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zIO+y^*ndO)cu-1P8I8hsWkLh)xT)zvcCRKfBq|M+E;~84ks~-FAIrN!xTlI=F(H#sYdjwlQDvy mb>jcUxTtg1tj?9I7WJ?Cc7NZJPCVejpsI7ql-@6$*7;xE@yEIV diff --git a/doc/doc.tex b/doc/doc.tex index 3286aec..36a6ad9 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -132,7 +132,7 @@ In Abschnitt 3 werden wir uns mit der approximativen Berechnung des Doppelintegr \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.\\ +beschäftigen. $\kappa(\bs x, \bs y)$ 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. @@ -269,16 +269,17 @@ achsenorientiertes Rechteck. Ferner heißt \end{align*} die zu $T$ zugehörige Parametrisierung. \end{defi} +\todo{ \begin{bem} -Weiterhin werden wir die vier Ecken des Rechtecks -\begin{align*} - \K_T&:=\{k = \gamma_T(x,y) ~|~ x,y \in \{0,1\}\} -\end{align*} -als Menge der Knoten des Rechtecks $T$ bezeichnen und als Kanten von $T$ nennen wir die Menge -\begin{align*} - \E_T&:= \{e = [k_T, \tilde k_T] ~|~ k_T,\tilde k_T\in \K_T, k_T\neq\tilde k_T\}. -\end{align*}\todo{Diagonalen sind dabei!!!} +Hier Knoten und Seiten definieren mit Text und Skizze \end{bem} +\begin{figure}[ht] +\centering +\subfloat[gültige Partition]{\includegraphics[width=0.5\textwidth]{fig/net_single}} +\caption{Rechteck} +\label{fig:net:single} +\end{figure}} + \begin{defi}Sei $a,b\in\R$ für $T$ definiert wie in Def. \ref{math:def:T}, dann heißt \begin{align*} \diam (T) &= (a^2+b^2)^{1/2} @@ -291,10 +292,10 @@ die Seitenlänge des Rechtecks $T$ in Richtung $\bs a$. Die Seitenlänge in Rich \end{defi} \begin{defi}Der Abstand zweier Rechtecke $T_j$ und $T_k$ sei definiert durch \begin{align*} - \dist(T_j,T_k) &= \inf \{\abs{\bs x-\bs y} ~|~ \bs x \in T_j,\bs y \in T_k\}. + \dist(T_j,T_k) &= \min \{\abs{\bs x-\bs y} ~|~ \bs x \in T_j,\bs y \in T_k\}, \end{align*} +wobei das Minimum aufgrund der Kompaktheit von $T_j, T_k$ angenommen wird. \end{defi} -\todo{Unterscheiden zwischen parallel und orthogonal und dafür $\dist$ definieren} \begin{defi} Sei $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine endliche Menge von achsenorientierten Rechtecken. Es bezeichne $\K_{\ell}:=\bigcup_{T\in\T} k_T$ die Menge der Knoten von $\T_{\ell}$ und $\E_{\ell}:=\bigcup_{T\in\T} e_T$ die Menge der Kanten. Wir nennen $\T_{\ell}$ eine Partition von $\Gamma$, falls \begin{itemize} @@ -320,7 +321,7 @@ Ferner liegen auf jeder Kante von $\T_{\ell}$ maximal 3 Knoten. \subsection{Verfeinern} \begin{defi}[Lokale Verfeinerung] -Ein Element $T \in \T$ wird isotrop in vier Elemente $T_1,\ldots,T_4$ geteilt, wenn $T = \bigcup_{n=1}^4 T_n$ gilt und die Seitenlängen der Elemente $T_1,\ldots,T_4$ gleich groß sind. Weiterhin wird ein Element $T \in \T$ anisotrop in zwei Elemente $T_1,T_2$ geteilt, wenn ebenfalls $T = T_1 \cup T_2$ gilt und $T_1,T_2$ gleich große Seitenlängen haben. Hierbei kann $T$ entweder horizontal oder vertikal geteilt werden, wie in Abb.~\ref{fig:refType} gezeigt ist. +Ein Element $T \in \T$ wird isotrop in vier Elemente $T_1,\ldots,T_4$ geteilt, wenn $T = \bigcup_{n=1}^4 T_n$ gilt und alle Söhne $T_1,\ldots,T_4$ ähnlich sind zum Vaterelement $T$. Weiterhin wird ein Element $T \in \T$ anisotrop in zwei Elemente $T_1,T_2$ geteilt, wenn ebenfalls $T = T_1 \cup T_2$ gilt und $T_1,T_2$ gleich große Seitenlängen haben. Hierbei kann $T$ entweder horizontal oder vertikal geteilt werden, wie in Abb.