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Die Seitenlänge in Rich \begin{align*} \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. +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\}, +\end{align*} +mit dem Skalarprodukt $\cdot$ und $\bs a \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$. \end{defi} Mit diesen Vorüberlegungen definieren wir uns die Diskretisierung des Randes $\Gamma$. \begin{defi} @@ -543,7 +547,7 @@ Weiterhin wollen wir uns kurz die Ableitung der asymptotisch glatten Kernfunktio 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 \begin{align*} \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))} + &= \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))} \end{align*} für jeden Multiindex $\alpha = (\alpha_1, \alpha_2, \alpha_3, \alpha_4)$, mit $\abs{\alpha} \geq 1$. \end{lem} @@ -636,8 +640,12 @@ Mit diesen Vorüberlegungen können wir nun den folgenden Satz über die Interpo C_{\zeta_Q,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{2}{c_2\zeta_Q} \right) \end{align*} 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+c_2\zeta_Q\right)^{-(p+1)}. +% \end{align*} \begin{align*} - \norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot))&-\I_p^4\kappa(\gamma_j(\cdot),\gamma_k(\cdot))}_{\infty,[0,1]^4} + \norm{\kappa(\cdot,\cdot)&-\I_p^4\kappa(\cdot,\cdot)}_{\infty,T_j \times T_k} \leq C_{\zeta_Q,j,k}\Lambda_p^4 (p+1)\left(1+c_2\zeta_Q\right)^{-(p+1)}. \end{align*} \end{sat} @@ -661,7 +669,7 @@ Bezeichne $g_{jk} : [0,1]^4 \rightarrow (T_j \times T_k) \subset \R^6$ die Param % \Abs{\partial_x^{\alpha}\partial_y^{\beta}\kappa} &= \Abs{\partial_\lambda^{\alpha}\kappa(\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4))} &=\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))}\\ + &= \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_{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*} @@ -799,6 +807,7 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{beweis} + \noindent \todo{ \begin{itemize} @@ -808,6 +817,229 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{itemize} } +\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{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)\}. + \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{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) + \end{align*} +die Abschätzung + \begin{align*} + \sup_{x_2 \in [0,1]\atop\bs y \in T_k}\norm{\kappa(\gamma_j(\cdot,x_2),\bs y)&-\I_p\kappa(\gamma_j(\cdot,x_2),\bs y)}_{\infty,[0,1]} + \leq C_{\zeta_S,j,k}\Lambda_p (p+1)\left(1+2 c_2\zeta_S\right)^{-(p+1)}. + \end{align*} +\end{sat} + + +\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} +\end{align*} +und können dann die Konstante $C_{\zeta_S,j,k}$ kurz +\begin{align*} + C_{\zeta_S,j,k} &= 8eC_{\kappa} (1+\rho_{\kappa}) +\end{align*} +schreiben.\\ +Sei $\bs y \in T_k$ und $x_2 \in [0,1]$ fest gewählt. Sei weiterhin $\gamma_j$ die Parametrisierung zu $T_j$ aus Definition \ref{math:def:T}. So gilt mithilfe von Lemma \ref{thm:sem:kett} +\begin{align*} + \Abs{\partial_\lambda^{\alpha}\kappa(\gamma_j(\lambda,x_2),\bs y)} + &= \diam_{\bs a}(T_j)^{\abs{\alpha}}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda,x_2),\bs y)}, +\end{align*} +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& 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 +\begin{align*} + \norm{\kappa(\gamma_j(\cdot,x_2),\bs y)&-\I_p\kappa(\gamma_j(\cdot,x_2),\bs y)}_{\infty,[0,1]}\\ + &\leq C_{\kappa} 8e(1+\rho_{\kappa})\Lambda_p(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ +% &= C_{\zeta,j,k}\Lambda_p^4p\left(1+\frac{2}{\rho_{\kappa}}\right)^{-p}\\ + &= C_{\zeta_S,j,k}\Lambda_p(p+1)\left(1+2 c_2\zeta_S\right)^{-(p+1)}. +\end{align*} +\end{beweis} + +\subsection{Quadratur über ein Element} + + +\begin{defi} Seien $T_j,T_k \in \R^3$ achsenorientierte Rechtecke und sei $\zeta_E > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_E$-zulässig, genau dann wenn + \begin{align}\label{math:sem:zetaE} + \dist(T_j, T_k)&\geq \zeta_E \min\{ \diam(T_j) , \diam(T_k)\}. + \end{align} + Andernfalls heißt das Paar $(T_j,T_k)$ $\zeta_E$-unzulässig. +\end{defi} + + +\begin{sat} \label{thm:sem:pol:E} Seien $T_j,T_k \subseteq \R^3$ zwei $\zeta_E$-zulässige Rechtecke mit $\diam(T_j) \leq \diam(T_k)$. 