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a/doc/doc.tex b/doc/doc.tex index ec061c3..585f089 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -92,7 +92,8 @@ \newtheorem{bew}[defi]{Beweis} \newtheorem{bem}[defi]{Bemerkung} \newtheorem{alg}[defi]{Algorithmus} -\newcommand{\beweis}{{\it Beweis. }} +%\newcommand{\beweis}{{\it Beweis. }} +\newenvironment{beweis}{{\it Beweis. }}{\hfill$\square$} % \numberwithin{lem}{section} \numberwithin{defi}{section} @@ -258,13 +259,13 @@ Abschließend werden wir kurz die numerische Umsetzung der Techniken vorstellen \subsection{Netze} Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. Zunächst wollen wir ein achsenorientiertes Rechteck beschreiben, dessen Seiten parallel zu den Achsen des kartesischen Koordinatensystems liegen. -\begin{defi}\label{math:def:T} Sei $\bs v \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $\alpha, \beta \in \R$ mit $\alpha, \beta > 0$. Dann heißt +\begin{defi}\label{math:def:T} Sei $\bs v \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $a, b \in \R$ mit $a, b > 0$. Dann heißt \begin{align*} -T := \{\bs v + \lambda_1 \alpha {\bs a} + \lambda_2 \beta {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\} +T := \{\bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\} \end{align*} achsenorientiertes Rechteck. Ferner heißt \begin{align*} - \gamma_T := [0,1]^2 \to T: \lambda_1,\lambda_2 \mapsto \bs v + \lambda_1 \alpha {\bs a} + \lambda_2 \beta {\bs b} + \gamma_T := [0,1]^2 \to T: \lambda_1,\lambda_2 \mapsto \bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} \end{align*} die zu $T$ zugehörige Parametrisierung. \end{defi} @@ -388,7 +389,7 @@ A_{jk} &=\frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y unter bestimmten Voraussetzungen an die affinen Randstücke $T_j,T_k$ und den asymptotisch glatten Integranden $\kappa : \R^3 \times \R^3 \to \R$. \subsection{Interpolation} -An dieser stelle werden wir zunächst den Interpolationsoperator auf dem Intervall $[0,1]$ definieren. Ferner wollen wir mithilfe von Chebyshev-Knoten einen Fehlerschätzer für die Chebyshev'sche Interpolation auf Intervallen $[0,1]^d$ mit $d \in \N$ definieren. Im Folgenden bezeichnet $\P^p$ die Menge aller Polynome vom Grad $p$. +An dieser stelle werden wir zunächst den Interpolationsoperator auf dem Intervall $[0,1]$ definieren. Ferner wollen wir mithilfe von Chebyshev-Knoten einen Fehlerschätzer für die Chebyshev'sche Interpolation auf Intervallen $[0,1]^d$ mit $d \in \N$ definieren. Im Folgenden bezeichnet $\P^p$ die Menge aller Polynome vom Grad $p$ auf $[0,1]$. \begin{defi} Für einen festen Grad $p \in \N$ und paarweise verschiedene Knoten $x_j \in [0,1]$ lautet das Lagrange'sche Interpolationsproblem: @@ -433,7 +434,7 @@ Durch affine Transformation der Chebyshev-Knoten ergibt sich die Fehlerabschätz \quad \text{ für alle }u \in \C^p([0,1]) \end{align} für Chebyshev-Polynome.