~\ref{fig:refType} gezeigt ist. \end{defi} \begin{figure}[ht] \centering @@ -335,14 +336,15 @@ Ein Element $T \in \T$ wird isotrop in vier Elemente $T_1,\ldots,T_4$ geteilt, w \begin{defi} Für eine Partition $\T_{\ell}$ bezeichnen wir mit $\widehat \T_{\ell}$ jene Partition die entsteht, wenn alle Elemente isotrop verfeinert werden. \end{defi} - -\begin{alg}[Verfeinern] \label{alg:refine} Sei $\T_{\ell}$ eine Partition und $\sqcap_{\ell} \subseteq \{e_T ~|~ T \in \T_{\ell},e\in\E_T \}:=\S_{\ell}$ eine Menge markierter Kanten. Nun sei $\sqcap_{\ell}^{(0)}:=\sqcap_{\ell}$ und $i=0$, dann gehe so vor: +\todo{ +Text der die Idee des Alg beschreibt + Vorüberlegungen} +\begin{alg}[Verfeinern] \label{alg:refine} Sei $\T_{\ell}$ eine Partition und $\sqcap_{\ell} \subseteq \S_{\ell}$ eine Menge markierter Kanten. Nun sei $\sqcap_{\ell}^{(0)}:=\sqcap_{\ell}$ und $i=0$. Dann gehe so vor: \begin{enumerate} \renewcommand{\theenumi}{(\roman{enumi})} \item \label{alg:refine:first} - $\sqcap_{\ell}^{(i+1)} := \sqcap_{\ell}^{(i)} \cup \{ e_{T'}' \in \S_{\ell}\backslash\sqcap_{\ell}^{(i)} ~|~ \exists e_T \in \sqcap_{\ell}^{(i)}: e' \supsetneq e \} \cup \{\tilde e_{T'} \in \S_{\ell} ~|~ \todo{ \exists \tilde e\in\E_{T'} :} e'\cap\tilde e=\emptyset\}$ + $\sqcap_{\ell}^{(i+1)} := \sqcap_{\ell}^{(i)} \cup \{ e_{T'} \in \S_{\ell}\backslash\sqcap_{\ell}^{(i)} ~|~ \exists e_T \in \sqcap_{\ell}^{(i)}: e_{T'} \supsetneq e_T \}\\ \hfill \cup \{e_{T'}' \in \S_{\ell} ~|~ \exists \tilde e_{T'}\in\E_{T'} : e'_{T'}\cap e_{T'}=\emptyset\}$ \item - Falls $\sqcap_{\ell}^{(i)} \subsetneq \sqcap_{\ell}^{(i+1)}$, $i = i+1$ und gehe zu Schritt \ref{alg:refine:first} + Falls $\sqcap_{\ell}^{(i)} \subsetneqq \sqcap_{\ell}^{(i+1)}$, erhöhe Zähler $i \mapsto i+1$ und gehe zu Schritt \ref{alg:refine:first} \item Teile alle Elemente aus $\T_{\ell}$ bezüglich der markierten Kanten $\sqcap_{\ell}^{(i)}$, wie in der lokalen Verfeinerung vorgegeben \end{enumerate} \end{alg} @@ -367,7 +369,7 @@ analytisch/semidiskret/volldiskret \noindent Ziel dieses Abschnitts ist die approximative Berechnung des Integrals \begin{align}\label{math:gal:kap} -A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}. +A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}, \end{align} beziehungsweise als Spezialfall davon die Berechnung des Integrals \begin{align}\label{math:gal:frac} @@ -382,20 +384,20 @@ An dieser stelle werden wir zunächst den Interpolationsoperator auf dem Interva Für einen festen Grad $p \in \N$ und paarweise verschiedene Knoten $x_j \in [0,1]$ lautet das Lagrange'sche Interpolations Problem: Zu gegebenen Funktionswerten $y_j \in \R$ finde ein Polynom $q \in \P^{p-1}$ so, dass \begin{align*} - q(x_j) &=y_j \quad \text{ für alle } j=0,\ldots,p. + q(x_j) &=y_j \quad \text{ für alle } j=1,\ldots,p. \end{align*} \end{defi} Für die Lagrange-Polynome \begin{align*} -L_j(x) &= \prod_{i=1,i\neq j}^p \frac{x-x_i}{x_j-xi}. +L_j(x) &= \prod_{i=1,i\neq j}^p \frac{x-x_i}{x_j-x_i}. \end{align*} wissen wir aus \todo{cite}, dass für das Lagrange'sche Interpolations Problem eine eindeutige Lösung gegeben ist, durch \begin{align*} -q&=\sum_{j=0}^py_jL_j. +q&=\sum_{j=1}^py_jL_j. \end{align*} Mit diesem Wissen definieren wir uns nun den Interpolationsoperator $\I_p u : \C[0,1]\to\P^{p-1}$ \begin{align} -\I_pu &:= \sum_{j=0}^p u(x_j)L_j. +\I_pu &:= \sum_{j=1}^p u(x_j)L_j. \end{align} % Ferner definieren wir noch die Lebesgue Konstante % \begin{align} @@ -406,11 +408,11 @@ Wie an der einfachen Fehlerabschätzung für den Interpolationsoperator \norm{u-\I_pu}_{\infty,[0,1]} &\leq 2\frac{\norm{u^{(p)}}_{\infty,[0,1]}}{p!