From d45700653f329df8756d25067f1683722c0c705b Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Wed, 21 Nov 2012 10:39:44 +0100 Subject: [PATCH] [doc] Kapitel 3... --- doc/doc.pdf | Bin 308428 -> 315350 bytes doc/doc.tex | 71 ++++++++++++++++++++++++++++++++++++---------------- 2 files changed, 50 insertions(+), 21 deletions(-) diff --git a/doc/doc.pdf b/doc/doc.pdf index 0f6a2a4c819760b0046768b05de0c913b6fc73e1..924d372de13b9ba8f2807e82cce555ca7005a59f 100644 GIT binary patch delta 159868 zcmZ5`Q;a1Hux8t~F>QO=wrxz?wolu(ZQHhO+qQfEP43Mm`%HReqL6OO?OL1t4R1_|KjJ;_Wm3g zyEw9(cTunK>~wEc7)0cRt018F$f`)f)Y7a-vBR*$x^+2LMOsjuY#g5aa9EVecS#ak zK9#KuCPuaP$;hb1X)5~LqsLy2GIROC3*c8J_3>G55~D1ZSS7^84+=-S+=K~~V5ff&x_99N;m zr%BuEp;=in+z-~Sz;CIAvZFn2C5arhx?>0$ty(=Y-^OruMubm_kV8rMmQ%_N15jKg z9N}C%LhTVID|Su^{xhJk_56VKTKQ?FB{@60@nZlhn4IW|!xHfx@T#ar9q7gW#}z~# z7vBxO^qHurFyLs*Bcxr;;$hul2Di`wv}km2NMhK3T~mHEo5B7%3iT9{*qVHBC4c&B zZZ2W6c`4x~O|vra zUlxE#04ln80D*1wuT&CQwy^i3Beaz%>Du230_rk@l93Mx6i4>2xW63bTQjtjMyZid 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zbi^YAW>RRl5#s*u@BWUjPNGwR+ZYOIBrN1cGqXARdq4mJ&j9A|8aUBp(1qL&9T{uT4;2xI-c_hF=&6UJriXgW!kAn8hJ! z3uVLhNyGzFz#4DCH4upc1af~(anuqacp_4O`&IH{291&M@Supm9X5c`E%cwWr$7;C zqI0HDImDwUcnk>9hy-b*yA#~AkOB#~10qbmBkZbhlS3j1Zj(4!6!2ao0h#2m1|m`h zis{$G5292$=2VX;yK|`K<}9iS#e~8!VVRkknwh|%A7;wL%#^v1f||@T{QqLC!V8&i a3|$=(8UcS;F9{wJZbBv1)y \diam(T_k)$ können wir $A_{jk}$ approximieren indem wir mithilfe von $A_{jk} = A_{kj}$ und Satz (\ref{thm:sem:quad:V}), $A_{kj}$ berechnen. \end{bem} + +\todo{\begin{bem} + $\Lambda_p$ wächst für Chebyshev-Polynome im wesentlichen logarithmisch in $p$. Daher konvergiert $(A_p)_{jk}$ für wachsenden Quadraturgrad $p$ exponentiell schnell gegen $A_{jk}$ + Zusammenhang von Quadraturgrad und Konvergenz +\end{bem} +} \subsection{Approximierende Matrix} -\begin{defi} +\begin{defi} \label{thm:sem:quad:Ap} Seien $T_1,T_2,\ldots,T_n$ Rechtecke und $A \in \R^{n \times n}$ gegeben durch \eqref{math:gal:kap}. Dann ist die $A$ approximierende Matrix $A_p$ folgendermaßen definiert. \begin{itemize} \item Sind $T_j$ und $T_k$ $\zeta$-unzulässig, so ist @@ -642,7 +643,35 @@ Mithilfe von Lemma \ref{thm:sem:pol:V} erhalten wir die Behauptung \end{itemize} \end{defi} - +\noindent +Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius-Norm für wachsenden Quadraturgrad exponentiell schnell gegen die gegebene Matrix konvergiert. Die Frobenius-Norm ist hierbei gegeben durch +\begin{align*} + \norm{A}_F := \left( \sum_{i=1}^n\sum_{j=1}^n A_{ij}^2 \right)^{1/2} \quad \text{für} \quad A \in \R^{n \times n}. +\end{align*} +\todo{Dieser Satz ist wahrscheinlich unnötig +\begin{sat} + Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^2\times \R^2 \to \R$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2$ und Singularitätsordung $s \geq 0$. Sei $\tilde C_{\zeta_Q,j,k}$ wie in \eqref{math:sem:zetaQ:c}. Seien $A\in \R^{n \times n}$ eine Matrix, deren Einträge gegeben sind durch + \begin{align*} + A_{jk} + &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}. + \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)}. + \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)} +% \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. +\end{align*} +Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\square$ +} % \subsection{Quadratur über eine Achse} % \subsubsection{Quadratur} % \begin{defi} Seien $T_j,T_k$ Rechtecke und sei $\zeta_A > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_A$-zulässig, genau dann wenn @@ -789,7 +818,7 @@ g(-3/2;y;x;\lambda) &= \frac{y-x}{\lambda^2}\{(y-x)^2+\lambda^2\}^{-1/2} \end{align*} benötigen, welche alle in \cite[Seite 2-3]{mai:3dbem} bewiesen wurden. \end{lem} -Hierbei wollen wir darauf hinweisen, dass wir für $p=-1$ nicht die in \cite[Seite 3]{mai:3dbem} vorgeschlagene Formel verwenden, sondern die mit Hilfe von Substitution durch $z = \frac{y-x}{\abs{\lambda}}$, beziehungsweise $dz = \frac 1 {\abs{\lambda}} dy$ selbst hergeleitete +Hierbei wollen wir darauf hinweisen, dass wir für $p=-1$ nicht die in \cite[Seite 3]{mai:3dbem} vorgeschlagene Formel verwenden, sondern die selbst hergeleitete. Mithilfe von Substitution durch $z = \frac{y-x}{\abs{\lambda}}$ und $dz = \frac 1 {\abs{\lambda}} dy$ gilt \begin{align*} \int \frac 1 {(x-y)^2 +\lambda^2} dy &= \int \frac 1 {\lambda^2((\frac{y-x}{\abs{\lambda}})^2 +1)} dy = \frac 1 {\abs{\lambda}} \int \frac 1 {z^2+1} dz\\ @@ -808,7 +837,7 @@ benötigen, welches sich für den den Fall $p = -3/2$ lässt als G(-3/2;y_1,y_2;x_1,x_2,\lambda) &=- \frac{\sgn\{(y_1-x_1)(y_2-x_2)\}}{2|\lambda|} \quad \text{ für } \lambda \neq 0\\ &\cdot \arccos\left( \frac{-2(y_1-x_1)^2(y_2-x_2)^2}{\{(y_1-x_1)^2+\lambda^2\}\{(y_2-x_2)^2+\lambda^2\}} +1 \right). \end{align*} -Für alle weiteren relevanten Fälle $p = -3/2 +k mit k \in \N$ können wir folgende Rekursionsformel aufstellen +Für alle weiteren relevanten Fälle $p = -3/2 +k$ mit $k \in \N$ können wir folgende Rekursionsformel aufstellen \begin{align*} (2p+2)G(p;y_1,y_2;x_1,x_2,\lambda) &= 2p\lambda^2 G(p-1;y_1,y_2;x_1,x_2,\lambda)\\ &+ (y_1-x_1)g(p;y_2;x_2;\sqrt{(y_1-x_1)^2+\lambda^2})\\ @@ -841,7 +870,7 @@ ${\bs a} = \tilde {\bs a}$ und ${\bs b} = \tilde{\bs b}$ gilt. Wir setzen $\bs{\ \begin{lem} Für die Stammfunktion auf parallelen Elementen \begin{align*} - F_{p}&(x1,x_2,y_1,y_2,\bs \delta) :=\\ &\int \int \int \int \left\{(x_1-y_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+\delta_3^2\right\}^{-1/2}dy_2dy_1 dx_2 dx_1 + F_{p}&(x_1,x_2,y_1,y_2,\bs \delta) :=\\ &\int \int \int \int \left\{(x_1-y_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+\delta_3^2\right\}^{-1/2}dy_2dy_1 dx_2 dx_1 \end{align*} gilt nach \cite[Seite 13]{mai:3dbem} \begin{align} -- 2.47.3