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kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über $T_k$ wird nun mittels einer doppelten Gauss-Quadratur berechnet. -\begin{defi} +\begin{defi}\label{thm:sem:glatt} 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 + \begin{align} % 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} @@ -636,32 +636,36 @@ 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.\\ -Bezeichnet $\gamma_j$ die Parametrisierung von $T_j$, so gilt aufgrund der Kettenregel +Bezeichne $g_{jk} : [0,1]^4 \rightarrow D \subset \R^6$ die Parametrisierung +\begin{align} + g_{jk}(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)) +\end{align} + mit der jeweiligen Parametrisierung $\gamma_j, \gamma_k$ zu $T_j, T_k$ aus Definition \ref{math:def:T}. So gilt Aufgrund von Lemma \ref{thm:sem:kett} \begin{align*} % \Abs{\partial_x^{\alpha}\partial_y^{\beta}\kappa} &= - \Abs{\partial_\lambda^{\alpha}\partial_y^{\beta}\kappa(\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_1,\lambda_2))} - &= \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} - \Abs{(\partial_x^{\alpha}\partial_y^{\beta}\kappa)(\gamma_j(x),\gamma_k(y))}. + \Abs{\partial_\lambda^{\alpha}\kappa(\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_1,\lambda_2))} + &=\Abs{\partial_\lambda^{\alpha}\kappa(g_{jk}(\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_{jk}(\lambda))}\\ + &\leq\diam(T_j)^{\alpha_1 + \alpha_2}\diam(T_k)^{\alpha_3 + \alpha_4} \Abs{\partial^{t_{jk}(\alpha)}(\kappa)(g_{jk}(\lambda))}\\ + &\leq \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} \Abs{\partial^{t_{jk}(\alpha)}(\kappa)(g_{jk}(\lambda))}, \end{align*} -Ferner gilt mit \eqref{math:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung +mit Multiindex $\alpha = (\alpha_1, \ldots , \alpha_4)$, wobei $\abs{\alpha}>0$ sei. +Ferner gilt mit Definition \ref{thm:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung \begin{align*} - \diam(T_j)^{\abs \alpha}& \diam(T_k)^{\abs \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} c_1 (c_2 \abs{\gamma_j(x) - \gamma_k(y)})^{-(\abs{\alpha}+\abs{\beta}+s)}(\abs{\alpha}+\abs{\beta})!\\ - \leq & \max(\diam(T_j),\diam(T_k))^{\abs{\alpha+\beta}} c_1 (c_2 \abs{\bs x - \bs y})^{-(\abs \alpha+\abs \beta+s)}(\abs \alpha+\abs \beta)!\\ - = & \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)^{\abs \alpha+\abs \beta}(\abs{\alpha}+\abs{\beta})!\\ - \leq& C_{\kappa}\rho_{\kappa}^nn!. +\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 \abs{\bs x - \bs y})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ + = & \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)^{\abs{\alpha}}\abs{\alpha}!\\ + \leq& C_{\kappa}\rho_{\kappa}^{\abs{\alpha}}\abs{\alpha}!, \end{align*} -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} +wobei $\bs x \in T_j$ und $\bs y \in T_k$ sei, mit $\zeta_Q$-zulässigen $T_j, T_k$ . Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt, und es gilt 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_{\kappa} 8e(1+2\rho_{\kappa})\Lambda_p^4(p+1)\left(1+\frac{1}{\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_Q,j,k}\Lambda_p^4(p+1)\left(1+\frac{c_2}{\zeta_Q}\right)^{-(p+1)}. + &\leq C_{\kappa} 8e(1+2\rho_{\kappa})\Lambda_p^4p\left(1+\frac{1}{\rho_{\kappa}}\right)^{-p}\\ +% &= C_{\zeta,j,k}\Lambda_p^4p\left(1+\frac{2}{\rho_{\kappa}}\right)^{-p}\\ + &= C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-p}. \end{align*} -Womit der Beweis abgeschlossen ist. +Damit ist der Beweis abgeschlossen. \end{beweis} % MatrixEINTRAG ------------------------------------------------- \begin{sat}\label{thm:sem:quad:V} @@ -669,42 +673,42 @@ Seien $T_j,T_k$ zwei $\zeta_Q$-zulässige Rechtecke mit zugehörigen Parametrisi \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\zeta_Q}{c_2} \right) \end{align} -Dann gilt für das Integral +Dann gilt mit $\bs \lambda_j = (\lambda_1,\lambda_2)$ und $\bs \lambda_k = (\lambda_3,\lambda_4)$ für das Integral \begin{align} A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}\\ &= \abs{T_j}\abs{T_k} - \int_{[0,1]^2} \int_{[0,1]^2} \kappa(\gamma_j(\bs