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_E,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\sqrt 2}{c_2\zeta_E} \right) + \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} + \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} + + +\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{1}{c_2\zeta_E} +\end{align*} +und können dann die Konstante $C_{\zeta_E,j,k}$ kurz +\begin{align*} + C_{\zeta_E,j,k} &= 8eC_{\kappa} (1+\sqrt 2\rho_{\kappa}) +\end{align*} +schreiben.\\ +Sei $\bs y \in T_k$ fest gewählt. Sei weiterhin $\gamma_j$ die Parametrisierung zu $T_j$ aus Definition \ref{math:def:T}. So gilt mithilfe von Lemma \ref{thm:sem:kett} +\begin{align*} + \Abs{\partial_\lambda^{\alpha}\kappa(\gamma_j(\lambda_1,\lambda_2),\bs y)} + &= \diam_{\bs a}(T_j)^{\alpha_1}\diam_{\bs b}(T_j)^{\alpha_2}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,\lambda_2),\bs y)},\\ +% &\leq \diam(T_j)^{\alpha_1 + \alpha_2}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,\lambda_2),\bs y)},\\ + &\leq \diam(T_j)^{\abs{\alpha}}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,\lambda_2),\bs y)}, +\end{align*} +mit Multiindex $\alpha = (\alpha_1, \alpha_2, 0, 0)$, 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}}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,\lambda_2),\bs y)} + \leq& \diam(T_j)^{\abs{\alpha}} c_1 (c_2 \abs{\gamma_j(\lambda_1,\lambda_2) - \bs y})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ + \leq & \diam(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(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 $\gamma_j(\lambda_1,\lambda_2) \in T_j$ sei. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt, und es gilt abschließend die Abschätzung +\begin{align*} + \norm{\kappa(\gamma_j(\cdot),\bs y)-\I_p^2\kappa(\gamma_j(\cdot),\bs y)}_{\infty,[0,1]^2} + &\leq C_{\kappa} 8e(1+\sqrt 2 \rho_{\kappa})\Lambda_p^2(p+1)\left(1+\frac{\sqrt 2}{\rho_{\kappa}}\right)^{-(p+1)}\\ + &= C_{\zeta_E,j,k}\Lambda_p^2(p+1)\left(1+\sqrt 2 c_2\zeta_E\right)^{-(p+1)}. +\end{align*} +\end{beweis} + + +\subsection{Quadratur über eine Achse} + +\todo{$\bs a$ von $T_j$ muss nicht gleich von $T_k$ sein, ändert aber nichts am Schätzer, oder ist das ausreichend Definiert. Sollte aber jeweils die kleinste Seite sein. Und interessanter weise brauch ich keine Aussage über parrallele oder orthogonale Lage.} + +\begin{defi} Seien $T_j,T_k \in \R^3$ achsenorientierte Rechtecke mit $\diam_{\bs a}(T_i) \leq \diam_{\bs b}(T_i)$, $i \in \{j,k\}$ und sei $\zeta_A > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_A$-zulässig, genau dann wenn + \begin{align}\label{math:sem:zetaA} + \dist(T_j, T_k)&\geq \zeta_A \max\{ \diam_{\bs a}(T_j) , \diam_{\bs a}(T_k)\}. + \end{align} + Andernfalls heißt das Paar $(T_j,T_k)$ $\zeta_A$-unzulässig. +\end{defi} + + +\begin{sat} \label{thm:sem:pol:A} Seien $T_j,T_k \subseteq \R^3$ zwei $\zeta_A$-zulässige Rechtecke mit $\diam_{\bs a}(T_i) \leq \diam_{\bs b}(T_i)$, wobei $i \in \{j,k\}$. Sei $\kappa(\cdot,\cdot)$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit + \begin{align*} + C_{\zeta_A,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\sqrt 2}{c_2\zeta_A} \right) + \end{align*} +die Abschätzung + \begin{align*} + &\sup_{x_2,y_2 \in [0,1]}\norm{\kappa(\gamma_j(\cdot,x_2),\gamma_k(\cdot,y_2))-\I_p^2\kappa(\gamma_j(\cdot,x_2),\gamma_k(\cdot,y_2))}_{\infty,[0,1]^2}\\ + &\leq C_{\zeta_A,j,k}\Lambda_p^2 (p+1)\left(1+\sqrt 2 c_2\zeta_A\right)^{-(p+1)}. + \end{align*} +\end{sat} + + +\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{1}{c_2\zeta_A} +\end{align*} +und können dann die Konstante $C_{\zeta_A,j,k}$ kurz +\begin{align*} + C_{\zeta_A,j,k} &= 8eC_{\kappa} (1+\sqrt 2\rho_{\kappa}) +\end{align*} +schreiben.