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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 - \begin{align}\label{math:sem:zetaS} - \dist (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 $\diam_{\bs a}(T_j) \leq \diam_{\bs a}(T_k)$. 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_S,j,k}&:=8e\frac{c_1}{(c_2\dist(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(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{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)}, -\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(T_j,T_k))^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ - =& \frac{c_1}{(c_2 \dist(T_j,T_k))^s} \left( \frac{\diam_{\bs a}(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 $\gamma_j(\lambda_1,\lambda_2) \in T_j$ 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_S,j,k}\Lambda_p(p+1)\left(1+2 c_2\zeta_S\right)^{-(p+1)}. -\end{align*} -\end{beweis} - - - -\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} - \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}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*} - (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*} -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*} - \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*} -Mithilfe von Satz \ref{thm:sem:pol:S} für den Grad $2p+1$ erhalten wir die Behauptung -\begin{align*} - \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} - -\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{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))},\\ - &\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:zetaD} - \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:D} 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(\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_{\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{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))},\\ - &\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 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 +% \begin{align}\label{math:sem:zetaS} +% \dist (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 $\diam_{\bs a}(T_j) \leq \diam_{\bs a}(T_k)$. 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_S,j,k}&:=8e\frac{c_1}{(c_2\dist(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(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{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)}, +% \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(T_j,T_k))^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ +% =& \frac{c_1}{(c_2 \dist(T_j,T_k))^s} \left( \frac{\diam_{\bs a}(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 $\gamma_j(\lambda_1,\lambda_2) \in T_j$ 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_S,j,k}\Lambda_p(p+1)\left(1+2 c_2\zeta_S\right)^{-(p+1)}. +% \end{align*} +% \end{beweis} +% +% +% +% \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} +% \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}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*} +% (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*} +% 