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zSp(gF}6!Yk6A51Oi@lchLRq}24Z^&txmTu1#Euf_qY~{ zc-1@hkkrD7AuOILTHJ%Xii}fuB)qyUPhhmoz_)ZQi5`xNef)(o^-IK37w3R6sa66r zg|36)g^pos*FsY@IA=~XcOR^+(s&q_GBtT)wCnR^cA}iE4E;n0`rQv+77sD6t|Gy``eZXH4e&O2X=dZo|{NYmv3^*tX4(`13!LL1Z F@IN+sb?yKF diff --git a/doc/doc.tex b/doc/doc.tex index 10530db..db41a91 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -811,7 +811,7 @@ Mithilfe von Satz \ref{thm:sem:pol:V} für den Grad $2p+1$ erhalten wir die Beha & = 8 e \frac{c_1}{(c_2 \zeta_Q)^s} \max\{ \diam(T_j) , \diam(T_k)\}^{2-s} \left( 1 + \frac{2}{c_2\zeta_Q} \right)\sqrt{\abs{T_j}\abs{T_k}} , \end{align*} - welche unabhängig von der im Netz auftretenden Distanzen ist, können wir also eine gute Aussage über die Stabilität der Zulässigkeitsbedingung im Zuge der Netzverfeinerung treffen, sofern $s<2$. Für das Modellproblem \eqref{math:gal:kap+} gilt beispielsweise $s=1$. + welche unabhängig von der im Netz auftretenden Distanzen ist, können wir also eine gute Aussage über die Stabilität der Zulässigkeitsbedingung im Zuge der Netzverfeinerung treffen, sofern $s\leq2$. Für das Modellproblem \eqref{math:gal:kap+} gilt beispielsweise $s=1$. \end{beweis} @@ -853,10 +853,10 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \end{align*} 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 8 e\frac{c_1(c_2\zeta_Q+2)}{(c_2\zeta_Q)^{s+1}}\Lambda_{2p+1}^4 \frac{2(p+1)}{\left(1+c_2\zeta_Q\right)^{2(p+1)}} \max_{j,k=1,\ldots,n} \frac{\abs{\T}}{\max\{\diam(T_j),\diam(T_k)\}^{s-2}}. + \norm{A-A_p}_F &\leq 8 e\frac{c_1(c_2\zeta_Q+2)}{(c_2\zeta_Q)^{s+1}}\Lambda_{2p+1}^4 \frac{2(p+1)}{\left(1+c_2\zeta_Q\right)^{2(p+1)}} \abs{\Omega} \max_{j,k=1,\ldots,n} {\max\{\diam(T_j),\diam(T_k)\}^{2-s}}. \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 mit $\tilde C_{\zeta_Q,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, können wir Satz \ref{thm:sem:quad:V} anwenden. Damit erhalten wir \begin{align*} @@ -868,7 +868,7 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \\ &= \left( 8 e \frac{c_1}{c_2^s\zeta_Q^s} \left(1+\frac{2}{c_2\zeta_Q} \right) \Lambda_{2p+1}^4 \frac{2(p+1)}{\left(1+c_2\zeta_Q\right)^{2(p+1)}}\right)^2 \sum_{j,k=1}^n \frac{\abs{T_j}\abs{T_k}}{\max\{\diam(T_j),\diam(T_k)\}^{2s-4}} \\ - &\leq \left( 8 e \frac{c_1(c_2\zeta_Q+2)}{(c_2\zeta_Q)^{s+1}} \Lambda_{2p+1}^4 \frac{2(p+1)}{\left(1+c_2\zeta_Q\right)^{2(p+1)}}\right)^2 \max_{j,k=1,\ldots,n} \frac{\abs{\T}^2}{\max\{\diam(T_j),\diam(T_k)\}^{2s-4}}. + &\leq \left( 8 e \frac{c_1(c_2\zeta_Q+2)}{(c_2\zeta_Q)^{s+1}} \Lambda_{2p+1}^4 \frac{2(p+1)}{\left(1+c_2\zeta_Q\right)^{2(p+1)}}\right)^2 \abs{\Omega}^2 \max_{j,k=1,\ldots,n} {\max\{\diam(T_j),\diam(T_k)\}^{4-2s}}. \end{align*} Durch Ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{beweis} @@ -952,6 +952,10 @@ 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} +Weiterhin gilt für die Konstante $\tilde C_{\zeta_E,j,k}$ +\begin{align} + \tilde C_{\zeta_E,j,k}&\leq 8 e \frac{c_1 }{(c_2 \zeta_E)^s} \min\{ \diam(T_j) , \diam(T_k)\}^{1-s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right)\sqrt{\max\{\abs{T_j},\abs{T_k}\}}. +\end{align} \end{sat} @@ -976,18 +980,19 @@ Mithilfe von Satz \ref{thm:sem:pol:E} für den Grad $2p+1$ erhalten wir die Beha &= \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*} + \noindent Da die Konstante $\tilde C_{\zeta_E,j,k}$ durch die