} \max_{x\in[0,1]} \prod_{j=1}^n \abs{x-x_j} \quad \text{ für alle }u \in \C^p([0,1]) \end{align} - zu sehen ist, hängt diese noch von der Wahl der Knoten ab. Um den Fehler weiter zu verbessen, werden wir das Produkt + zu sehen ist \todo{cite}, hängt diese noch von der Wahl der Knoten ab. Um den Fehler weiter zu verbessern, werden wir das Produkt \begin{align*} \max_{x\in[0,1]} \prod_{j=1}^n \abs{x-x_j} \end{align*} -durch benutzen von Chebyshev-Knoten minimieren. Diese sind für das Intervall $[-1,1]$ gegeben durch +durch benutzen von Chebyshev-Knoten minimieren \todo{cite}. Diese sind für das Intervall $[-1,1]$ gegeben durch \begin{align*} x_j &= \cos \left( \frac{2j-1}{p} \frac{\pi}{2} \right) \quad \text{ für } j=1,\ldots,p. \end{align*} @@ -421,29 +423,35 @@ Durch affine Transformation der Chebyshev-Knoten, ergibt sich der Fehlerschätze \quad \text{ für alle }u \in \C^p([0,1]) \end{align} für Chebyshev-Polynome.\\ -Ferner wollen wir nun die Interpolation auf Intervallen der Gestalt $[0,1]^d$ mit $d\in\N$ einführen. -\begin{sat} -\todo{Multi InterPol} +Ferner wollen wir nun die Interpolation auf Boxen der Gestalt $[0,1]^d$ mit $d\in\N$ einführen. Hierzu definieren wir uns mit dem Tensor Produkt den Interpolationsoperator \begin{align} - \norm{u-\I_pu}_{\infty,[0,1]^d} + \I_k^{[0,1]^d} := \bigotimes_{i=1}^d \I_k^{[0,1]} \quad \text{ für Boxen } [0,1]^d. +\end{align} +Weiterhin benötigen wir für die Abschätzung die Lebesgue-Konstante +\begin{align} + \Lambda_k := \max_{x\in[0,1]} \sum_{j=1}^p \abs{L_j(x)}. +\end{align} +Dann können wir den Fehlerschätzer für den Interpolationsoperator $\I_k^{[0,10]^d}$ anschreiben \todo{cite} +\begin{align} + \norm{u-\I_p^{[0,1]^d}u}_{\infty,[0,1]^d} &\leq 4\frac{ 4^{-p}}{p!}\lambda_p^{d-1}\sum_{j=1}^d \norm{\partial_j^d u^{(p)}}_{\infty,[0,1]^d} - \quad \text{ für alle }u \in \C^{p+1}([0,1]^d) + \quad \text{ für alle }u \in \C^{p+1}([0,1]^d). \end{align} -\end{sat} + \subsection{Gauss-Quadratur} -Im Folgenden wollen wir die klassische Gauss-Quadratur definieren.\\ -Unter einer Quadratur verstehen wir die approximative Berechnung eines Integrals der Form: +Im Folgenden wollen wir die klassische Gauss-Quadratur definieren, welche durch Verwendung der 1-Funktion statt der Gewichtungsfunktion entsteht. +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 +durch die 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 heißt exakt für eine Funktion $f$, falls $\Q(f) = \int_0^1 f(x) dx$ ist.\\ +Eine Quadratur hat den Exaktheitsgrad $m$, wenn sie für alle 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) @@ -461,19 +469,19 @@ gilt, wobei $p$ das Interpolationspolynom von $f$ vom Grad $n$ zu den Knoten $x_ % \int_{T_j} \int_{T_k} &\approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2}\\ % \int_{T_j} \int_{T_k} &\approx \q_{T_j} \q_{J_1} \int_{J_2} % \end{align}} -\subsection{voll analytische Berechnung des Doppelintegrals} +\subsection{Analytische Berechnung des Doppelintegrals} % \subsubsection{Glatter Kern} % Das Integral über $T_j$ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über $T_k$ wird nun mittels einer doppelten Gauss-Quadratur berechnet. \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 + 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}\label{math:sem:glatt} % Glatter KERN \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)}(\abs{\alpha} + \abs{\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}: +Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktionen gut durch Polynome interpolieren. Hierzu benötigen wir das folgende Lemma: % \begin{lem}\label{math:sem:ipol} %Interpol über Glatten KERN 1D % Sei $J \subseteq \R$ ein abgeschlossenes und beschränktes Intervall und angenommen die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ % \begin{align*} @@ -487,32 +495,32 @@ Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktione % \hfill$\square$ \begin{lem}\label{thm:sem:ipolnD} %Interpol über Glatten KERN nD -Sei $B \subseteq \R^d$ ein abgeschlossenes und beschränktes Intervall und angenommen die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ +Sei ${[0,1]^d} \subseteq \R^d$ eine achsenorientierte Box wobei die Intervalle $[0,1]$ abgeschlossen sind, und vorausgesetzt die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ \begin{align} - \norm{\partial_j^nu}_{\infty,B} &\leq C_u \rho_u^nn!\quad\text{~für alle~} j \in \{1,\ldots,d\} \text{~und~} n \in \N_0. + \norm{\partial_j^nu}_{\infty,{[0,1]^d}} &\leq C_u \rho_u^nn!\quad\text{~für alle~} j \in \{1,\ldots,d\} \text{~und~} n \in \N_0. \end{align} Dann gilt für alle $k\in \N_0$ \begin{align}\label{math:sem:ipolnD} - \norm{u-\I_k^Bu}_{\infty,B} &\leq C_u 8e(1+\rho_u\diam (B))\Lambda_k^d(k+1)\left(1+\frac{2}{\rho_u\diam(B)}\right)^{-(k+1)}. + \norm{u-\I_k^{[0,1]^d}u}_{\infty,{[0,1]^d}} &\leq C_u 8e(1+\rho_u)\Lambda_k^d(k+1)\left(1+\frac{2}{\rho_u}\right)^{-(k+1)}. \end{align} \end{lem} -\hfill$\square$ +\todo{\beweis Steht eigentlich in ``Low-rank Approximation of integral operators by interpolation'' von Steffen Börm und Lars Grasedyck: Theorem 3.2 \hfill$\square$} % Quadratur ----------------------------------------- -\begin{defi} Seien $T_j,T_k$ Rechtecke und sei $\zeta_Q > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_Q$-zulässig, genau dann wenn +\begin{defi} Seien $T_j,T_k \in \R^3$ achsenorientierte Rechtecke und sei $\zeta_Q > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_Q$-zulässig, genau dann wenn \begin{align}\label{math:sem:zetaQ} \dist (T_j, T_k)&\geq \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\}. \end{align} Andernfalls heißt das Paar $(T_j,T_k)$ $\zeta_Q$-unzulässig. \end{defi} -\begin{sat} \label{thm:sem:pol:V} Seien $T_j,T_k$ zwei $\zeta_Q$-zulässige Rechtecke. Sei $\kappa$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit +\begin{sat} \label{thm:sem:pol:V} Seien $T_j,T_k \in \R^3$ zwei $\zeta_Q$-zulässige Rechtecke. Sei $\kappa(\cdot,\cdot)$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit \begin{align*} C_{\zeta_Q,j,k}&:=2^3e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\zeta_Q}{c_2} \right) \end{align*} die Abschätzung \begin{align*} - \norm{\kappa(\cdot,\cdot)-\I_k^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} - &\leq C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}. + \norm{\kappa(\cdot,\cdot)-\I_p^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} + &\leq C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}. \end{align*} \end{sat} \beweis Zunächst definieren wir die Konstanten @@ -524,11 +532,11 @@ und können dann die Konstante $C_{\zeta_Q,j,k}$ kurz C_{\zeta_Q,j,k} &= 8eC_{\kappa} (1+\rho_{\kappa}) \end{align*} schreiben.