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des Rechtecks $T$ in Richtung $\bs a$. Die Seitenlänge in Rich \end{align*} wobei das Minimum aufgrund der Kompaktheit von $T_j, T_k$ angenommen wird. Weiterhin definieren wir auch den Abstand in einer bestimmten Richtung $\bs a$ durch \begin{align*} - \dist_{\bs a}(T_j,T_k) &= \min \{\bs a^T \cdot (\bs x-\bs y) ~|~ \bs x \in T_j,\bs y \in T_k\}, + \dist_{\bs a}(T_j,T_k) &= \min \{\abs{\bs a^T \cdot (\bs x-\bs y)} ~|~ \bs x \in T_j,\bs y \in T_k\}, \end{align*} mit dem Skalarprodukt $\cdot$ und $\bs a \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$. \end{defi} @@ -441,7 +441,7 @@ Wie an der einfachen Fehlerabschätzung aus \cite[Theorem 1.17]{pla:nummat} für \end{align*} durch benutzen von Chebyshev-Knoten minimieren \cite[Definition 1.22]{pla:nummat}. 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=0,\ldots,p. + x_j &= \cos \left( \frac{2j-1}{p} \frac{\pi}{2} \right) \quad \text{ für } j=1,\ldots,p. \end{align*} Durch affine Transformation der Chebyshev-Knoten \cite[Theorem 1.25]{pla:nummat} ergibt sich die Fehlerabschätzung \begin{align*} @@ -467,7 +467,7 @@ Dann können wir die Fehlerabschätzung für den Interpolationsoperator $\I_p^d$ \subsection{Gauss-Quadratur} -Im Folgenden wollen wir die klassische Gauss-Quadratur definieren, welche durch Verwendung der 1-Funktion statt der Gewichtungsfunktion entsteht. +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 @@ -513,7 +513,7 @@ Zunächst wollen wir zeigen, dass wir die vier Integrale aus \eqref{math:gal:kap \noindent -Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktionen gut durch Polynome interpolieren. Hierzu benötigen wir aus \cite[Theorem 3.2]{bor:errbox} das folgende Lemma: +Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktionen außerhalb des Singularitätsbereichs gut durch Polynome interpolieren. Hierzu benötigen wir aus \cite[Theorem 3.2]{bor:errbox} das folgende Lemma: \begin{lem}\label{thm:sem:ipolnD} %Interpol über Glatten KERN nD Die Funktion $u \in \C^{\infty}([0,1])^d$ erfülle für Konstanten $C_u,\rho_u > 0$ @@ -531,7 +531,7 @@ Dann gilt für alle $p\in \N_0$ Weiterhin wollen wir uns kurz die Ableitung der asymptotisch glatten Kernfunktion mit einer Parametrisierung anschauen. \begin{lem} \label{thm:sem:kett} - Sei $\kappa(\cdot , \cdot) : \R^3\times\R^3 \rightarrow \R$ eine asymptotisch glatte Kernfunktion und sei weiterhin $ g: \R^4 \rightarrow \R^6$ die Parametrisierung zweier achsenorientierten Rechtecke $T_j$ und $T_k$, mit $T_j \cap T_k = \emptyset$ + Sei $\kappa(\cdot , \cdot) : \R^3\times\R^3 \rightarrow \R$ eine asymptotisch glatte Kernfunktion und sei weiterhin $ g: \R^4 \rightarrow \R^6$ die Parametrisierung zweier achsenorientierten Rechtecke $T_j$ und $T_k$, mit $T_j \cap T_k = \emptyset$, sowie \begin{align} g(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)), \end{align} @@ -540,11 +540,11 @@ Weiterhin wollen wir uns kurz die Ableitung der asymptotisch glatten Kernfunktio \begin{align} t_{jk}(\alpha) = \begin{pmatrix} - \alpha_1 \cdot \bs a_j + \alpha_2 \cdot \bs b_j \\ - \alpha_3 \cdot {\bs a_k} + \alpha_4 \cdot {\bs b_k} + \alpha_1 \cdot \bs a + \alpha_2 \cdot \bs b \\ + \alpha_3 \cdot \tilde{\bs a} + \alpha_4 \cdot \tilde {\bs