Da wir später einen 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 sind. 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 sicher gestellt, dass das arithmetische Mittel über die Elemente immer gültig auszuführen ist.
(Siehe Figur:\ref{exmpl3:f2s})
+$[\T_{1/2}, F2S ] = refineQuad(\T, marked);$\\
$[COO_{fine}, ELE_{fine}, NEI_{fine}, F2S ] = refineQuad(COO, ELE, NEI, marked);$
\begin{figure}[ht]
\end{figure}
-
-
\section{Fehlerschätzer}
\begin{eqnarray*}
%\mu_h^2 & = & \norm{\varrho_h^{1/2}(\phi_{\frac h 2}-\Pi_h\phi_{\frac h 2})}_{\L^2(\T)}^2 \\
&=&\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_e(T_j)^2 & = & \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2}
+\mu_h(T_j)^2 & = & \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2}
\end{eqnarray*}
+$mu = computeEstSlpMuTilde(x_{fine}, \T, F2S);$\\
$mu = computeEstSlpMuTilde(x_{fine}, COO, ELE, F2S);$
\end{eqnarray*}
$xF2S := x_{fine}[F2S]$\\
-$marked = computeEstSlpMuTilde(xF2S, mu, theta, eta);$
+$marked = mark(xF2S, mu, theta, eta);$
\section{V Aufbauen}
-Es seien \Ea, \Eb ~$\subseteq$~ $\R^3$ zwei beschränkte, achsenorientierte rechteckige Seiten in $\R^3$.
-Berechnet werden soll:
+%Es seien \Ea, \Eb ~$\subseteq$~ $\R^3$ zwei beschränkte, achsenorientierte rechteckige Seiten in $\R^3$.
+%Berechnet werden soll:
\begin{eqnarray*}
--\frac{1}{4\pi} \int_E \int_{\tilde E} \frac{1}{|x-y|} ds_y ds_x \in \R^3
+V(j,k) &=& -\frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|x-y|} ds_y ds_x \in \R^3,T_j,T_k\in\T
\end{eqnarray*}
+$V = mex\_build\_AU(\T,mu,type)$\\
+$V = mex\_build\_AU(COO,ELE,mu,type)$
+
-$V = mex_build_AU(COO,ELE,mu,type)$
+\section{Vorgehensweise}
+Mithilfe der oben Definierten Funktionen ist es uns nun möglich den Ablauf der Verfeinerungen zusammen zu fassen.
+Sei das Netz $\T^{(0)}$ mit $\#{\T^{(0)}}$ Elementen gegeben und iterieren wir über $i \in \N$:
+
+\begin{enumerate}
+ \item $(\T^{(i)}_{1/2},f2s^{(i)}_{1/2}) = refineQuad(\T^{(i)},2)$
+ \item $V^{(i)}_{1/2} = mex\_build\_AU(\T^{(i)}_{1/2},$ vollAnalytisch $)$
+ \item $b^{(i)}_{1/2} = \left\{b_k=size(T_k) | T_k\in\T^{(i)}_{1/2}, k \in \{1,2 \dots\#\T^{(i)}_{1/2}\} \right\}$
+ \item Löse: $V^{(i)}_{1/2} \cdot \phi^{(i)}_{1/2} = b^{(i)}_{1/2}$
+ \item $\mu(\T^{(i)})^2 = \left\{\frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(\phi_{1/2,j}^{(i)(k)}-\frac 1 4\sum_{l=1}^4\phi_{1/2,j}^{(i)(l)})^2}, j \in \{1,2 \dots\#\T^{(i)}\} \right\}$\\$ = computeEstSlpMuTilde(\phi^{(i)}_{1/2},\T^{(i)},f2s^{(i)}_{1/2})$
+ \item $marked^{(i)} = mark(\phi^{(i)}_{1/2}(f2s^{(i)}_{1/2}),\mu^{(i)}(\T^{(i)})^2$,theta, eta$)$
+ \item $\T^{(i+1)} = refineQuad(\T^{(i)},marked)$
+\end{enumerate}
+Zum Plotten wird zusätzlich auch folgendes berechnet:
+\begin{itemize}
+ \item $V^{(i)} = mex\_build\_AU(\T^{(i)},$ vollAnalytisch $)$