\\ -Bezeichnet $\gamma$ die Parametrisierung von $T$, so gilt aufgrund der Kettenregel +Bezeichnet $\gamma_j$ die Parametrisierung von $T_j$, so gilt aufgrund der Kettenregel \begin{align*} \Abs{\partial_x^{\alpha}\partial_y^{\beta}\kappa} &=\Abs{\partial_x^{\alpha}\partial_y^{\beta}\kappa(\gamma_j(x),\gamma_k(y))}\\ - &=\diam(T_j)^{\alpha} \diam(T_k)^{\beta} + &\leq\diam(T_j)^{\alpha} \diam(T_k)^{\beta} \Abs{(\partial_x^{\alpha}\partial_y^{\beta}\kappa)(\gamma_j(x),\gamma_k(y))}. \end{align*} Ferner gilt mit \eqref{math:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung @@ -539,17 +547,17 @@ Ferner gilt mit \eqref{math:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kap = & \frac{c_1}{(c_2 \abs{\bs x - \bs y})^s} \left( \frac{\max(\diam(T_j),\diam(T_k))}{c_2 \abs{\bs x - \bs y}} \right)^{\alpha+\beta}(\abs{\alpha}+\abs{\beta})!\\ \leq& C_{\kappa}\rho_{\kappa}^nn!. \end{align*} -Laut Voraussetzung sind $T_j,T_k$ $\zeta_Q$-zulässig, weshalb $\kappa$ auf $T_j,T_k$ definitionsgemäß glatt ist. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt und es gilt die Abschätzung mithilfe von \eqref{math:sem:zetaQ} +Laut Voraussetzung sind $T_j,T_k$ $\zeta_Q$-zulässig, weshalb $\kappa(\cdot,\cdot)$ auf $T_j\times T_k$ definitionsgemäß glatt ist. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt, und es gilt die Abschätzung mithilfe von \eqref{math:sem:zetaQ} \begin{align*} \norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot))&-\I_p^{[0,1]^4}\kappa(\gamma_j(\cdot),\gamma_k(\cdot))}_{\infty,[0,1]^4}\\ - &\leq C_{\kappa} 8e(1+\rho_{\kappa})\Lambda_k^4(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ -% &= C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ - &= C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}. + &\leq C_{\kappa} 8e(1+\rho_{\kappa})\Lambda_p^4(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ +% &= C_{\zeta,j,k}\Lambda_p^4(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ + &= C_{\zeta,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}. \end{align*} Weiterhin folgt für $T_j \times T_k$ \begin{align*} - \norm{\kappa(\cdot,\cdot)-\I_k^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} - &\leq C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}, + \norm{\kappa(\cdot,\cdot)-\I_p^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} + &\leq C_{\zeta,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}, \end{align*} womit der Beweis abgeschlossen ist. \hfill$\square$ @@ -576,7 +584,7 @@ und für den durch die Gauss-Quadratur von $A_{jk}$ zum Grad $p$ entstehenden Te \end{align*} die Abschätzung \begin{align} - \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)} + \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)} \end{align} \end{sat} \beweis Da die Gauss-Quadratur ist interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist gilt: @@ -595,13 +603,13 @@ wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ in $x_{\b Mithilfe von Lemma \ref{thm:sem:pol:V} erhalten wir die Behauptung \begin{align*} \abs{A_{jk} - (A_p)_{jk}} - &\leq