b} \end{pmatrix}, \end{align} - wobei $\bs a_j,\bs b_j \in \R^3$ die paarweise verschiedenen Einheitsvektoren zur Parametrisierung $\gamma_j$ sind und für den Index $k$ analog, gilt dann für die Verknüpfung $\kappa \circ g$ die Kettenregel + wobei $\bs a,\bs b,\tilde{\bs a}, \tilde{\bs b} \in \R^3$ die paarweise verschiedenen Einheitsvektoren zu Parametrisierungen $\gamma_j,\gamma_k$ sind, gilt dann für die Verknüpfung $\kappa \circ g$ die Kettenregel \begin{align*} \abs{\partial^{\alpha}(\kappa \circ g)(\lambda)} &= \diam_{\bs a}(T_j)^{\alpha_1}\diam_{\bs b}(T_j)^{\alpha_2} \diam_{\bs a}(T_k)^{\alpha_3}\diam_{\bs b}(T_k)^{\alpha_4} \abs{\partial^{t_{jk}(\alpha)} \kappa (g(\lambda))} @@ -555,7 +555,7 @@ Weiterhin wollen wir uns kurz die Ableitung der asymptotisch glatten Kernfunktio \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}. + \partial(\kappa \circ g)(\lambda) = \partial \kappa(g(\lambda)) \circ \partial g(\lambda) \qquad \in \R^{1\times 4}. \end{align*} Mithilfe der Jacobimatrizen $A := \partial \kappa(g(\lambda)) \in \R^{1\times 6}$, $B := \partial g(\lambda) \in\R^{6\times 4}$ untersuchen wir zunächst die partiellen Ableitungen \begin{align} @@ -645,7 +645,7 @@ die Abschätzung % \leq C_{\zeta_Q,j,k}\Lambda_p^4 (p+1)\left(1+c_2\zeta_Q\right)^{-(p+1)}. % \end{align*} \begin{align*} - \norm{\kappa(\cdot,\cdot)&-\I_p^4\kappa(\cdot,\cdot)}_{\infty,T_j \times T_k} + \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+c_2\zeta_Q\right)^{-(p+1)}. \end{align*} \end{sat} @@ -678,7 +678,7 @@ Ferner gilt mit Definition \ref{thm:sem:glatt} und den Konstanten $C_{\kappa},\r \begin{align*} \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} &\Abs{\partial^{t_{jk}(\alpha)}\kappa(g_{jk}(\lambda))}\\ \leq& \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} c_1 (c_2 \abs{\gamma_j(\lambda_1,\lambda_2) - \gamma_k(\lambda_3,\lambda_4)})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ - = & \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} c_1 (c_2 \dist(T_j,T_k))^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ + \leq & \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} c_1 (c_2 \dist(T_j,T_k))^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ = & \frac{c_1}{(c_2 \dist(T_j,T_k))^s} \left( \frac{\max(\diam(T_j),\diam(T_k))}{c_2 \dist(T_j,T_k)} \right)^{\abs{\alpha}}\abs{\alpha}!\\ \leq& C_{\kappa}\rho_{\kappa}^{\abs{\alpha}}\abs{\alpha}!, \end{align*} @@ -701,7 +701,7 @@ Da sich wie gezeigt die Kernfunktion besonders gut durch Polynome Interpolieren \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 \begin{align} \label{math:sem:zetaQ:c} - \tilde C_{\zeta_Q,j,k}&:=2^4e\frac{c_1\abs{T_j}\abs{T_k}}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{2}{c_2\zeta_Q} \right) + \tilde C_{\zeta_Q,j,k}&:=2^3e\frac{c_1\abs{T_j}\abs{T_k}}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{2}{c_2\zeta_Q} \right) \end{align} Dann gilt mit $\bs \lambda_j = (\lambda_1,\lambda_2)$ und $\bs \lambda_k = (\lambda_3,\lambda_4)$ für das Integral \begin{align} @@ -716,12 +716,12 @@ 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_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)} + \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4 2(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)} \end{align} \end{sat} -\begin{beweis}Wir