x),\gamma_k(\bs y)) ds_{\bs y} ds_{\bs x} + \int_{[0,1]^2} \int_{[0,1]^2} \kappa(\gamma_j(\bs \lambda_j),\gamma_k(\bs \lambda_k)) d{\bs \lambda_k} d{\bs \lambda_j} \end{align} und für den durch die Gauss-Quadratur von $A_{jk}$ zum Grad $p$ entstehenden Term \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)) + (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})) \end{align*} die Abschätzung \begin{align} - \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)} + \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p} \end{align} \end{sat} -\begin{beweis} Da die Gauss-Quadratur ist interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist gilt: +\begin{beweis} Da die Gauss-Quadratur 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))\\ - &=\abs{T_j}\abs{T_k} \int_{[0,1]^2}\int_{[0,1]^2} \I_{2p+1}^4 \kappa(\gamma_j(\bs x),\gamma_k(\bs y)) ds_{\bs y} ds_{\bs x}, + &=\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}))\\ + &=\abs{T_j}\abs{T_k} \int_{[0,1]^2}\int_{[0,1]^2} \I_{2p+1}^4 \kappa(\gamma_j(\bs \lambda_j),\gamma_k(\bs \lambda_k)) d{\bs \lambda_k} d{\bs \lambda_j}, \end{align*} -wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ in $x_{\bs a}$ bezeichnet. Wegen der Additivität des Integrals und durch Hineinziehen des Betrags erhält man +wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ bezeichnet. Wegen der Additivität des Integrals und durch Hineinziehen des Betrags erhält man \begin{align*} - \abs{A&_{jk} - (A_p)_{jk}}\\ - &=\abs{T_j}\abs{T_k} \Abs{\int_{[0,1]^2}\int_{[0,1]^2}\kappa(\gamma_j(\bs x),\gamma_k(\bs y)) - \int_{[0,1]^2}\int_{[0,1]^2} \I_{2p+1}^4 \kappa(\gamma_j(\bs x),\gamma_k(\bs y)) ds_{\bs y} ds_{\bs x}}\\ - &\leq \abs{T_j}\abs{T_k}\int_{[0,1]^2}\int_{[0,1]^2} \Abs{\kappa(\gamma_j(\bs x),\gamma_k(\bs y)) - \I_{2p+1}^4 \kappa(\gamma_j(\bs x),\gamma_k(\bs y))} ds_{\bs y} ds_{\bs x}\\ - &\leq \abs{T_j}\abs{T_k} \norm{\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)-\I_{2p+1}^4\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)}_{\infty,[0,1]^4} + \abs{A_{jk} -& (A_p)_{jk}}\\ + &=\abs{T_j}\abs{T_k} \Big|\int_{[0,1]^2}\int_{[0,1]^2}\kappa(\gamma_j(\bs \lambda_j),\gamma_k(\bs \lambda_k)) d{\bs \lambda_k} d{\bs \lambda_j}\\ &\quad- \int_{[0,1]^2}\int_{[0,1]^2} \I_{2p+1}^4 \kappa(\gamma_j(\bs \lambda_j),\gamma_k(\bs \lambda_k)) d{\bs \lambda_k} d{\bs \lambda_j}\Big|\\ + &\leq \abs{T_j}\abs{T_k}\int_{[0,1]^2}\int_{[0,1]^2} \Abs{\kappa(\gamma_j(\bs \lambda_j),\gamma_k(\bs \lambda_k)) - \I_{2p+1}^4 \kappa(\gamma_j(\bs \lambda_j),\gamma_k(\bs \lambda_k))} d{\bs \lambda_k} d{\bs \lambda_j}\\ + &\leq \abs{T_j}\abs{T_k} \norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot))-\I_{2p+1}^4\kappa(\gamma_j(\cdot),\gamma_k(\cdot))}_{\infty,[0,1]^4} \end{align*} -Mithilfe von Lemma \ref{thm:sem:pol:V} erhalten wir die Behauptung +Mithilfe von Satz \ref{thm:sem:pol:V} erhalten wir die Behauptung \begin{align*} \abs{A_{jk} - (A_p)_{jk}} - &\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)}. + &\leq \abs{T_j}\abs{T_k} C_{\zeta_Q,j,k}\Lambda_p^42p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p}\\ + &= \tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p}. \end{align*} -\todo{}\end{beweis} +\end{beweis} \begin{lem} \label{thm:sem:switch} Seien $Tj,T_k$ zwei Randstücke, $\kappa(\bs x,\bs y)$ eine asymptotisch glatte Kernfunktion und @@ -812,7 +816,7 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{bewei % &=\diam_{\bs a}(T_j)^{\alpha} \diam_{\bs a}(T_k)^{\beta} % \Abs{(\partial_x^{\alpha}\partial_y^{\beta}\kappa)(\gamma_j(x,x_b),\gamma_k(y,y_b))}. % \end{align*} -% Ferner gilt mit \eqref{math:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung +% Ferner gilt mit \eqref{thm:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung % \begin{align*} % \diam_{\bs a}(T_j)^{\alpha}& \diam_{\bs a}(T_k)^{\beta} \Abs{(\partial_x^{\alpha}\partial_y^{\beta}\kappa)(\gamma_j(x,x_b),\gamma_k(y,y_b))}\\ % \leq& \diam_{\bs a}(T_j)^{\alpha} \diam_{\bs a}(T_k)^{\beta} c_1 (c_2 \abs{\gamma_j(x,x_b)-\gamma_k(y,y_b))})^{-(\abs{\alpha}+\abs{\beta}+s)}(\abs{\alpha}+\abs{\beta})!\\ -- 2.47.3