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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)\}. @@ -492,14 +492,14 @@ Dann gilt für alle $k\in \N_0$ Andernfalls heißt das Paar $(T_j,T_k)$ $\zeta_Q$-unzulässig. \end{defi} -\begin{sat} Seien $T_j,T_k$ zwei $\zeta_Q$-zulässige Rechtecke. Sei $\kappa$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit +\begin{sat} \label{thm:sem:pol:V} Seien $T_j,T_k$ zwei $\zeta_Q$-zulässige Rechtecke. Sei $\kappa$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit \begin{align*} - C_{\zeta_Q,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\zeta_Q}{c_2} \right) + C_{\zeta_Q,j,k}&:=2^3e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\zeta_Q}{c_2} \right) \end{align*} die Abschätzung \begin{align*} \norm{\kappa(\cdot,\cdot)-\I_k^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} - &\leq C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}. + &\leq C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}. \end{align*} \end{sat} \beweis Zunächst definieren wir die Konstanten @@ -540,16 +540,15 @@ Weiterhin folgt für $T_j \times T_k$ \end{align*} womit der Beweis abgeschlossen ist. \hfill$\square$ - -\subsubsection{Matrix} -\begin{sat} +% 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 \begin{align} - \tilde C_{\zeta,j,\kappa} + D_{j,k} &= \diam_{\bs a}(T_j)\diam_{\bs b}(T_j)\diam_{\bs a}(T_k)\diam_{\bs b}(T_k) \end{align} und \begin{align} - D_{j,k} &= \diam_{\bs a}(T_j)\diam_{\bs b}(T_j)\diam_{\bs a}(T_k)\diam_{\bs b}(T_k). + \tilde C_{\zeta_Q,j,k}&:=2^4e\frac{c_1D_{j,k}}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\zeta_Q}{c_2} \right) \end{align} Dann gilt für das Integral \begin{align} @@ -564,7 +563,7 @@ 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}}&= \tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_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: @@ -578,14 +577,17 @@ wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ in $x_{\b \abs{A_{jk} &- (A_p)_{jk}}\\ &=D_{j,k} \Abs{\kappa(\gamma_j(x),\gamma_k(y)) - \int_{[0,1]^2}\int_{[0,1]^2} \I_{2p+1}^2\I_{2p+1}^2 \kappa(\gamma_j(x),\gamma_k(y)) dx dy}\\ &\leq D_{j,k}\int_{[0,1]^2}\int_{[0,1]^2} \Abs{\kappa(\gamma_j(x),\gamma_k(y)) - \I_{2p+1}^2\I_{2p+1}^2 \kappa(\gamma_j(x),\gamma_k(y))} dx dy\\ - &=\why{\sup_{\bs y \in T_k,x_b} \norm{\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)-\I_{2p+1}\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)}_{\infty,[0,1]}} + &\leq D_{j,k} \norm{\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)-\I_{2p+1}^4\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)}_{\infty,[0,1]^4} +\end{align*} +Mithilfe von Lemma \ref{thm:sem:pol:V} erhalten wir die Behauptung +\begin{align*} + \abs{A_{jk} - (A_p)_{jk}} + &\leq D_{j,k} C_{\zeta_Q,j,k}\Lambda_k^42(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}\\ + &= \tilde C_{\zeta_Q,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2(p+1)}. \end{align*} - \todo{\hfill$\square$\\} - - -\begin{lem} +\begin{lem} \label{thm:sem:switch} Seien $Tj,T_k$ zwei Randstücke, $\kappa$ eine asymptotisch glatte Kernfunktion und \begin{align} A_{jk} @@ -595,10 +597,15 @@ wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ in $x_{\b \begin{align} A_{jk} = A_{k_j}. \end{align} +\end{lem} \todo{\beweis \hfill$\square$} -\end{lem} +\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. +\end{bem} +\subsection{Approximierende Matrix} + \subsection{Quadratur über eine Achse} \subsubsection{Quadratur} -- 2.47.3