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Abschließend werden wir kurz die numerische Umsetzung der Techniken vorstellen \subsection{Netze} Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. Zunächst wollen wir ein achsenorientiertes Rechteck beschreiben, dessen Seiten parallel zu den Achsen des kartesischen Koordinatensystems liegen. -\begin{defi}\label{math:def:T} Sei $\bs v \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $a, b \in \R$ mit $a, b > 0$. Dann heißt +\begin{defi}\label{thm:def:T} Sei $\bs v \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $a, b \in \R$ mit $a, b > 0$. Dann heißt \begin{align*} T := \{\bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\} \end{align*} @@ -299,7 +299,7 @@ die zu $T$ zugehörige Parametrisierung. \label{fig:net:single} \end{figure} Des Weiteren werden wir für die Berechnungen noch Aussagen über die Größe eines Elements, sowie über den Abstand zweier Elemente festhalten. -\begin{defi}Sei $a,b\in\R$ für $T$ definiert wie in Definition~\ref{math:def:T}, dann heißt +\begin{defi}\label{thm:def:diam}Sei $a,b\in\R$ für $T$ definiert wie in Definition~\ref{thm:def:T}, dann heißt \begin{align*} \diam (T) &= (a^2+b^2)^{1/2} \end{align*} @@ -309,7 +309,7 @@ Durchmesser von T. Weiterhin nennen wir \end{align*} die Seitenlänge des Rechtecks $T$ in Richtung $\bs a$. Die Seitenlänge in Richtung $\bs b$ definieren wir analog. \end{defi} -\begin{defi}Der Abstand zweier Rechtecke $T_j$ und $T_k$ sei definiert durch +\begin{defi}\label{thm:def:dist}Der Abstand zweier Rechtecke $T_j$ und $T_k$ sei definiert durch \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*} @@ -535,7 +535,7 @@ Weiterhin wollen wir uns kurz die Ableitung der asymptotisch glatten Kernfunktio \begin{align} g(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)), \end{align} - wobei $\gamma_j,\gamma_k$ die Parametrisierungen aus Definition \ref{math:def:T} seien. + wobei $\gamma_j,\gamma_k$ die Parametrisierungen aus Definition \ref{thm:def:T} seien. Mit der Hilfsfunktion $t_{jk} : \N^4 \rightarrow \N^6$ \begin{align} t_{jk}(\alpha) = @@ -664,7 +664,7 @@ Bezeichne $g_{jk} : [0,1]^4 \rightarrow (T_j \times T_k) \subset \R^6$ die Param \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} + mit der jeweiligen Parametrisierung $\gamma_j, \gamma_k$ zu $T_j, T_k$ aus Definition \ref{thm: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}\kappa(\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4))} @@ -762,7 +762,7 @@ Mithilfe von Satz \ref{thm:sem:pol:V} für den Grad $2p+1$ erhalten wir die Beha \noindent Im Folgenden wollen wir nun annehmen, dass $T_1,T_2,\ldots,T_n$ Randelemente sind. Dann betrachten wir die Matrizen $A, A_p$, wobei die Einträge von $A$ genau durch \eqref{math:gal:kap+} gegeben sind und $A_p$ eine $A$ approximierende Matrix ist. Bei der Definition der approximierenden Matrix unterscheiden wir hier zwischen zwei Fällen. -\begin{defi} \label{thm:sem:quad:Ap} +\begin{defi} \label{thm:sem:quad:AV} Seien $T_1,T_2,\ldots,T_n$ Rechtecke und $A \in \R^{n \times n}$ gegeben durch \eqref{math:gal:kap}. Dann ist die $A$ approximierende Matrix $A_p$ folgendermaßen definiert. \begin{itemize} \item Sind $T_j$ und $T_k$ $\zeta_Q$-unzulässig, so ist @@ -784,13 +784,13 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \end{align*} -\begin{sat} +\begin{sat}\label{thm:sem:quad:EAV} 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 + und sei $A_p$ die $A$ approximierende Matrix gemäß Definition \ref{thm:sem:quad:AV}. 