\\ -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 +Ferner wollen wir nun die Interpolation auf Boxen der Gestalt $[0,1]^d$ mit $d\in\N$ einführen. Hierzu definieren wir uns mit dem Tensorprodukt den Interpolationsoperator \begin{align} \I_p^{d} := \bigotimes_{i=1}^d \I_p \quad \text{ für Boxen } [0,1]^d. \end{align} @@ -441,7 +442,7 @@ Weiterhin benötigen wir für die Abschätzung die Lebesgue-Konstante \begin{align} \Lambda_p := \max_{x\in[0,1]} \sum_{j=1}^p \abs{L_j(x)}. \end{align} -Dann können wir eine Fehlerabschätzung für den Interpolationsoperator $\I_k^d$ anschreiben \todo{cite} +Dann können wir eine Fehlerabschätzung für den Interpolationsoperator $\I_p^d$ anschreiben \todo{cite} \begin{align} \norm{u-\I_p^d u}_{\infty,[0,1]^d} &\leq 4\frac{ 4^{-p}}{p!}\lambda_p^{d-1}\sqrt p\sum_{j=1}^d \norm{\partial_j^d u^{(p)}}_{\infty,[0,1]^d} @@ -492,19 +493,19 @@ gilt, wobei $p$ das Interpolationspolynom von $f$ vom Grad $n$ zu den Knoten $t_ 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 gut durch Polynome interpolieren. Hierzu benötigen wir das folgende Lemma: -\begin{lem}[BG, Theorem 3.2]\label{thm:sem:ipolnD} %Interpol über Glatten KERN nD -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{lem}\label{thm:sem:ipolnD} %Interpol über Glatten KERN nD +Die Funktion $u \in \C^{\infty}([0,1])^d$ erfüllt für Konstanten $C_u,\rho_u > 0$ \begin{align} \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 $p\in \N_0$ \begin{align}\label{math:sem:ipolnD} - \norm{u-\I_p^d u}_{\infty,{[0,1]^d}} &\leq C_u 8e(1+\sqrt d\rho_u)\Lambda_k^d(p+1)\left(1+\frac{2}{\sqrt d\rho_u}\right)^{-(p+1)}. -\end{align}\hfill$\square$ + \norm{u-\I_p^d u}_{\infty,{[0,1]^d}} &\leq C_u 8e(1+\sqrt d\rho_u)\Lambda_p^d p\left(1+\frac{2}{\sqrt d\rho_u}\right)^{-p}. +\end{align} \end{lem} % Quadratur ----------------------------------------- -\begin{lem} +\begin{lem} \label{thm:sem:kett} Sei $\kappa(\cdot , \cdot) : \R^6 \rightarrow \R$ eine asymptotisch glatte Kernfunktion und sei weiterhin $ g: \R^4 \rightarrow \R^6$ die Parametrisierung der achsenorientierten Rechtecke $T_j$ und $T_k$ \begin{align} g(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)), @@ -512,111 +513,94 @@ Dann gilt für alle $p\in \N_0$ wobei $\gamma_j$ die Parametrisierung des achsenorientierten Rechtecks $T_j$ aus Definition \ref{math:def:T} ist. Mit der Hilfsfunktion $t_{jk} : \N^4 \rightarrow \N^6$ \begin{align} - t_{jk}(\bs z) = + t_{jk}(\alpha) = \begin{pmatrix} - \bs z_1 \cdot \bs a_j + \bs z_2 \cdot \bs b_j \\ - \bs z_3 \cdot {\bs a_k} + \bs z_4 \cdot {\bs b_k} + \alpha_1 \cdot \bs a_j + \alpha_2 \cdot \bs b_j \\ + \alpha_3 \cdot {\bs a_k} + \alpha_4 \cdot {\bs b_k} \end{pmatrix}, \end{align} - wobei $\bs a_j,\bs b_j \in \R^3$ die paarweise verschiedenen Einheitsvektoren zur Parametrisierung aus Definition \ref{math:def:T} sind, gilt dann für die partiellen Ableitungen der Verknüpfung $\kappa \circ g$ die Kettenregel + wobei $\bs a_j,\bs b_j \in \R^3$ die paarweise verschiedenen Einheitsvektoren zur Parametrisierung $\gamma_j$ sind, gilt dann für die Verknüpfung $\kappa \circ g$ die Kettenregel \begin{align*} - \abs{\partial^z(\kappa \circ g)(x)} - &= \diam(T_j)_{\bs a}^{z_1}\diam(T_j)_{\bs b}^{z_2} \diam(T_k)_{\bs a}^{z_3}\diam(T_k)_{\bs b}^{z_4} \abs{\partial^{t_{jk}(z)}(\kappa)(g(x))}. + \abs{\partial^{\alpha}(\kappa \circ g)(\lambda)} + &= \diam(T_j)_{\bs a}^{\alpha_1}\diam(T_j)_{\bs b}^{\alpha_2} \diam(T_k)_{\bs a}^{\alpha_3}\diam(T_k)_{\bs