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\zeta \min\{ \diam(T_j) , \diam(T_k)\}\\ - \dist (T_j,T_k) &\geq \zeta_1 \max\{ \diam_{|\bs a}(T_j) , \diam_{|\bs a}(T_k)\}\\ - \dist (T_j, T_k)&\geq \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\}\\ -\end{align} -\begin{align} - \int_{T_j} \int_{T_k} &\approx \q_{T_j} \int_{T_k}\\ - \int_{T_j} \int_{T_k} &\approx \q_{I_1} \q_{J_1} \int_{I_2}\int_{J_2}\\ - \int_{T_j} \int_{T_k} &\approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2}\\ - \int_{T_j} \int_{T_k} &\approx \q_{T_j} \q_{J_1} \int_{J_2} -\end{align} - -\subsection{Glatter Kern} +% \todo{\subsection{---} +% \begin{align} +% \dist (T_j, T_k)&\geq \zeta \min\{ \diam(T_j) , \diam(T_k)\}\\ +% \dist (T_j,T_k) &\geq \zeta_1 \max\{ \diam_{|\bs a}(T_j) , \diam_{|\bs a}(T_k)\}\\ +% \dist (T_j, T_k)&\geq \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\}\\ +% \end{align} +% \begin{align} +% \int_{T_j} \int_{T_k} &\approx \q_{T_j} \int_{T_k}\\ +% \int_{T_j} \int_{T_k} &\approx \q_{I_1} \q_{J_1} \int_{I_2}\int_{J_2}\\ +% \int_{T_j} \int_{T_k} &\approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2}\\ +% \int_{T_j} \int_{T_k} &\approx \q_{T_j} \q_{J_1} \int_{J_2} +% \end{align}} +\subsection{voll analytische Berechnung des Doppelintegrals} +\subsubsection{Glatter Kern} % Das Integral über $T_j$ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über $T_k$ wird nun mittels einer doppelten Gauss-Quadratur berechnet. \begin{defi} @@ -484,7 +483,7 @@ Dann gilt für alle $k\in \N_0$ \end{align} \end{lem} \hfill$\square$ -\subsection{volle Quadratur} +\subsubsection{volle Quadratur} \begin{defi} Seien $T_j,T_k$ 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)\}. @@ -595,7 +594,7 @@ womit der Beweis abgeschlossen ist. % Da $\bs y$ und $x_{\bs b}$ beliebig war und die rechte Seite unabhängig von $\bs y,x_{\bs b}$ ist, folgt die Behauptung für das Supremum. % \hfill$\square$ -\subsection{Matrix} +\subsubsection{Matrix} \begin{sat} Seien $Tj,Tk$ zwei $\zeta_1$-zulässige affine Randstücke mit zugehörigen Parametrisierungen $\gamma_j,\gamma_k$ und $\diam(T_j) \leq \diam(T_k)$. 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} -- 2.47.3