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 2(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}. \end{align*} @@ -817,6 +817,181 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{itemize} } + +\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(\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} + + +\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{thm: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} + + + +\begin{sat}\label{thm:sem:quad:E} +Seien $T_j,T_k$ zwei $\zeta_E$-zulässige Rechtecke, wobei $\diam(T_j) \leq \diam(T_k)$ sei , 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:zetaE:c} + \tilde C_{\zeta_E,j,k}&:=2^3e\frac{c_1\abs{T_j}}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\sqrt 2}{c_2\zeta_E} \right) +\end{align} +Dann gilt 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} + \int_0^1 \int_0^1 \int_{T_k} \kappa(\gamma_j(\lambda_1,\lambda_2),\bs y) d{\bs y} d{\lambda_2} d{ \lambda_1} +\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}\sum_{a=0}^p w_a \sum_{b=0}^p w_b \int_{T_k} \kappa(\gamma_j( \lambda_{a},\lambda_{b}),\bs y)) d{\bs y} +\end{align*} +die Abschätzung +\begin{align} + \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_E,j,k}\Lambda^2_{2p+1} 2(p+1)\left(1+ \sqrt 2 c_2\zeta_E\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 beginnend mit 0 statt 1. Deshalb gilt mit $\bs \lambda = (\lambda_1,\lambda_2)$ +\begin{align*} + (A_p)_{jk} + &=\abs{T_j}\sum_{a=0}^p w_a \sum_{b=0}^p w_b \int_{T_k} \kappa(\gamma_j( \lambda_{a},\lambda_{b}),\bs y)) d{\bs y}\\ + &=\abs{T_j}\int_0^1 \int_0^1 \int_{T_k} \I_{2p+1} \kappa(\gamma_j( \lambda_{1},\lambda_{2}),\bs y)) d{\bs y} d{\lambda_2} d{\lambda_1}\\ + &=\abs{T_j}\int_{[0,1]^2}\int_{T_k} \I_{2p+1} \kappa(\gamma_j(\bs \lambda),\bs y)) d{\bs y} d{\bs \lambda}, +\end{align*} +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} \Big|\int_{[0,1]^2}\int_{T_k}\kappa(\gamma_j(\bs \lambda),\bs y) d{\bs y} d{\bs \lambda_j} - \int_{[0,1]^2}\int_{T_k} \I_{2p+1} \kappa(\gamma_j(\bs \lambda),\bs y) d{\bs y} d{\bs \lambda}\Big|\\ + &\leq \abs{T_j}\int_{[0,1]^2}\int_{T_k}\Big|\kappa(\gamma_j(\bs \lambda),\bs y) - \I_{2p+1} \kappa(\gamma_j(\bs \lambda),\bs y) \Big|d{\bs y} d{\bs \lambda}\\ + &\leq \abs{T_j} \sup_{ \bs y \in T_k} \norm{\kappa(\gamma_j(\cdot,\cdot),\bs y)-\I_{2p+1}\kappa(\gamma_j(\cdot,\cdot),\bs y)}_{\infty,[0,1]^2} +\end{align*} +Mithilfe von Satz \ref{thm:sem:pol:E} für den Grad $2p+1$ erhalten wir die Behauptung +\begin{align*} + \abs{A_{jk} - (A_p)_{jk}} + &\leq \abs{T_j}C_{\zeta_E,j,k}\Lambda^2_{2p+1} 2(p+1)\left(1+ \sqrt 2 c_2\zeta_E\right)^{-2(p+1)}\\ + &= \tilde C_{\zeta_E,j,k}\Lambda^2_{2p+1} 2(p+1)\left(1+ \sqrt 2 c_2\zeta_E\right)^{-2(p+1)}. +\end{align*} +\end{beweis} + + +\begin{lem} \label{thm:sem:switch} + Seien $T_j,T_k$ zwei Randstücke mit $\dist(T_j,T_k) > 0$, $\kappa(\bs x,\bs y)$ eine asymptotisch glatte Kernfunktion mit + \begin{align*} + A_{jk} + &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}. + \end{align*} + Dann gilt + \begin{align*} + A_{jk} = A_{k_j}, + \end{align*} + insbesondere ist auch die Integrationsreihenfolge