b}^{\alpha_4} \abs{\partial^{t_{jk}(\alpha)}(\kappa)(g(\lambda))} \end{align*} + mit dem Multiindizes $\alpha \in \N_0^4$, wobei $\abs{\alpha} \geq 1$ gilt. \end{lem} -\beweis - -\begin{align} - \kappa \circ g : \R^4 \to \R^6 \to \R -\end{align} - Kettenregel -\begin{align} - D(\kappa \circ g)(x) = D\kappa(g(x)) \circ D g(x) \quad \in \R^{1\times 4} -\end{align} - -\begin{align} - A_{1\ell} &= \partial_{\ell} \kappa(g(x))\\ - B_{\ell m} &= \partial_m g_{\ell}(x) -\end{align} - -\begin{align} - AB_{1m} & = \sum_{\ell=1}^6 A_{1\ell} B_{\ell m} = \sum_{\ell=1}^6 \underbrace{\partial_{\ell} \kappa(g(x))}_{\text{Asymp Glatt}} \cdot \underbrace{ \partial_m g_{\ell}(x)}_{\text{?}} -\end{align} - -\begin{align} - \partial_m (\kappa \circ g)(x) = \sum_{\ell=1}^6 \partial_{\ell} \kappa(g(x)) \cdot \partial_m g_{\ell}(x) -\end{align} - - -\begin{align} - g(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)) -\end{align} - -Für $\ell \in \{1,2,3\}$ -\begin{align} - g_{\ell}(\lambda) &= \gamma_j(\lambda_1,\lambda_2) = \bs v + \lambda_1 \alpha {\bs a} + \lambda_2 \beta {\bs b}\\ - \partial_{34} g_{\ell}(\lambda) &= 0\\ - \partial_1 g_{\ell}(\lambda) &= \alpha {\bs a}_{\ell}\\ - \partial_2 g_{\ell}(\lambda) &= \beta {\bs b}_{\ell} -\end{align} - -Für $\ell \in \{4,5,6\}$ +\begin{beweis} Die Ableitung der Funktion $\kappa \circ g : \R^4 \to \R^6 \to \R$ ist +\begin{align*} + D(\kappa \circ g)(\lambda) = D\kappa(g(\lambda)) \circ D g(\lambda) \qquad \in \R^{1\times 4}. +\end{align*} +Mithilfe der Jacobimatrizen $A \in \R^{1\times 6}$, $B \in\R^{6\times 4}$ und $m,\ell \in \N$ +\begin{align*} + A_{1\ell} &= \partial_{\ell} \kappa(g(\lambda)) \\ + B_{\ell m} &= \partial_m g_{\ell}(\lambda) +\end{align*} +untersuchen wir zunächst die Ableitungen \begin{align} - g_{\ell}(\lambda) &= \gamma_k(\lambda_3,\lambda_4) = \tilde {\bs v} + \lambda_3 \tilde \alpha \tilde{ \bs a} + \lambda_4 \tilde \beta \tilde{ \bs b}\\ - \partial_{12} g_{\ell}(\lambda) &= 0\\ - \partial_3 g_{\ell}(\lambda) &= \tilde \alpha \tilde{ \bs a}_{\ell -3}\\ - \partial_4 g_{\ell}(\lambda) &= \tilde \beta \tilde{ \bs b}_{\ell -3} + \partial_m (\kappa \circ g)(\lambda) &= (AB)_{1m} = \sum_{\ell=1}^6 A_{1\ell} B_{\ell m} = \sum_{\ell=1}^6 \partial_{\ell} \kappa(g(\lambda)) \partial_m g_{\ell}(\lambda). \end{align} - - - +Hierzu treffen wir einige Vorüberlegungen.\\ +Für die ersten drei Komponenten von $g$ gilt +\begin{align*} + g_{1,2,3}(\lambda) &= \gamma_j(\lambda_1,\lambda_2) = \bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b}, +\end{align*} +mit $\bs a, \bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$, $\bs a \neq \bs b$ und $a,b \in \R$ mit $a,b > 0$, +wodurch sich die folgenden Ableitungen ergeben. +\begin{align*} + \partial_1 g_{1,2,3}(\lambda) &= a {\bs a} & + \partial_2 g_{1,2,3}(\lambda) &= b {\bs b} & + \partial_{3,4} g_{1,2,3}(\lambda) &= 0 +\end{align*} +Für die letzten drei Komponenten +\begin{align*} + g_{4,5,6}(\lambda) &= \gamma_k(\lambda_3,\lambda_4) = \tilde {\bs v} + \lambda_3 \tilde a \tilde{ \bs a} + \lambda_4 \tilde b \tilde{ \bs b} +\end{align*} +können wir die Ableitungen analog anschreiben. Hierbei müssen wir nur beachten, dass sich die Indizierung von $ \tilde {\bs a}, \tilde {\bs b}$ ändert, weshalb wir hier die Komponenten +\begin{align*} + \partial_{1,2} g_{\ell}(\lambda) &= 0 & + \partial_3 g_{\ell}(\lambda) &= \tilde a \tilde{ \bs a}_{\ell -3} & + \partial_4 g_{\ell}(\lambda) &= \tilde b \tilde{ \bs b}_{\ell -3} +\end{align*} +für $\ell \in \{4,5,6\}$ erhalten.