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*} +% \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*} +% Mithilfe von Satz \ref{thm:sem:pol:S} für den Grad $2p+1$ erhalten wir die Behauptung +% \begin{align*} +% \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} +% +% \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{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))},\\ +% &\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:zetaD} +% \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:D} 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(\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_{\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{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))},\\ +% &\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} @@ -1370,7 +1370,7 @@ g(p;y;x;\lambda) &:= \int \{(x-y)^2 + \lambda^2 \}^p dy \end{align*} mit $\lambda = 0$ gilt die Formel mit beliebigen Skalaren $x,y,p \in \R$ \begin{align*} -(2p+1)g(p;y;x;0) = \begin{cases} \sgn(y-x) \log \abs{y-x}^{2p} & p=-1/2 \\(y-x) \abs{y-x}^{2p} & sonst \end{cases} +(2p+1)g(p;y;x;0) = \begin{cases} \sgn(y-x) \log \abs{y-x}^{2p} & p=-1/2 \\(y-x) \abs{y-x}^{2p} & \text{sonst} \end{cases} \end{align*} falls $x$ für $p\leq-1/2$ außerhalb des betrachteten Integrationsbereichs liegt. Für beliebige $x,p,y, \lambda \in \R$ mit $\lambda \neq 0$ gilt die Rekursionsformel @@ -1646,7 +1646,7 @@ Damit jedem Element $T_i$ eine Teilungsart zugeordnet werden kann, haben wir ein Relevant zum Verfeinern eines Netzes sind also die Koordinaten $COO$, Elemente $ELE$ sowie die Nachbarschaftsrelationen $NEI$ und der Markierungs\-vektor $marked$. Da wir später den Fehlerschätzer berechnen wollen, ist es wichtig sich zu jedem Element seine Teilelemente zu merken. Dazu legen wir während der Teilung eine $M \times 4$ Matrix an, in der die maximal vier Elementindizes gespeichert werden. Wenn wir also ein Element in vier gleich große Teile verfeinern, so wird das neue Element links unten das erste sein und alle weiteren folgen gegen den Uhrzeigersinn. Teilen wir ein Element in zwei gleich große Elemente, so werden die doppelt belegten Quadranten auch doppelt eingetragen. Ein gar nicht geteiltes Element wird also vier mal den alten Indizes speichern. Dadurch wird sichergestellt, dass das arithmetische Mittel über die Elemente immer gültig auszuführen ist. -(Siehe Abbildung \ref{exmpl13:f2s}) +(Siehe Abbildung \ref{exmpl13}) \begin{align*} [COO_{fine}, ELE_{fine}, NEI_{fine}, F2S ] = refineQuad(COO, ELE, NEI, marked); \end{align*} @@ -1681,78 +1681,89 @@ In diesem Abschnitt definieren wir die a-posteriori Fehlerschätzer, die wir zur Wir verwenden dazu die $h-h/2$ Strategie aus \cite{fer:errbem}. Im Folgenden bezeichnen wir mit $\hat \T_{\ell}$ das Gitter welches entsteht, wenn das Gitter $\T_{\ell}$ uniform, also entlang aller Kanten geteilt wird. Weiterhin bezeichne $\phi$ die exakte Lösung des Galerkin-Verfahrens und $\phi_{\ell}$ die Lösung zum Gitter $\T_{\ell}$, sowie $\hat \phi_{\ell}$ die Lösung zum uniformen Gitter $\hat \T_{\ell}$. \begin{defi}Es bezeichne $\phi$ die Lösung von Formel \todo{ref}, $\phi_{\ell}$ die Galerkin-Lösung auf dem Gitter $\T_{\ell}$ und $\hat \phi_{\ell}$ die Lösung auf dem uniform verfeinerten Gitter $\hat \T_{\ell}$. Dann gilt, der Schätzer -\begin{align} +\begin{align*} \eta_{\ell} &:= \enorm{\hat \phi_{\ell} - \phi_{\ell}} -\end{align} +\end{align*} ist effizient -\begin{align} +\begin{align*} \eta_{\ell} &\leq \enorm{\phi - \phi_{\ell}} -\end{align} +\end{align*} und unter der Saturationsannahme -\begin{align} -\norm{\phi -\hat \phi_{\ell}} &\leq C_{sat} \cdot \norm{\phi - \phi_{\ell}} & 0 < C_{sat} < 1 -\end{align} +\begin{align*} +\norm{\phi -\hat \phi_{\ell}} &\leq C_{sat} \cdot \norm{\phi - \phi_{\ell}} \qquad \text{mit }0 < C_{sat} < 1 +\end{align*} zuverlässig -\begin{align} +\begin{align*} \enorm{\phi - \phi_{\ell}}&\leq \frac 1{\sqrt{1-C_{sat}}}\eta_{\ell}. -\end{align} - +\end{align*} \end{defi} -Der Schätzer ist berechenbar, liefert aber keine lokalen Beiträge und kann daher nicht unmittelbar verwendet werden, um einen adaptiven Algorithmus zu steuern. +Der Schätzer ist berechenbar, liefert aber keine lokalen Beiträge und kann daher nicht unmittelbar verwendet werden, um einen adaptiven Algorithmus zu steuern. Hierzu definieren wir den $\tilde \mu$ Schätzer. -\begin{sat}[A-posteriori Fehlerschätzer] Seien also: +\begin{sat}[A-posteriori Fehlerschätzer] Seien aus \cite[Theorem 3.2 und 3.4]{fer:errbem}: \begin{align} \eta_{\ell} &= \enorm{\hat \phi_{\ell} - \phi_{\ell}}\\ \tilde\eta_{\ell} &= \enorm{\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell}}\\ -\mu_{\ell} &= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \phi_{\ell})}_{L^2(\Gamma)}\\ -\tilde\mu_{\ell} &= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell})}_{L^2(\Gamma)} +\mu_{\ell} &= \norm{\varrho^{\ell/2}(\hat \phi_{\ell} - \phi_{\ell})}_{L^2(\Gamma)}\\ +\tilde\mu_{\ell} &= \norm{\varrho^{\ell/2}(\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell})}_{L^2(\Gamma)} \end{align} wobei $\Pi_{\ell}$ die $L_2$ Projektion auf $P^0(\T_{\ell})$ ist. \end{sat} -$\mu_{\ell}$ ist da $\mu_{\ell} \approx \eta_{\ell}$ gilt noch immer zuverlässig und effizient.\\ -Dann gilt auf isotropen Netzen: +Weiterhin ist $\mu_{\ell}$ noch immer zuverlässig und effizient, da $\mu_{\ell} \approx \eta_{\ell}$ gilt.\\ +Dadurch gilt auf isotropen Netzen: \begin{itemize} \item Schätzer sind equivalent\\ $\tilde \mu_{\ell} \leq \mu_{\ell} \leq \sqrt{2}C_3 \eta_{\ell}$ \item sie sind effizient \item sie sind unter Saturationsannahme auch zuverlässig \end{itemize} - -Siehe \cite[Theorem 3.2 \& 3.4]{fer:errbem}. - - - - -\begin{align*} -%\mu_{\ell}^2 &= \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ -%&=\sum_{T\in \T_{\ell}}\mu_{\ell}(T)^2\\ -\mu_{\ell}(T)^2 &= \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ -&= h_{min}(T)\norm{\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ -&& T_j \in \tau_{\ell}, t_j^{(1)},\dots,t_j^{(4)} \in \tau_{\frac l 2} \\ -\phi_{\frac l 2}|_{T_j} &=x_j^{(1)}\chi_{t_j^{(1)}}+\dots+x_j^{(4)}\chi_{t_j^{(4)}}\\ -\Pi_{\ell}\phi_{\frac l 2}|_{T_j}&=\frac 1 {\abs{T_j}}\int_{T_j}\phi_{\frac l 2}d\Gamma\\ -&=\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\int_{t^{(k)}}1d\Gamma\\ -&=\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\abs{t_j^{(k)}}\\ -&=\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\frac {\abs{T_j}} 4 \\ -&=\frac 1 4 (x_j^{(1)} +\dots+x_j^{(4)} )=: m_j\\ -\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2(T_j)}^2 -&=\sum_{k=1}^4\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2\left(t_j^{(k)}\right)}^2\\ -&=\sum_{k=1}^4 \int_{t_j^{(k)}}(\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2})^2d\Gamma\\ -&=\sum_{k=1}^4 \int_{t_j^{(k)}}(x_j^{(k)}-m_j)^2d\Gamma\\ -&=\sum_{k=1}^4 \abs{t_j^{(k)}}(x_j^{(k)}-m_j)^2\\ -&=\frac {\abs{T_j}} 4\sum_{k=1}^4 (x_j^{(k)}-m_j)^2\\ -\mu_{\ell}(T_j)^2 &= \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} -\end{align*} - +% +% +% Siehe \cite[Theorem 3.2 \& 3.4]{fer:errbem}. +% +% +% +% +% \begin{align*} +% %\mu_{\ell}^2 &= \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ +% %&=\sum_{T\in \T_{\ell}}\mu_{\ell}(T)^2\\ +% \mu_{\ell}(T)^2 &= \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ +% &= h_{min}(T)\norm{\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ +% && T_j \in \tau_{\ell}, t_j^{(1)},\dots,t_j^{(4)} \in \tau_{\frac l 2} \\ +% \phi_{\frac l 2}|_{T_j} &=x_j^{(1)}\chi_{t_j^{(1)}}+\dots+x_j^{(4)}\chi_{t_j^{(4)}}\\ +% \Pi_{\ell}\phi_{\frac l 2}|_{T_j}&=\frac 1 {\abs{T_j}}\int_{T_j}\phi_{\frac l 2}d\Gamma\\ +% &=\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\int_{t^{(k)}}1d\Gamma\\ +% &=\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\abs{t_j^{(k)}}\\ +% &=\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\frac {\abs{T_j}} 4 \\ +% &=\frac 1 4 (x_j^{(1)} +\dots+x_j^{(4)} )=: m_j\\ +% \norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2(T_j)}^2 +% &=\sum_{k=1}^4\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2\left(t_j^{(k)}\right)}^2\\ +% &=\sum_{k=1}^4 \int_{t_j^{(k)}}(\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2})^2d\Gamma\\ +% &=\sum_{k=1}^4 \int_{t_j^{(k)}}(x_j^{(k)}-m_j)^2d\Gamma\\ +% &=\sum_{k=1}^4 \abs{t_j^{(k)}}(x_j^{(k)}-m_j)^2\\ +% &=\frac {\abs{T_j}} 4\sum_{k=1}^4 (x_j^{(k)}-m_j)^2\\ +% \mu_{\ell}(T_j)^2 &= \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} +% \end{align*} +% -\subsection{Markieren} \cite{dor:adapt} -Bestimme $M_{\ell} \subseteq T_{\ell}$ mit minimaler Kardinalität -\begin{align*} -\theta \sum_{T\in T_{\ell}} \mu_{\ell}(T)^2 &\leq \sum_{T\in M_{\ell}} \mu_{\ell}(T)^2 -\end{align*} -Zur anisotropen Verfeinerung wird weiterhin berechnet: +\subsection{Markieren} +Im Adaptiven Algorithmus werden wir die Elemente abhängig vom Fehlerschätzer $\tilde \mu$ verfeinern. Dazu wählen wir mithilfe der Dörfler-Markierung \cite{dor:adapt} eine Teilmenge aus. +\begin{defi}[Dörfler-Markierung] + Bestimme für feste Konstante $\theta \in (0,1)$ die Menge $M_{\ell} \subseteq T_{\ell}$ mit minimaler Kardinalität + \begin{align*} + \theta \sum_{T\in T_{\ell}} \tilde \mu_{\ell}(T)^2 &\leq \sum_{ T\in M_{\ell}} \tilde \mu_{\ell}(T)^2. + \end{align*} + Um die Symmetrie des Netzes zu erhalten bestimme weiterhin die kleinste Teilmenge $\tilde M_{\ell} \subseteq T_{\ell}$ für die gilt + \begin{align*} + \sum_{T\in M_{\ell}} \tilde \mu_{\ell}(T)^2 - \sum_{T\in \tilde M_{\ell}} \tilde \mu_{\ell}(T)^2 / \sum_{T\in M_{\ell}} \tilde \mu_{\ell}(T)^2 &> 10^{-2}. + \end{align*} +\end{defi} +Da wir im Adaptiven Algorithmus auch anisotrope Verfeinerungen zulassen werden definieren wir an dieser Stelle eine Auswahlstrategie zum bestimmen der Verfeinerungsart für jedes Element. +\begin{defi} +Seien $\phi_j^{(1)},\ldots, \phi_j^{(4)}$ die Lösungen der isotropen Verfeinerung $T_j^{(1)},\ldots, T_j^{(4)} \in \hat \T_{\ell}$ von $T_j \in T_{\ell}$, +% das heißt $T_j = \sum_{k=1,\ldots,4} T_j^{(k)}$, +dann sei $C_j$ zum Element $T_j $ definiert durch \begin{align*} \begin{pmatrix} C_j^{(1)}\\ C_j^{(2)}\\C_j^{(3)}\\ C_j^{(4)} @@ -1765,17 +1776,25 @@ C_j^{(1)}\\ C_j^{(2)}\\C_j^{(3)}\\ C_j^{(4)} 1 & -1 & -1 & 1\\ \end{pmatrix} \begin{pmatrix} -x_j^{(1)}\\ x_j^{(2)}\\x_j^{(3)}\\ x_j^{(4)} -\end{pmatrix} +\phi_j^{(1)}\\ \phi_j^{(2)}\\\phi_j^{(3)}\\ \phi_j^{(4)} +\end{pmatrix}. \end{align*} - +Dann ist mit fester Konstante $\nu \in (0,1)$ die Art der Markierung $marked_j$ zu einem Element $T_j \in \tilde M_{\ell}$ bestimmt durch \begin{align*} -\nu \abs{ C_j^{(3)}} &\geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(4)}}^2}\\ -\nu \abs{ C_j^{(4)}} &\geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(3)}}^2} +marked_j := + \begin{cases} % + 3 \qquad (\text{vertikal}) &\text{falls } \nu \abs{ C_j^{(3)}} \geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(4)}}^2}\\ % + 4 \qquad (\text{horizontal}) &\text{falls } \nu \abs{ C_j^{(4)}} \geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(3)}}^2}\\ % + 2 \qquad (\text{isotrop}) & \text{sonst.} + \end{cases} \end{align*} - -$xF2S := x_{fine}[F2S]$\\ -$marked = mark(xF2S, mu, theta, nu);$ +Weiterhin sei $marked_j = 1$ für alle $T_j \in \T_{\ell} \backslash \tilde M_{\ell}$. +\end{defi} +Die Funktion +\begin{align*} +marked = mark(xF2S, tmu, theta, nu); \qquad \text{mit } xF2S := x_{fine}[F2S] +\end{align*} +implementiert die Definitionen zum Bestimmen der Markierung. @@ -1795,55 +1814,54 @@ $marked = mark(xF2S, mu, theta, nu);$ \subsection{Adaptiver Algorithmus} Mithilfe der oben Definierten Funktionen ist es uns nun möglich den Ablauf der Berechnungen zusammen zu fassen. -\begin{alg}[Adaptives Verfahren] -$\theta \in (0,1),i =0$ +\begin{alg}[Adaptives Verfahren] Sei $\theta,\nu \in (0,1)$ fest gewählt und das Netz $\T_{\ell}$ gegeben. \begin{enumerate} \renewcommand{\theenumi}{(\roman{enumi})} -\item Verfeinere $T_{\ell}^{(i)}$ um $\hat T_{\ell}$ zu erhalten +\item Verfeinere $T_{\ell}$ um $\hat T_{\ell}$ zu erhalten \label{alg:adapt:begin} \item Berechne die Galerkinlösung $\hat \phi_{\ell} \in P^0(\hat T_{\ell})$ \item Berechne Fehlerschätzer $\tilde \mu_{i} := \norm{\varrho^{\ell/2}(\hat \phi_{\ell} - \Pi_{\ell} \hat \phi_{\ell} )}$ -\item Wähle $M_{\ell} \subseteq T_{\ell}^{(i)}$ mit minimaler Kardinalität, so dass -\begin{align} -\theta \sum_{T\in \T^{(i)}_{\ell}} \tilde\mu_{i}^2 & \leq \sum_{T\in M_{\ell}} \tilde\mu_{i}^2 -\end{align} -\item Verfeinere