beliebig. +\end{lem} + +\begin{beweis} Aus der Definition \ref{thm:def:dist} für den Abstand zweier Rechtecke folgt $T_j \cap T_k = \emptyset$. +Da $\kappa(\bs x,\bs y)$ asymptotisch glatt ist, ist sie auch glatt auf $T_j\times T_k$. Da die Menge beschränkt und abgeschlossen ist, folgt die Kompaktheit aufgrund vom Satz von Heine-Borel. Weiterhin ist $\kappa$ als stetige Abbildung auf einem kompakten Intervall auf $T_j \times T_k$ integrierbar. Daher gilt der Satz von Fubini +\begin{align*} + \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x} = \int_{T_k} \int_{T_j} \kappa(\bs x,\bs y) ds_{\bs x} ds_{\bs y}, +\end{align*} +weshalb die Behauptung folgt. +\end{beweis} + + + +\begin{bem} + Seien $T_j,T_k$ zwei $\zeta_E$ zulässige Rechtecke. Ist $\diam(T_j) \leq \diam(T_k)$ so kann Satz (\ref{thm:sem:quad:E}) angewendet werden um $A_{jk}$ zu approximieren. Im Fall $\diam(T_j) > \diam(T_k)$ können wir, da für $\zeta_E$-zulässige Elemente auch $\dist(T_j,T_k)>0$ gilt, $A_{jk}$ approximieren indem wir mithilfe von $A_{jk} = A_{kj}$ und Satz (\ref{thm:sem:quad:E}), $A_{kj}$ berechnen. +\end{bem} + +\begin{defi} \label{thm:sem:quad:Ap} + Seien $T_1,T_2,\ldots,T_n$ Rechtecke und $A \in \R^{n \times n}$ gegeben durch \eqref{math:gal:kap}. Dann ist die $A$ approximierende Matrix $A_p$ folgendermaßen definiert. + \begin{itemize} + \item Sind $T_j$ und $T_k$ $\zeta_E$-unzulässig, so ist + \begin{align*} + (A_p)_{jk} := A_{jk}. + \end{align*} + \item Sind $T_j$ und $T_k$ $\zeta_E$-zulässig und $\diam(T_j) \leq \diam (T_k)$, so ist + \begin{align*} + (A_p)_{jk} := \abs{T_j}\sum_{a=0}^p w_a \sum_{b=0}^p w_b \int_{T_k} \kappa(\gamma_j( \lambda_{a},\lambda_{b}),\bs y)d\bs y, + \end{align*} + \item Sind $T_j$ und $T_k$ $\zeta_E$-zulässig und $\diam(T_j) > (T_k)$, so ist + \begin{align*} + (A_p)_{jk} := \abs{T_k}\sum_{c=0}^p w_c \sum_{d=0}^p w_d \int_{T_j} \kappa(\bs x,\gamma_k( \lambda_{c},\lambda_{d}))d\bs x. + \end{align*} + \end{itemize} +\end{defi} + +\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 $\tilde C_{\zeta_E,j,k}$ wie in \eqref{math:sem:zetaE:c} für $\zeta_E$-zulässsige Rechtecke $T_j,T_k$ und für unzulässige sei $\tilde C_{\zeta_E,j,k} = 0$ . 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_E,j,k}\Lambda_{2p+1}^2 2(p+1)\left(1+ \sqrt 2 c_2\zeta_E\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 $0$ und die Abschätzung für $\tilde C_{\zeta_E,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, unterscheiden wir zwei Fälle. Ist $\diam(T_j) \leq \diam (T_k)$, können wir Satz \ref{thm:sem:quad:E} anwenden. Andernfalls ist durch Lemma \ref{thm:sem:switch} $A_{jk}=A_{kj}$, worauf wir dann Satz \ref{thm:sem:quad:E} anwenden können und dadurch die selbe Abschätzung erhalten. Daraus folgt +\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_E,j,k}\Lambda_{2p+1}^2 2(p+1)\left(1+\sqrt 2 c_2\zeta_E\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_E,j,k}\Lambda_{2p+1}^2 2(p+1)\left(1+ \sqrt 2 c_2\zeta_E\right)^{-2(p+1)}\right)^2\\ + &= n^2 \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_E,j,k}\Lambda_{2p+1}^2 2(p+1)\left(1+ \sqrt 2 c_2\zeta_E\right)^{-2(p+1)}\right)^2. +\end{align*} +Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. +\end{beweis} + + + \subsection{Quadratur über eine Seite} \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 @@ -848,7 +1023,7 @@ und können dann die Konstante $C_{\zeta_S,j,k}$ kurz 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} +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{thm: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)}, @@ -866,72 +1041,63 @@ wobei $\gamma_j(\lambda_1,\lambda_2) \in T_j$ und $\gamma_k(\lambda_3,\lambda_4) \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*} +\begin{sat}\label{thm:sem:quad:S} +Seien $T_j,T_k$ zwei $\zeta_S$-zulässige Rechtecke, wobei $\diam(T_j) \leq \diam(T_k)$ sei , 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_S,j,k}&:=2^3e\frac{c_1\abs{T_j}}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{1}{c_2\zeta_S} \right) +\end{align} +Dann gilt 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} + \int_0^1 \int_0^1 \int_{T_k} \kappa(\gamma_j(\lambda_1,\lambda_2),\bs y) d{\bs y} d{\lambda_2} d{ \lambda_1} +\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}\sum_{a=0}^p w_a \int_0^1 \int_{T_k} \kappa(\gamma_j( \lambda_{a},\lambda_{2}),\bs y)) d{\bs y} d{\lambda_2} +\end{align*} die Abschätzung - \begin{align*} - \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*} +\begin{align} + \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_S,j,k}\Lambda_{2p+1} 2(p+1)\left(1+ 2 c_2\zeta_S\right)^{-2(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{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*} - \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)}, + (A_p)_{jk} + &=\abs{T_j}\sum_{a=0}^p w_a \int_0^1 \int_{T_k} \kappa(\gamma_j( \lambda_{a},\lambda_{2}),\bs y)) d{\bs y} d{\lambda_2}\\ + &=\abs{T_j}\int_0^1 \int_0^1 \int_{T_k} \I_{2p+1} \kappa(\gamma_j( \lambda_{1},\lambda_{2}),\bs y)) d{\bs y} d{\lambda_2} d{\lambda_1}, \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 +wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ bezeichnet. Wegen der Additivität des Integrals und durch Hineinziehen des Betrags erhält man mit $\bs \lambda = (\lambda_1,\lambda_2)$ \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}!, + \abs{A_{jk} -& (A_p)_{jk}}\\ + &=\abs{T_j} \Big|\int_{[0,1]^2}\int_{T_k}\kappa(\gamma_j(\bs \lambda),\bs y) d{\bs y} d{\bs \lambda} \quad- \int_{[0,1]^2}\int_{T_k} \I_{2p+1} \kappa(\gamma_j(\bs \lambda),\bs y) d{\bs y} d{\bs \lambda}\Big|\\ + &\leq \abs{T_j}\Big|\int_{[0,1]^2}\int_{T_k}\kappa(\gamma_j(\bs \lambda),\bs y) d{\bs y} d{\bs \lambda} \quad- \int_{[0,1]^2}\int_{T_k} \I_{2p+1} \kappa(\gamma_j(\bs \lambda),\bs y) d{\bs y} d{\bs \lambda}\Big|\\ + &\leq \abs{T_j} \sup_{\lambda_2 \in [0,1] \atop \bs y \in T_k} \norm{\kappa(\gamma_j(\cdot,\lambda_2),\bs y)-\I_{2p+1}\kappa(\gamma_j(\cdot,\lambda_2),\bs y)}_{\infty,[0,1]} \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 +Mithilfe von Satz \ref{thm:sem:pol:S} für den Grad $2p+1$ erhalten wir die Behauptung \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)}. + \abs{A_{jk} - (A_p)_{jk}} + &\leq \abs{T_j}C_{\zeta_S,j,k}\Lambda_{2p+1} 2(p+1)\left(1+2c_2\zeta_S\right)^{-2(p+1)}\\ + &= \tilde C_{\zeta_S,j,k}\Lambda_{2p+1} 2(p+1)\left(1+2c_2\zeta_S\right)^{-2(p+1)}. \end{align*} \end{beweis} +\begin{bem} + Seien $T_j,T_k$ zwei $\zeta_S$ zulässige Rechtecke. Ist $\diam(T_j) \leq \diam(T_k)$ so kann Satz (\ref{thm:sem:quad:S}) angewendet werden um $A_{jk}$ zu approximieren. Im Fall $\diam(T_j) > \diam(T_k)$ können wir, da für $\zeta_S$-zulässige Elemente auch $\dist(T_j,T_k)>0$ gilt, $A_{jk}$ approximieren indem wir mithilfe von $A_{jk} = A_{kj}$ und Satz (\ref{thm:sem:quad:S}), $A_{kj}$ berechnen. +\end{bem} + \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)\}. @@ -961,7 +1127,7 @@ und können dann die Konstante $C_{\zeta_A,j,k}$ kurz 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} +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{thm: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))},\\ @@ -1016,7 +1182,7 @@ und können dann die Konstante $C_{\zeta_D,j,k}$ kurz 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} +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{thm: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))},\\ @@ -1042,25 +1208,6 @@ wobei $\gamma_j(\lambda_1,\lambda_2) \in T_j$ sei. Daher sind die Voraussetzunge % \subsection{Quadratur über eine Achse} -% \begin{lem} \label{thm:sem:switch} -% Seien $T_j,T_k$ zwei Randstücke, $\kappa(\bs x,\bs y)$ eine asymptotisch glatte Kernfunktion und -% \begin{align} -% A_{jk} -% &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}. -% \end{align} -% Dann gilt -% \begin{align} -% A_{jk} = A_{k_j}. -% \end{align} -% \end{lem} -% \todo{\begin{beweis} -% \end{beweis}} -% -% \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} - - % % % \begin{defi} \label{thm:sem:quad:Ap} @@ -1258,7 +1405,7 @@ Bei der Berechnung von \eqref{math:analy:int} werden wir Aufgrund der speziellen % \subsubsection{Parallele Elemente} \noindent -Das heißt für parallele Elemente $T_j,T_k$ können wir o.B.d.A annehmen, dass mit Definition \ref{math:def:T} +Das heißt für parallele Elemente $T_j,T_k$ können wir o.B.d.A annehmen, dass mit Definition \ref{thm:def:T} \begin{align*} T_j &= \{\bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\}\\ T_k &= \{\tilde {\bs v} + \tilde \lambda_1 \tilde a \tilde {\bs a} + \tilde \lambda_2 \tilde b \tilde {\bs b} ~|~ \tilde \lambda_1,\tilde \lambda_2 \in[0,1]\} @@ -1294,7 +1441,7 @@ ${\bs a} = \tilde {\bs a}$ und ${\bs b} = \tilde{\bs b}$ gilt. Wir setzen $\bs{\ % \subsubsection{Orthogonale Elemente} \noindent -Analog können wir orthogonal liegende Elemente mit Definition \ref{math:def:T} schreiben als +Analog können wir orthogonal liegende Elemente mit Definition \ref{thm:def:T} schreiben als \begin{align*} T_j &= \{\bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\}\\ T_k &= \{\tilde {\bs v} + \tilde \lambda_2 \tilde b \tilde {\bs b} + \tilde \lambda_1 \tilde a \tilde {\bs a} ~|~ \tilde \lambda_1,\tilde \lambda_2 \in[0,1]\} @@ -1444,7 +1591,7 @@ Die Elemente $\T_{\ell}$ werden wir ebenfalls Zeilenweise in einer $N \times 4$ \begin{align} ELE[i,1:4] = T_i := (j,k,l,m). \end{align} -Die Knoten wollen wir gegen den Uhrzeigersinn anordnen und der Knoten $C_j$ sei der Punkt $\bs v$ aus Definition \ref{math:def:T}, also der kleinste Bezüglich des Koordinatensystems. +Die Knoten wollen wir gegen den Uhrzeigersinn anordnen und der Knoten $C_j$ sei der Punkt $\bs v$ aus Definition \ref{thm:def:T}, also der kleinste Bezüglich des Koordinatensystems. \noindent Für die bessere Handhabung der Elemente beim Verfeinern der Partition, wollen wir auch die Nachbarschaftsrelationen geeignet abspeichern. Aufgrund der Netzstabilität wollen wir maximal zwei Nachbarn pro Kante eines Elements zulassen. -- 2.47.3