D_{j,k} C_{\zeta_Q,j,k}\Lambda_k^42(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\\ - &= \tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}. + &\leq D_{j,k} C_{\zeta_Q,j,k}\Lambda_p^42(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\\ + &= \tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}. \end{align*} \todo{\hfill$\square$\\} \begin{lem} \label{thm:sem:switch} - Seien $Tj,T_k$ zwei Randstücke, $\kappa$ eine asymptotisch glatte Kernfunktion und + Seien $Tj,T_k$ zwei Randstücke, $\kappa(\bs x,\bs y)$ eine asymptotisch glatte Kernfunktion und \begin{align} A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}. @@ -656,18 +664,18 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \end{align*} und sei $A_p$ die $A$ approximierende Matrix gemäß Definition \ref{thm:sem:quad:Ap}. Dann gilt \begin{align*} - \norm{A-A_p}_F \leq n \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}. + \norm{A-A_p}_F \leq n \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}. \end{align*} \end{sat} \beweis Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig ist die Differenz laut Definition $0$. Sind $T_j$ und $T_k$ hingegen zulässig so gilt nach Satz \ref{thm:sem:quad:V} % \begin{align} -% \abs{A_{jk} - (A_p)_{jk}}&= \tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)} +% \abs{A_{jk} - (A_p)_{jk}}&= \tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)} % \end{align} \begin{align*} \norm{A-A_p}_F^2 &= \sum_{j,k=1}^n (A_{jk} - (A_p)_{jk})^2\\ - &\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\right)^2\\ - &\leq \sum_{j,k=1}^n \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\right)^2\\ - &= n^2 \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\right)^2. + &\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\right)^2\\ + &\leq \sum_{j,k=1}^n \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\right)^2\\ + &= n^2 \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\right)^2. \end{align*} Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\square$ @@ -686,8 +694,8 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\sq % \end{align*} % die Abschätzung % \begin{align*} -% \norm{\kappa(\cdot,\cdot)-\I_k^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} -% &\leq C_{\zeta,j,k}\Lambda_k^2(p+1)\left(1+\frac{2c_2}{\zeta_A}\right)^{-(p+1)}. +% \norm{\kappa(\cdot,\cdot)-\I_p^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} +% &\leq C_{\zeta,j,k}\Lambda_p^2(p+1)\left(1+\frac{2c_2}{\zeta_A}\right)^{-(p+1)}. % \end{align*} % \end{sat} % \beweis Zunächst definieren wir die Konstanten @@ -717,14 +725,14 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\sq % Laut Voraussetzung sind $T_j,T_k$ $\zeta_A$-zulässig, weshalb $\kappa$ auf $T_j,T_k$ definitionsgemäß glatt ist. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt und es gilt die Abschätzung mithilfe von \eqref{math:sem:zetaQ} % \begin{align*} % \norm{\kappa(\gamma_j(\cdot,x_b),\gamma_k(\cdot,y_b))&-\I_p^{[0,1]^2}\kappa(\gamma_j(\cdot,x_b),\gamma_k(\cdot,y_b))}_{\infty,[0,1]^2}\\ -% &\leq C_{\kappa} 8e(1+\rho_{\kappa})\Lambda_k^2(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ -% % &= C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ -% &= C_{\zeta,j,k}\Lambda_k^2(p+1)\left(1+\frac{2c_2}{\zeta_A}\right)^{-(p+1)}. +% &\leq C_{\kappa} 