wissen aus \cite[Korollar 6.38]{pla:nummat}, dass die Gauss-Quadratur interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist, unter Berücksichtigung, der hier verwendeten Indizierung von 0 statt 1. Deshalb gilt +\begin{beweis}Wir wissen aus \cite[Korollar 6.38]{pla:nummat}, dass die Gauss-Quadratur interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist, unter Berücksichtigung, der hier verwendeten Indizierung beginnend mit 0 statt 1. Deshalb 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( \lambda_{a},\lambda_{b}),\gamma_k(\lambda_{c},\lambda_{d}))\\ @@ -738,18 +738,18 @@ Mithilfe von Satz \ref{thm:sem:pol:V} für den Grad $2p+1$ erhalten wir die Beha \begin{align*} \abs{A_{jk} - (A_p)_{jk}} &\leq \abs{T_j}\abs{T_k} C_{\zeta_Q,j,k}\Lambda_{2p+1}^42(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\\ - &= \tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}. + &= \tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4 2(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}. \end{align*} \end{beweis} \noindent Da die Konstante $\tilde C_{\zeta_Q,j,k}$ durch die Netzverfeinerung, aufgrund der Distanz sehr groß werden könnte, untersuchen wir sie an dieser Stelle noch einmal etwas genauer. - Mitithilfe der folgenden Abschätzung + Mitithilfe der folgenden Abschätzung für $\zeta_Q$-zulässige Rechtecke $T_j,T_k$ \begin{align*} - \tilde C_{\zeta_Q,j,k} & = 2^4 e \frac{c_1 \abs{T_j}\abs{T_k}}{(c_2 \dist(T_j,T_k))^s}\left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ - & \leq 2^4 e \frac{c_1 \abs{T_j}\abs{T_k}}{(c_2 \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ - & \leq 2^4 e \frac{c_1 \max\{ \diam(T_j) , \diam(T_k)\}^4}{(c_2 \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ - & = 2^4 e \frac{c_1}{(c_2 \zeta_Q)^s} \max\{ \diam(T_j) , \diam(T_k)\}^{4-s} \left( 1 + \frac{2}{c_2\zeta_Q} \right), + \tilde C_{\zeta_Q,j,k} & = 2^3 e \frac{c_1 \abs{T_j}\abs{T_k}}{(c_2 \dist(T_j,T_k))^s}\left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ + & \leq 2^3 e \frac{c_1 \abs{T_j}\abs{T_k}}{(c_2 \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ + & \leq 2^3 e \frac{c_1 \max\{ \diam(T_j) , \diam(T_k)\}^4}{(c_2 \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ + & = 2^3 e \frac{c_1}{(c_2 \zeta_Q)^s} \max\{ \diam(T_j) , \diam(T_k)\}^{4-s} \left( 1 + \frac{2}{c_2\zeta_Q} \right), \end{align*} welche unabhängig von der im Netz auftretenden Distanzen ist, können wir also eine gute Aussage über die Stabilität der Zulässigkeitsbedingung im Zuge der Netzverfeinerung treffen. @@ -785,24 +785,24 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \begin{sat} - Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^3\times \R^3 \to \R$ 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}. Sei $A\in \R^{n \times n}$ eine Matrix, deren Einträge gegeben sind durch + Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^3\times \R^3 \to \R$ 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} für $\zeta_Q$-zulässsige Rechtecke $T_j,T-k$ und für unzulässige sei $\tilde C_{\zeta_Q,j,k} = 0$ . Sei $A\in \R^{n \times n}$ 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_{2p+1}^4(p+1)\left(1+c_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_{2p+1}^4 