} \end{align} \subsection{Semianalytisch} @@ -364,6 +437,26 @@ Das Integral über \Tb~ kann durch die analytische Doppelintegral Funktion erset \end{align} +\begin{defi} + Die Kern-Funktion $\kappa(\bs x,\bs y)$ heißt asymptotisch glatt, falls sie glatt ist für $\bs x \neq \bs y$ und Konstanten $c_1,c_2>0$ und eine Ordnung der Singularität $s \in \R$ existieren, sodass + \begin{align} + \abs{\partial_{\bs x}^{\alpha}\partial_{\bs y}^{\beta}\kappa(\bs x, \bs y)} &\leq c_1(c_2(\abs{\bs x - \bs y})^{-(\abs{\alpha}-\abs{\beta}+s)}(\alpha + \beta)! + \end{align} +für alle Multiindizes $\alpha,\beta \in \N_0^d$ mit $\abs{\alpha} + \abs{\beta}\geq 1$ gilt. +\end{defi} +Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktionen besonders gut durch Polynome interpolieren. Hierzu benötigen wir das Lemma \todo{cite}: +\begin{lem} + Sei $J \subseteq \R$ ein abgeschlossenes und beschränktes Intervall und angenommen die Funktion $u$ erfüllt für Konstanten $C_u,\gamma_u > 0$ + \begin{align} + \norm{u^{(n)}}_{\infty,J} &\leq C_u \gamma_u^nn!\text{~~~für alle} n \in \N_0. + \end{align} +Dann gilt für alle $k\in \N_0$ +\begin{align} + \min_{v \in \P_p} \norm{u-v}_{\infty,J} &\leq C_u 4e(1+\gamma_u\abs{J})(p+1)\left(1+\frac{2}{\gamma_u\abs{J}}\right)^{-(p+1)}. +\end{align} +\end{lem} +\hfill$\square$ + \section{Analytische Berechnung vom Integral im Fall des Einfachschichtpotentials} \todo{ @@ -810,9 +903,9 @@ Dann gilt auf isotropen Netzen: \item sie sind unter Saturationsannahme auch zuverlässig \end{itemize} -\begin{bew} + Siehe S.F. Paper $\mapsto$ THM 3.2 \& 3.4 -\end{bew} + \begin{align*} -- 2.47.3