\\ +Mit diesen Vorüberlegungen können wir, da $\kappa$ asymptotisch Glatt ist, $AB \in \R^{1 \times 4}$ anschreiben {% \newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}} \begin{align} - \partial_m (\kappa \circ g)(x) = + AB = \begin{pmatrix} - \partial_1 \kappa(g(x)) & \cdots & \partial_6 \kappa(g(x)) + \partial_1 \kappa(g(\lambda)) & \cdots & \partial_6 \kappa(g(\lambda)) \end{pmatrix} \cdot \begin{pmatrix} - \alpha {\bs a} & \beta {\bs b} & \mc{2}{c}{\bs 0}\\ - \mc{2}{c}{\bs 0} & \tilde \alpha \tilde{ \bs a} &\tilde \beta \tilde{ \bs b} - \end{pmatrix} + a {\bs a} & b {\bs b} & \mc{2}{c}{\bs 0}\\ + \mc{2}{c}{\bs 0} & \tilde a \tilde{ \bs a} &\tilde b \tilde{ \bs b} + \end{pmatrix}. \end{align} - }% - -\begin{align} +Wenn wir die Summen der Einträge $(AB)_{1m}$ genauer betrachten, sehen wir, dass jeweils nur ein Eintrag $\neq 0$ ist, da $\bs a, \bs b, \tilde {\bs a}, \tilde {\bs b} $ Einheitsvektoren sind.\\ +Mit der Hilfsfunktion +\begin{align*} ind : \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\} &\rightarrow \{ 1 , 2 , 3 \}\\ - x &\mapsto (1, 2, 3) \cdot x -\end{align} - -\begin{align} - \partial_1 (\kappa \circ g)(x) &= \alpha \cdot \partial_{ind(\bs a)}\kappa(g(x))\\ - \partial_2 (\kappa \circ g)(x) &= \beta \cdot \partial_{ind(\bs b)}\kappa(g(x))\\ - \partial_3 (\kappa \circ g)(x) &= \tilde \alpha \cdot \partial_{ind(\tilde{\bs a})+3}\kappa(g(x))\\ - \partial_4 (\kappa \circ g)(x) &= \tilde \beta \cdot \partial_{ind(\tilde{\bs b})+3}\kappa(g(x)) -\end{align} - - \begin{align} - t_{jk} : \N^4 &\rightarrow \N^6\\ - \bs z &\mapsto - \begin{pmatrix} - z_1 \cdot \bs a + z_2 \cdot \bs b \\ - z_3 \cdot \tilde{\bs a} + z_4 \cdot \tilde{\bs b} - \end{pmatrix} -\end{align} - -\begin{align} - \partial^z (\kappa \circ g)(x) &= \alpha^{z_1} \beta^{z_2} \tilde \alpha^{z_3} \tilde \beta^{z_4} \cdot \partial^{t_{jk}(z)}\kappa(g(x)) - \quad z \in \N^4 -\end{align} - - - - -\hfill$\square$ - + \bs x &\mapsto (1, 2, 3) \cdot \bs x +\end{align*} +für die Einheitsvektoren mit dem Skalarprodukt $\cdot$, können wir den Gradient +% \begin{equation} +\begin{align*} +% \begin{split} + \partial_1 (\kappa \circ g)(\lambda) &= a \cdot \partial_{ind(\bs a)}\kappa(g(\lambda))& + \partial_3 (\kappa \circ g)(\lambda) &= \tilde a \cdot \partial_{ind(\tilde{\bs a})+3}\kappa(g(\lambda))\\ + \partial_2 (\kappa \circ g)(\lambda) &= b \cdot \partial_{ind(\bs b)}\kappa(g(\lambda))& + \partial_4 (\kappa \circ g)(\lambda) &= \tilde b \cdot \partial_{ind(\tilde{\bs b})+3}\kappa(g(\lambda)) +% \end{split} +\end{align*} +% \end{equation} +genau anschreiben, wobei auch hier wieder auf die richtige Indizierung geachtet werden muss.