8e(1+\rho_{\kappa})\Lambda_p^2(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ +% % &= C_{\zeta,j,k}\Lambda_p^4(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ +% &= C_{\zeta,j,k}\Lambda_p^2(p+1)\left(1+\frac{2c_2}{\zeta_A}\right)^{-(p+1)}. % \end{align*} % Weiterhin folgt für $T_j \times T_k$ % \begin{align*} -% \norm{\kappa(\cdot,\cdot)-\I_k^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} -% &\leq C_{\zeta,j,k}\Lambda_k^2(p+1)\left(1+\frac{2c_2}{\zeta_A}\right)^{-(p+1)}, +% \norm{\kappa(\cdot,\cdot)-\I_p^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} +% &\leq C_{\zeta,j,k}\Lambda_p^2(p+1)\left(1+\frac{2c_2}{\zeta_A}\right)^{-(p+1)}, % \end{align*} % womit der Beweis abgeschlossen ist. % \hfill$\square$ diff --git a/src/A_plots.m b/src/A_plots.m index e6e4fc7..9faddea 100644 --- a/src/A_plots.m +++ b/src/A_plots.m @@ -2,7 +2,7 @@ function A_plots(files,printt) % type2str = {'Analytisch ' 'Quad Element' 'Quad Achse ' 'Quad Seite '}; type2str = {'A ' 'QA' 'QEQA' 'QA ' 'QS '}; -type2sym = {'-o' '-x' '-' '-'}; +type2sym = {'-o' '-x' '-*' '-+' '-.'}; G_D=[]; X = []; @@ -121,6 +121,7 @@ loglog(repmat(X(:,i+1),1,3),[G_D(:,2+rows*i) ... ... G_D(:,2+10+rows*i)... ],type2sym{i+1}); hold on + for i = 1:step-1 loglog(repmat(X(:,i+1),1,3),[G_D(:,2+rows*i) ... ... G_D(:,2+8+rows*i)... diff --git a/src/export_exmpl.m b/src/export_exmpl.m index eb30722..6ca7a5d 100644 --- a/src/export_exmpl.m +++ b/src/export_exmpl.m @@ -35,6 +35,14 @@ axis off; print('-r600','-depsc',['../doc/fig/net_wrong.eps']) % system(['epstopdf ../doc/fig/net_wrong.eps']); +%% Example Single Element +load exmpl_2DQuad2 + +plotShape(coordinates,elements(:,:),'ben'); +% view(2); +% axis off; +print('-r600','-depsc',['../doc/fig/net_single.eps']) +% system(['epstopdf ../doc/fig/net_single.eps']); %% Example Objects load exmpl_2DLShape diff --git a/src/plotShape.m b/src/plotShape.m index c08e868..9cf6331 100644 --- a/src/plotShape.m +++ b/src/plotShape.m @@ -12,7 +12,7 @@ function plotShape(coordinates, elements, varargin) % s -> Flaechen werden gefaerbt, wobei VEC die Farben der Elemente angibt % t -> Element Text wird angezeigt (oder Element Index) % d -> Koordinaten Text wird angezeigt (oder Koordinaten Index) -% e -> thick lines for export +% e -> dicke linien fuer den Export % % P.Schaefer @@ -21,6 +21,7 @@ c = 0; e = 1; n = 0; t = 0; +f = 0; line = 1; optargin = size(varargin,2); if(optargin>2) @@ -126,7 +127,12 @@ if(n) end for idx = eles current = sum(coordinates(elements(idx,[2,4])',:),1)/2; - current = [current ; current+anorm(idx,:);coordinates(elements(idx,1)',:)]; + %Zeichnet auch den ersten Punkt ein ! + current = [current ; current+anorm(idx,:)]; + if(f==1) + current = [current ; coordinates(elements(idx,1)',:)]; + end + plot3(current(:,1),current(:,2),current(:,3),'r'); % Zeichnet Oberflaeche scatter3(current(2,1),current(2,2),current(2,3),'xr'); hold on @@ -134,7 +140,7 @@ if(n) end -if(t) +if(t==1) for idx = eles current = sum(coordinates(elements(idx,[2,4])',:),1)/2; if(e) diff --git a/src/test_sol.m b/src/test_sol.m index c513408..e1d29e2 100644 --- a/src/test_sol.m +++ b/src/test_sol.m @@ -9,7 +9,7 @@ mex mex_build_V.cpp slpRectangle.cpp CFLAGS="\$CFLAGS -fopenmp" CXXFLAGS="\$CXXF % Test ausführen %Anzahl der Schritte -steps = 10; +steps = 4; %Art der Berechnungen type = [2 2 2 2 2]; -- 2.47.3