2(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}. \end{align*} \end{sat} -\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, können wir Satz \ref{thm:sem:quad:V} anwenden. Damit erhalten wir +\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 $0$ und die Abschätzung für $\tilde C_{\zeta_Q,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, können wir Satz \ref{thm:sem:quad:V} anwenden. Damit erhalten wir \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_{2p+1}^4(p+1)\left(1+c_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_{2p+1}^4(p+1)\left(1+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_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\right)^2. + &\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4 2(p+1)\left(1+c_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_{2p+1}^4 2(p+1)\left(1+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_{2p+1}^4 2(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\right)^2. \end{align*} Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{beweis} @@ -818,19 +818,18 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. } \subsection{Quadratur über eine Seite} -\todo{$\bs a$ und die kleinste Seite der beiden Elemente, ist das Eindeutig?} -\begin{defi} Seien $T_j,T_k \in \R^3$ achsenorientierte Rechtecke mit $\diam_{\bs a}(T_i) \leq \diam_{\bs b}(T_i)$, wobei $i \in \{j,k\}$ und sei $\zeta_S > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_S$-zulässig, genau dann wenn +\begin{defi} Seien $T_j,T_k \in \R^3$ achsenorientierte Rechtecke und sei $\zeta_S > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_S$-zulässig, genau dann wenn \begin{align}\label{math:sem:zetaS} - \dist_{\bs a} (T_j, T_k)&\geq \zeta_S \min\{ \diam_{\bs a}(T_j) , \diam_{\bs a}(T_k)\}. + \dist (T_j, T_k)&\geq \zeta_S \min\{ \diam_{\bs a}(T_j) , \diam_{\bs a}(T_k)\}. \end{align} Andernfalls heißt das Paar $(T_j,T_k)$ $\zeta_S$-unzulässig. \end{defi} -\begin{sat} \label{thm:sem:pol:S} Seien $T_j,T_k \subseteq \R^3$ zwei $\zeta_S$-zulässige Rechtecke mit $\diam_{\bs a}(T_i) \leq \diam_{\bs b}(T_i)$, $i \in \{j,k\}$. Sei $\kappa(\cdot,\cdot)$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit +\begin{sat} \label{thm:sem:pol:S} Seien $T_j,T_k \subseteq \R^3$ zwei $\zeta_S$-zulässige Rechtecke mit $\diam_{\bs a}(T_j) \leq \diam_{\bs a}(T_k)$. 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_S,j,k}&:=8e\frac{c_1}{(c_2\dist_{\bs a}(T_j,T_k))^s} \left(1+\frac{1}{c_2\zeta_S} \right) + C_{\zeta_S,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{1}{c_2\zeta_S} \right) \end{align*} die Abschätzung \begin{align*} @@ -842,7 +841,7 @@ die Abschätzung \begin{beweis} Zunächst definieren wir die Konstanten \begin{align*} - C_{\kappa} &=\frac{c_1}{(c_2\dist_{\bs a}(T_j,T_k))^s} \quad \text{und} \quad \rho_{\kappa} = \frac{1}{c_2\zeta_S} + C_{\kappa} &=\frac{c_1}{(c_2\dist(T_j,T_k))^s} \quad \text{und} \quad \rho_{\kappa} = \frac{1}{c_2\zeta_S} \end{align*} und können dann die Konstante $C_{\zeta_S,j,k}$ kurz \begin{align*} @@ -858,9 +857,9 @@ mit Multiindex $\alpha = (\alpha_1, 0, 0, 0)$, wobei $\alpha_1>0$ sei. Ferner gilt mit Definition \ref{thm:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung \begin{align*} \diam_{\bs