die Markierten Elemente $M_{\ell}$ um $\T_{\ell}^{(i+1)}$ zu erhalten -\item $i \mapsto i+1$, gehe zu $(i)$ +\item Wähle $M_{\ell} \subseteq T_{\ell}$ mit minimaler Kardinalität, so dass +\begin{align*} +\theta \sum_{T\in \T_{\ell}} \tilde\mu_{\ell}^2 & \leq \sum_{T\in M_{\ell}} \tilde\mu_{\ell}^2 +\end{align*} +\item Verfeinere mindestens die Markierten Elemente $M_{\ell}$ um $\T_{\ell+1}$ zu erhalten +\item $\ell \mapsto \ell+1$, gehe zu \ref{alg:adapt:begin} \end{enumerate} \end{alg} -Zum Plotten (Abb.\ref{fig:exmplAA_2DQuad})werden noch folgende Schritte ausgeführt -\begin{itemize} -\item Berechne Galerkinlösung $\phi_{l} \in P^0(\T_{\ell}^{(i)})$ -% \item $\enorm{\hat \phi_{\ell}}$ -% \item $\enorm{\phi_{\ell}}$ -\item $error_{i} = \sqrt{\enorm{\phi}^2 - \enorm{\phi_{l}}^2}$ -\item $\mu_{i} = \norm{\varrho^{1/2}(\hat \phi_{\ell} - \phi_{l} )}$ -\item $\eta_{i} = \enorm{\hat \phi_{\ell} - \phi_{l}}$ -\item $\kappa_{i} = \enorm{\hat \phi_{\ell}^{(i)}}-\enorm{\hat \phi_{\ell}^{(i-1)}}$ -\item $\kappa2_{i} = \enorm{\phi_{l}^{(i)}}-\enorm{\phi_{l}^{(i-1)}}$ -\item $\kappa3_{i} = \enorm{\hat \phi_{\ell}^{(i)}-\hat \phi_{\ell}^{(i-1)}}$ -\end{itemize} - -\todo{Viele Bunte Bilder und Testergebnisse} -\begin{itemize} -\item gerechnet wurde $V\phi = 1$ -\item Beispiele für verschiedene Figuren -\begin{itemize} -\item 2D LShape -\item 2D Rechteck -\item 3D Würfel -\item Fischer Würfel -\item L Figur -\end{itemize} -\item Beispiele für verschiedene Berechnungsarten -\begin{itemize} -\item VollAnalytisch -\item SemiAnalytisch über Element -\item SemiAnalytisch über Achse -\item SemiAnalytisch über Seite -\end{itemize} -\item Fehlerschätzer untersuchen und prüfen ob sie sich wie erwartet Verhalten -\end{itemize} +% Zum Plotten (Abb.\ref{fig:exmplAA_2DQuad})werden noch folgende Schritte ausgeführt +% \begin{itemize} +% \item Berechne Galerkinlösung $\phi_{\ell} \in P^0(\T_{\ell})$ +% % \item $\enorm{\hat \phi_{\ell}}$ +% % \item $\enorm{\phi_{\ell}}$ +% \item $error_{i} = \sqrt{\enorm{\phi}^2 - \enorm{\phi_{\ell}}^2}$ +% \item $\mu_{i} = \norm{\varrho^{\ell/2}(\hat \phi_{\ell} - \phi_{\ell} )}$ +% \item $\eta_{i} = \enorm{\hat \phi_{\ell} - \phi_{\ell}}$ +% \item $\kappa_{i} = \enorm{\hat \phi_{\ell}}-\enorm{\hat \phi_{\ell-1}}$ +% \item $\kappa2_{i} = \enorm{\phi_{\ell}}-\enorm{\phi_{\ell-1}}$ +% \item $\kappa3_{i} = \enorm{\hat \phi_{\ell}-\hat \phi_{\ell-1}}$ +% \end{itemize} +% +% \todo{Viele Bunte Bilder und Testergebnisse} +% \begin{itemize} +% \item gerechnet wurde $V\phi = 1$ +% \item Beispiele für verschiedene Figuren +% \begin{itemize} +% \item 2D LShape +% \item 2D Rechteck +% \item 3D Würfel +% \item Fischer Würfel +% \item L Figur +% \end{itemize} +% \item Beispiele für verschiedene Berechnungsarten +% \begin{itemize} +% \item VollAnalytisch +% \item SemiAnalytisch über Element +% \item SemiAnalytisch über Achse +% \item SemiAnalytisch über Seite +% \end{itemize} +% \item Fehlerschätzer untersuchen und prüfen ob sie sich wie erwartet Verhalten +% \end{itemize} \showMesh[Beispiel 1.1]{exmpl11} -- 2.47.3

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