\\ +Sei $x_2,y_2 \in [0,1]$ fest gewählt. Seien weiterhin $\gamma_j,\gamma_k$ die Parametrisierungen zu $T_j$ und $T_k$ aus Definition \ref{math:def:T}. So gilt mithilfe von Lemma \ref{thm:sem:kett} +\begin{align*} + |\partial_\lambda^{\alpha}\kappa(\gamma_j(\lambda_1,x_2),&\gamma_k(\lambda_3,y_2))|\\ + &= \diam_{\bs a}(T_j)^{\alpha_1}\diam_{\bs a}(T_k)^{\alpha_3}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,x_2),\gamma_k(\lambda_3,y_2))},\\ + &\leq \max\{\diam_{\bs a}(T_j),\diam_{\bs a}(T_k)\}^{\abs{\alpha}}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,x_2),\gamma_k(\lambda_3,y_2))}, +\end{align*} +mit Multiindex $\alpha = (\alpha_1, 0, \alpha_3, 0)$, wobei $\abs{\alpha}>0$ sei. +Ferner gilt mit Definition \ref{thm:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung +\begin{align*} + \max\{\diam_{\bs a}(T_j),&\diam_{\bs a}(T_k)\}^{\abs{\alpha}}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,x_2),\gamma_k(\lambda_3,y_2))}\\ + \leq& \max\{\diam_{\bs a}(T_j),\diam_{\bs a}(T_k)\}^{\abs{\alpha}} c_1 (c_2 \abs{\gamma_j(\lambda_1,x_2) - \gamma_k(\lambda_3,y_2)})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ + \leq & \max\{\diam_{\bs a}(T_j),\diam_{\bs a}(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_{\bs a}(T_j),\diam_{\bs a}(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*} +wobei $\gamma_j(\lambda_1,\lambda_2) \in T_j$ sei. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt, und es gilt abschließend die Abschätzung +\begin{align*} + &\norm{\kappa(\gamma_j(\cdot,x_2),\gamma_k(\cdot,y_2))-\I_p^2\kappa(\gamma_j(\cdot,x_2),\gamma_k(\cdot,y_2))}_{\infty,[0,1]^2}\\ + &\leq C_{\kappa} 8e(1+\sqrt 2 \rho_{\kappa})\Lambda_p^2(p+1)\left(1+\frac{\sqrt 2}{\rho_{\kappa}}\right)^{-(p+1)}\\ + &= C_{\zeta_A,j,k}\Lambda_p^2(p+1)\left(1+\sqrt 2 c_2\zeta_A\right)^{-(p+1)}. +\end{align*} +\end{beweis} + + +\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} + \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{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]}\\ + &\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} + + +\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{1}{c_2\zeta_D} +\end{align*} +und können dann die Konstante $C_{\zeta_D,j,k}$ kurz +\begin{align*} + C_{\zeta_D,j,k} &= 8eC_{\kappa} (1+\sqrt 3\rho_{\kappa}) +\end{align*} +schreiben.\\ +Sei $y_2 \in [0,1]$ fest gewählt. Seien weiterhin $\gamma_j,\gamma_k$ die Parametrisierungen zu $T_j$ und $T_k$ aus Definition \ref{math:def:T}. So gilt mithilfe von Lemma \ref{thm:sem:kett} +\begin{align*} + |\partial_\lambda^{\alpha}\kappa(\gamma_j(\lambda_1,\lambda_2),&\gamma_k(\lambda_3,y_2))|\\ + &= \diam_{\bs a}(T_j)^{\alpha_1}\diam_{\bs b}(T_j)^{\alpha_2}\diam_{\bs a}(T_k)^{\alpha_3}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,y_2))},\\ + &\leq \max\{\diam(T_j),\diam_{\bs a}(T_k)\}^{\abs{\alpha}}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,y_2))}, +\end{align*} +mit Multiindex $\alpha = (\alpha_1, \alpha_2, \alpha_3, 0)$, wobei $\abs{\alpha}>0$ sei. +Ferner gilt mit Definition \ref{thm:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung +\begin{align*} + \max\{\diam(T_j)&,\diam_{\bs a}(T_k)\}^{\abs{\alpha}}\Abs{\partial^{t_{jk}(\alpha)} \kappa(\gamma_j(\lambda_1,x_2),\gamma_k(\lambda_3,y_2))}\\ + \leq& \max\{\diam(T_j),\diam_{\bs a}(T_k)\}^{\abs{\alpha}} c_1 (c_2 \abs{\gamma_j(\lambda_1,x_2) - \gamma_k(\lambda_3,y_2)})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ + \leq & \max\{\diam(T_j),\diam_{\bs a}(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_{\bs a}(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*} +wobei $\gamma_j(\lambda_1,\lambda_2) \in T_j$ sei. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt, und es gilt abschließend die Abschätzung +\begin{align*} + &\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_{\kappa} 8e(1+\sqrt 3 \rho_{\kappa})\Lambda_p^3(p+1)\left(1+\frac{2}{\sqrt 3\rho_{\kappa}}\right)^{-(p+1)}\\ + &= 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{beweis} + % \subsection{Quadratur über eine Achse} -- 2.47.3