\\ +Des Weiteren bestehen die ersten Ableitungen nur aus einem Skalar und dem asymptotisch Glatten Kern $\partial_{\ell}\kappa(g(\lambda))$, weshalb wir die Kettenregel beliebig oft anwenden können. +Mithilfe der Funktion $t_{jk} : \N^4 \rightarrow \N^6$ können wir abschließend für Multiindizes $\alpha \in \N_0^4$ die partiellen Ableitungen $\partial^{\alpha} (\kappa \circ g)(\lambda)$ anschreiben +\begin{align*} + \partial^\alpha (\kappa \circ g)(\lambda) &= a^{\alpha_1} b^{\alpha_2} \tilde a^{\alpha_3} \tilde b^{\alpha_4} \cdot \partial^{t_{jk}(\alpha)}\kappa(g(\lambda)). +\end{align*} +\end{beweis} \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)\}. @@ -631,10 +615,10 @@ Für $\ell \in \{4,5,6\}$ die Abschätzung \begin{align*} \norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot))&-\I_p^4\kappa(\gamma_j(\cdot),\gamma_k(\cdot))}_{\infty,[0,1]^4} - \leq C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{c_2}{\zeta_Q}\right)^{-(p+1)}. + \leq C_{\zeta_Q,j,k}\Lambda_p^4 p\left(1+\frac{c_2}{\zeta_Q}\right)^{-p}. \end{align*} \end{sat} -\beweis Zunächst definieren wir die Konstanten +\begin{beweis} Zunächst definieren wir die Konstanten \begin{align*} C_{\kappa} &=\frac{c_1}{(c_2\dist(T_j,T_k))^s} \quad \text{und} \quad \rho_{\kappa} = \frac{\zeta_Q}{c_2} \end{align*} @@ -643,16 +627,11 @@ und können dann die Konstante $C_{\zeta_Q,j,k}$ kurz C_{\zeta_Q,j,k} &= 8eC_{\kappa} (1+2\rho_{\kappa}) \end{align*} schreiben.\\ -Weiterhin können wir das Produkt der Seiten eines Rechtecks $T_j$ mithilfe der Diagonalen abschätzen -\begin{align*} - \diam_{\bs a}(T_j)^{\bs a \alpha}\diam_{\bs b}(T_j)^{\bs b \alpha} - \leq \max\{\diam_{\bs a}(T_j),\diam_{\bs b}(T_j)\}^{\abs \alpha} - \leq \diam(T_j)^{\abs \alpha}. -\end{align*} 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))}\\ +% \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)_{\bs a}^{\alpha_1}\diam(T_j)_{\bs b}^{\alpha_2} \diam(T_k)_{\bs a}^{\alpha_3}\diam(T_k)_{\bs b}^{\alpha_4} \abs{\partial^{t_{jk}(\alpha)}(\kappa)(g(x))}\\ % &=\diam_{\bs a}(T_j)^{\bs a \alpha}\diam_{\bs b}(T_j)^{\bs b \alpha}\diam_{\bs a}(T_k)^{\bs a \beta}\diam_{\bs b}(T_k)^{\bs b \beta} % \Abs{(\partial_x^{\alpha}\partial_y^{\beta}\kappa)(\gamma_j(x),\gamma_k(y))}\\ &\leq\diam(T_j)^{\abs \alpha}\diam(T_k)^{\abs \beta} @@ -674,7 +653,7 @@ Laut Voraussetzung sind $T_j,T_k$ $\zeta_Q$-zulässig, weshalb $\kappa(\cdot,\cd &= C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{c_2}{\zeta_Q}\right)^{-(p+1)}. \end{align*} Womit der Beweis abgeschlossen ist. -\hfill$\square$ +\end{beweis} % MatrixEINTRAG ------------------------------------------------- \begin{sat}\label{thm:sem:quad:V} Seien $T_j,T_k$ zwei $\zeta_Q$-zulässige Rechtecke mit zugehörigen Parametrisierungen $\gamma_j,\gamma_k$. Sei weiterhin $\kappa : \R^3\times\R^3\to \R : (\bs x, \bs y) \mapsto \kappa(\bs x,\bs y)$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2,s$. Sei @@ -697,7 +676,7 @@ die Abschätzung \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{c_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: +\begin{beweis} Da die Gauss-Quadratur ist interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist gilt: \begin{align*} (A_p)_{jk} &=\abs{T_j}\abs{T_k}\sum_{a=0}^p w_a \sum_{b=0}^p w_b \sum_{c=0}^p w_c \sum_{d=0}^p w_d \kappa(\gamma_j(x_a,x_b),\gamma_k(y_c,y_d))\\ @@ -716,7 +695,7 @@ Mithilfe