Netzverfeinerung, aufgrund der Distanz sehr groß werden könnte, untersuchen wir sie an dieser Stelle noch einmal etwas genauer. Mithilfe der folgenden Abschätzung für $\zeta_E$-zulässige Rechtecke $T_j,T_k$ \begin{align*} \tilde C_{\zeta_E,j,k} & = 8 e \frac{c_1 \abs{T_j}}{(c_2 \dist(T_j,T_k))^s}\left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right)\\ - & \leq 8 e \frac{c_1 \abs{T_j}\abs{T_k}}{\abs{T_k}(c_2 \zeta_E \min\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right) + & \leq 8 e \frac{c_1 \diam(T_j)}{(c_2 \zeta_E \min\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right)\sqrt{\abs{T_j}} \\ - & \leq 8 e \frac{c_1 \max\{ \diam(T_j) , \diam(T_k)\}^2}{\abs{T_k}(c_2 \zeta_E \min\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right)\sqrt{\abs{T_j}\abs{T_k}}\\ - & = 8 e \frac{c_1}{(c_2 \zeta_E)^s} \min\{ \diam(T_j) , \diam(T_k)\}^{2-s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right)\sqrt{\abs{T_j}\abs{T_k}} + & \leq 8 e \frac{c_1 }{(c_2 \zeta_E)^s} \min\{ \diam(T_j) , \diam(T_k)\}^{1-s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right)\sqrt{\max\{\abs{T_j},\abs{T_k}\}}\\ +% & = 8 e \frac{c_1}{(c_2 \zeta_E)^s} \min\{ \diam(T_j) , \diam(T_k)\}^{2-s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right)\sqrt{\abs{T_j}\abs{T_k}} , \end{align*} - welche unabhängig von der im Netz auftretenden Distanzen ist, können wir also eine gute Aussage über die Stabilität der Zulässigkeitsbedingung im Zuge der Netzverfeinerung treffen, sofern $s<4$. Für das Modellproblem \eqref{math:gal:kap+} gilt beispielsweise $s=1$. + welche unabhängig von der im Netz auftretenden Distanzen ist, können wir also eine gute Aussage über die Stabilität der Zulässigkeitsbedingung im Zuge der Netzverfeinerung treffen, sofern $s\leq 1$. Für das Modellproblem \eqref{math:gal:kap+} gilt beispielsweise $s=1$. \end{beweis} @@ -1044,17 +1049,22 @@ weshalb die Behauptung folgt. \end{align*} und sei $A_p$ die $A$ approximierende Matrix gemäß Definition \ref{thm:sem:quad:AE}. 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)}. + \norm{A-A_p}_F \leq 8 e \frac{c_1(c_2 \zeta_E+\sqrt{2}) }{(c_2 \zeta_E)^{s+1}} \frac{\Lambda_{2p+1}^2 2(p+1)\sqrt{2 \abs{\Omega}n}}{\left(1+\sqrt 2 c_2\zeta_E\right)^{2(p+1)}} \max_{j,k=1,\ldots,n} {\min\{ \diam(T_j) , \diam(T_k)\}^{1-s}}. \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 mit $\tilde C_{\zeta_E,j,k} = 0$ 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. + &\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 \left( 8 e \frac{c_1 \sqrt{\max\{\abs{T_j},\abs{T_k}\}}}{c_2^s \zeta_E^s \min\{ \diam(T_j) , \diam(T_k)\}^{s-1}} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right) \Lambda_{2p+1}^2 \frac{2(p+1)}{\left(1+\sqrt 2 c_2\zeta_E\right)^{2(p+1)}}\right)^2 + \\ + &= \left( 8 e \frac{c_1 }{c_2^s \zeta_E^s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right) \Lambda_{2p+1}^2 \frac{2(p+1)}{\left(1+\sqrt 2 c_2\zeta_E\right)^{2(p+1)}}\right)^2 \sum_{j,k=1}^n \frac{{\max\{\abs{T_j},\abs{T_k}\}}}{\min\{ \diam(T_j) , \diam(T_k)\}^{2s-2}} + \\ + &\leq \left( 8 e \frac{c_1(c_2 \zeta_E+\sqrt{2}) }{(c_2 \zeta_E)^{s+1}} \Lambda_{2p+1}^2 \frac{2(p+1)}{\left(1+\sqrt 2 c_2\zeta_E\right)^{2(p+1)}}\right)^2 2\abs{\Omega}n \max_{j,k=1,\ldots,n} {\min\{ \diam(T_j) , \diam(T_k)\}^{2-2s}}. \end{align*} Durch Ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{beweis} -- 2.47.3