a}(T_j)^{\abs{\alpha}}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda,x_2),\bs y)} - \leq& \diam_{\bs a}(T_j)^{\abs{\alpha}} c_1 (c_2 \abs{\gamma_j(\lambda,x_2) - \bs y})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ - \leq & \diam_{\bs a}(T_j)^{\abs{\alpha}} c_1 (c_2 \dist_{\bs a}(T_j,T_k))^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ - = & \frac{c_1}{(c_2 \dist_{\bs a}(T_j,T_k))^s} \left( \frac{\diam_{\bs a}(T_j)}{c_2 \dist_{\bs a}(T_j,T_k)} \right)^{\abs{\alpha}}\abs{\alpha}!\\ + \leq& \diam_{\bs a}(T_j)^{\abs{\alpha}} c_1 (c_2 \abs{\gamma_j(\lambda,x_2) - \bs y})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ + \leq& \diam_{\bs a}(T_j)^{\abs{\alpha}} c_1 (c_2 \dist(T_j,T_k))^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ + =& \frac{c_1}{(c_2 \dist(T_j,T_k))^s} \left( \frac{\diam_{\bs a}(T_j)}{c_2 \dist(T_j,T_k)} \right)^{\abs{\alpha}}\abs{\alpha}!\\ \leq& C_{\kappa}\rho_{\kappa}^{\abs{\alpha}}\abs{\alpha}!, \end{align*} wobei $\gamma_j(\lambda_1,\lambda_2) \in T_j$ und $\gamma_k(\lambda_3,\lambda_4) \in T_k$ sei. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt, und es gilt abschließend die Abschätzung @@ -889,7 +888,7 @@ wobei $\gamma_j(\lambda_1,\lambda_2) \in T_j$ und $\gamma_k(\lambda_3,\lambda_4) \end{align*} die Abschätzung \begin{align*} - \sup_{\bs y \in T_k}\norm{\kappa(\cdot,\bs y)&-\I_p^2\kappa(\cdot,\bs y)}_{\infty,T_j} + \sup_{\bs y \in T_k}\norm{\kappa(\gamma_j(\cdot),\bs y)&-\I_p^2\kappa(\gamma_j(\cdot),\bs y)}_{\infty,[0,1]^2} \leq C_{\zeta_E,j,k}\Lambda_p^2 (p+1)\left(1+\sqrt 2 c_2\zeta_E\right)^{-(p+1)}. \end{align*} \end{sat} @@ -989,20 +988,20 @@ wobei $\gamma_j(\lambda_1,\lambda_2) \in T_j$ sei. Daher sind die Voraussetzunge \subsection{Quadratur über drei Seiten} \begin{defi} Seien $T_j,T_k \in \R^3$ achsenorientierte Rechtecke mit $\diam(T_j) \leq \diam(T_k)$ und $\diam_{\bs a}(T_k) \leq \diam_{\bs b}(T_k)$. Sei $\zeta_D > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_D$-zulässig, genau dann wenn - \begin{align}\label{math:sem:zetaA} + \begin{align}\label{math:sem:zetaD} \dist(T_j, T_k)&\geq \zeta_D \max\{\diam(T_j),\diam_{\bs a}(T_k)\}. \end{align} Andernfalls heißt das Paar $(T_j,T_k)$ $\zeta_D$-unzulässig. \end{defi} -\begin{sat} \label{thm:sem:pol:A} Seien $T_j,T_k \subseteq \R^3$ zwei $\zeta_D$-zulässige Rechtecke mit $\diam(T_j) \leq \diam(T_k)$ und $\diam_{\bs a}(T_k) \leq \diam_{\bs b}(T_k)$. Sei $\kappa(\cdot,\cdot)$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit +\begin{sat} \label{thm:sem:pol:D} Seien $T_j,T_k \subseteq \R^3$ zwei $\zeta_D$-zulässige Rechtecke mit $\diam(T_j) \leq \diam(T_k)$ und $\diam_{\bs a}(T_k) \leq \diam_{\bs b}(T_k)$. 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_D,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\sqrt 3}{c_2\zeta_D} \right) \end{align*} die Abschätzung \begin{align*} - &\sup_{y_2 \in [0,1]}\norm{\kappa(\cdot,\gamma_k(\cdot,y_2))-\I_p^3\kappa(\cdot,\gamma_k(\cdot,y_2))}_{\infty,T_j\times[0,1]}\\ + &\sup_{y_2 \in [0,1]}\norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot,y_2))-\I_p^3\kappa(\gamma_j(\cdot),\gamma_k(\cdot,y_2))}_{\infty,[0,1]^3}\\ &\leq C_{\zeta_D,j,k}\Lambda_p^3(p+1)\left(1+\frac {2c_2\zeta_D} {\sqrt 3} \right)^{-(p+1)}. \end{align*} \end{sat} -- 2.47.3