von Lemma \ref{thm:sem:pol:V} erhalten wir die Behauptung &\leq \abs{T_j}\abs{T_k} C_{\zeta_Q,j,k}\Lambda_p^42(p+1)\left(1+\frac{c_2}{\zeta_Q}\right)^{-2(p+1)}\\ &= \tilde C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{c_2}{\zeta_Q}\right)^{-2(p+1)}. \end{align*} -\todo{\hfill$\square$\\} +\todo{}\end{beweis} \begin{lem} \label{thm:sem:switch} Seien $Tj,T_k$ zwei Randstücke, $\kappa(\bs x,\bs y)$ eine asymptotisch glatte Kernfunktion und @@ -729,8 +708,8 @@ Mithilfe von Lemma \ref{thm:sem:pol:V} erhalten wir die Behauptung A_{jk} = A_{k_j}. \end{align} \end{lem} -\todo{\beweis -\hfill$\square$} +\todo{\begin{beweis} +\end{beweis}} \begin{bem} Seien $T_j,T_k$ zwei $\zeta$ zulässige Rechtecke. Ist $\diam(T_j) \leq \diam(T_k)$ so kann Satz (\ref{thm:sem:quad:V}) angewendet werden um $A_{jk}$ zu approximieren. Im Fall $\diam(T_j) > \diam(T_k)$ können wir $A_{jk}$ approximieren indem wir mithilfe von $A_{jk} = A_{kj}$ und Satz (\ref{thm:sem:quad:V}), $A_{kj}$ berechnen. @@ -777,7 +756,7 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \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{c_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{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_p^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)} % \end{align} @@ -787,7 +766,7 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- &\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{c_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{c_2}{\zeta_Q}\right)^{-2(p+1)}\right)^2. \end{align*} -Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\square$ +Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{beweis} % \subsection{Quadratur über eine Achse} % \subsubsection{Quadratur} @@ -808,7 +787,7 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\sq % &\leq C_{\zeta,j,k}\Lambda_p^2(p+1)\left(1+\frac{2c_2}{\zeta_A}\right)^{-(p+1)}. % \end{align*} % \end{sat} -% \beweis Zunächst definieren wir die Konstanten +% \begin{beweis} Zunächst definieren wir die Konstanten % \begin{align*} % C_{\kappa} &=\frac{c_1}{(c_2\dist(T_j,T_k))^s} \quad \text{und} \quad \rho_{\kappa} = \frac{\zeta_A}{c_2} % \end{align*} @@ -845,7 +824,7 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\sq % &\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$\square$ +% \end{beweis} % % \subsubsection{Matrix} % \begin{sat} @@ -873,7 +852,7 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\sq % . % \end{align} % \end{sat} -% \beweis Da die Gauss-Quadratur ist interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist gilt: +% \begin{beweis} Da die Gauss-Quadratur ist interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist gilt: % \begin{align*} % (A_p)_{jk} % &=\abs{T_j}\abs{T_k}\sum_{a=0}^p w_a \sum_{b=0}^p w_b \sum_{c=0}^p w_c \sum_{d=0}^p w_d \kappa(\gamma_j(x_a,x_b),\gamma_k(y_c,y_d))\\ @@ -889,7 +868,7 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \hfill$\sq % &=\why{\sup_{\bs y \in T_k,x_b} \norm{\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)-\I_{2p+1}\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)}_{\infty,[0,1]}} % \end{align*} % -% \todo{\hfill$\square$\\} +% \todo{\end{beweis}\\} % -- 2.47.3