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\norm{u^{(n)}}_{\infty,J} &\leq C_u \rho_u^nn!\text{~~~für alle} n \in \N_0. -\end{align} +\end{align*} Dann gilt für alle $k\in \N_0$ -\begin{align} -\min_{v \in \P_p} \norm{u-v}_{\infty,J} &\leq C_u 4e(1+\rho_u\abs{J})(p+1)\left(1+\frac{2}{\rho_u\abs{J}}\right)^{-(p+1)}. +\begin{align}\label{math:sem:ipol} +\min_{v \in \P_k} \norm{u-v}_{\infty,J} &\leq C_u 4e(1+\rho_u\abs{J})(k+1)\left(1+\frac{2}{\rho_u\abs{J}}\right)^{-(k+1)}. \end{align} \end{lem} \hfill$\square$ @@ -504,7 +535,7 @@ und können die Konstante $C_{\zeta,j,\kappa}$ dann schreiben als: \end{align} Sei $\bs y \in T_k$ fest gewählt. Ist $\gamma_{j | x_{\bs b}}$ die Parametrisierung von $T_{j| x_{\bs b}}$, so gilt aufgrund $\abs{\gamma_{j|x_{\bs b}}'}=\abs{a_j\bs a_j}=\diam_{|\bs a}(T_j)$ und der Kettenregel \begin{align*} -\abs{\partial_{x_{\bs a}}^n\kappa} +\abs{\partial_{\bs x}^{\alpha}\kappa} &= \abs{\partial_{x_{\bs a}}^n(\kappa(\gamma_{j | x_{\bs b}}(x_{\bs a}),\bs y))} = (\diam_{|\bs a}(T_j))^n\abs{(\partial_{x_{\bs a}}^n\kappa)(\gamma_{j | x_{\bs b}}(x_{\bs a}),\bs y)}. \end{align*} @@ -712,7 +743,7 @@ Dann gilt nach \cite{mai:3dbem} &= \int_{T_j} \int_{T_k} \left ( ((x_1 + v_1)-\tilde v_1)^2+((x_2 + v_2)-(y_2 +\tilde v_2))^2+( v_3- (y_3 +\tilde v_3))^2 \right)^{-1/2} ds_{\bs y} ds_{\bs x}\\ - &= \int_0^{s_1}\int_0^{s_2}\int_0^{\tilde s_2}\int_0^{\tilde s_3} + &= \int_0^{a}\int_0^{b}\int_0^{\tilde b}\int_0^{\tilde a} \left ( (x_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+(y_3-\delta_3)^2 \right)^{-1/2} dy_3 dy_2 dx_2 dx_1 \end{align*} @@ -738,21 +769,21 @@ Dann gilt nach \cite{mai:3dbem} \todo{ \subsection{Bestimmtes Integral} \begin{align*} -& \int_T \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ + \int_T& \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ &\approx \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1}\int_0^{\tilde k_2} \dif{}{y_2} \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} F_{p/o}(x_1,x_2,y_1,y_2) dy_2 dy_1 dx_2 dx_1\\ % -&= \big( \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1} +=& \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1} \big( \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} - F_{p/o}(x_1,x_2,y_1,\tilde k_2) -- + F_{p/o}(x_1,x_2,y_1,\tilde k_2) \\ +&- \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} F_{p/o}(x_1,x_2,y_1,0) \big) dy_1 dx_2 dx_1\\ % -&= \big( \int_0^{k_1}\int_0^{k_2} +=& \int_0^{k_1}\int_0^{k_2}\big( \dif{}{x_2} \dif{}{x_1} F_{p/o}(x_1,x_2,\tilde k_1,\tilde k_2) - @@ -766,7 +797,7 @@ dy_2 dy_1 dx_2 dx_1\\ F_{p/o}(x_1,x_2,0,0)\big) dx_2 dx_1\\ % -&= \big( \int_0^{k_1} +=& \int_0^{k_1}\big( \dif{}{x_1} F_{p/o}(x_1,k_2,\tilde k_1,\tilde k_2) - @@ -792,7 +823,7 @@ dy_2 dy_1 dx_2 dx_1\\ F_{p/o}(x_1,0,0,0)\big) dx_1\\ % -&= \big( +=& F_{p/o}(k_1,k_2,\tilde k_1,\tilde k_2) - F_{p/o}(k_1,k_2,\tilde k_1,0) @@ -823,7 +854,7 @@ dy_2 dy_1 dx_2 dx_1\\ + F_{p/o}(0,0,0,\tilde k_2) - - F_{p/o}(0,0,0,0)\big) + F_{p/o}(0,0,0,0) \end{align*} } -- 2.47.3

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