+ \item $b^{(i)} = \left\{b_k=size(T_k) | T_k\in\T^{(i)}, k \in \{1,2 \dots\#\T^{(i)}\} \right\}$
+ \item Löse: $V^{(i)} \cdot \phi^{(i)} = b^{(i)}$
+ \item $\enorm{\phi^{(i)}_{1/2}} = \phi^{(i)\prime}_{1/2}\cdot V^{(i)}_{1/2} \cdot \phi^{(i)}_{1/2}$
+ \item $\enorm{\phi^{(i)}} = \phi^{(i)\prime}\cdot V^{(i)} \cdot \phi^{(i)}$
+ \item $error = \sqrt{\enorm{\phi} - \enorm{\phi^{(i)}}}$
+ \item $error_2 = \sqrt{\enorm{\phi} - \enorm{\phi^{(i)}_{1/2}}}$
+ \item $\eta = \sqrt{\enorm{\phi^{(i)}_{1/2}(f2s^{(i)}_{1/2}) - \phi^{(i)}}}$
+ \item $\mu_2(\T^{(i)})^2 = \left\{\frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(\phi_{1/2,j}^{(i)(k)}-\phi_{j}^{(i)})^2}, j \in \{1,2 \dots\#\T^{(i)}\} \right\}$
+\end{itemize}
+Siehe figur
+\begin{figure}[ht]
+\caption{VaterSohn}
+\centering
+\subfloat[Fehler aus]{\includegraphics[width=0.4\textwidth]{fig/Net_Neigh}}}
+\subfloat[Energie Norm]{{\includegraphics[width=0.4\textwidth]{fig/Net_Neigh}}}
+\end{figure}
\end{document}
\ No newline at end of file
leg1 = {};
sym = {};
-rows = 5;
+rows = 6;
for i = 1:length(files)
l0 = [files{i}(p1:p1+p2-2) ' '];
l1 = {type2str{data(1,[2+(0:step-1)*rows])}};
-% l2 ={};
-% l3 ={};
-% l4 ={};
for i = 1:step
- leg0 = {leg0{:} ['\mu ' l0 l1{i}] ['\eta ' l0 l1{i}] ['error ' l0 l1{i}] ['\mu2 ' l0 l1{i}]}';
+ leg0 = {leg0{:} ['\mu ' l0 l1{i}] ['\eta ' l0 l1{i}]...
+ ['error ' l0 l1{i}] ['\mu2 ' l0 l1{i}] ['error2 ' l0 l1{i}]}';
leg1 = {leg1{:} [ l0 l1{i}]}';
sym = {sym{:} type2sym{data(1,[2+(i-1)*rows])}}'
-% l2{i} = [ l0 l1{i}];
-% l3{i} = ['\mu ' l0 l1{i}];
-% l4{i} = ['\eta ' l0 l1{i}];
-% l5{i} = ['error ' l0 l1{i}];
end
-% leg0
-% leg1 = {leg1{:} l2{:}}'
-% leg2 = {leg2{:} l3{:}}'
-% leg3 = {leg3{:} l4{:}}'
-% leg4 = {leg4{:} l5{:}}'
-
+
end
[m n] = size(G_D);
disp ('Error: No Data to show.')
else
- data((end-8):end,[1 3 4 5])
+ data((end-8):end,[1 [3 4 5]])
% sol = interp1(1./X((3):(12),3)',G_D((1):(12),3)',0,'spline')
-% sol = 8.28466;
- sol = 4.609196
+% sol = 8.28466; % LShape
+ sol = 4.609196; % Quad
% G_D
figure(4)
i=0;
loglog(X(:,[2+(0:rows-2)]+rows*i),[G_D(:,2+rows*i) ...
- G_D(:,2+1+rows*i)*G_D(1,2)/G_D(1,2+1+rows*i) ...
- sqrt(abs(sol - G_D(:,2+2+rows*i)))*G_D(1,2)/sqrt(abs(sol - G_D(1,2+2+rows*i)))...
- G_D(:,2+3+rows*i)*G_D(1,2)/G_D(1,2+3+rows*i)] ...
- ,sym{i+1});
+ G_D(:,2+1+rows*i)...*G_D(1,2)/G_D(1,2+1+rows*i) ...
+ sqrt(abs(sol - G_D(:,2+2+rows*i)))...*G_D(1,2)/sqrt(abs(sol - G_D(1,2+2+rows*i)))...
+ G_D(:,2+3+rows*i)...*G_D(1,2)/G_D(1,2+3+rows*i) ...
+ G_D(:,2+4+rows*i)...*G_D(1,2)/G_D(1,2+3+rows*i) ...
+ ],sym{i+1});
hold on
for i = 1:step-1
loglog(X(:,[2+(0:rows-2)]+rows*i),[G_D(:,2+rows*i) ...
- G_D(:,2+1+rows*i)*G_D(1,2)/G_D(1,2+1+rows*i) ...
- sqrt(abs(sol - G_D(:,2+2+rows*i)))*G_D(1,2)/sqrt(abs(sol - G_D(1,2+2+rows*i)))...
- G_D(:,2+3+rows*i)*G_D(1,2)/G_D(1,2+3+rows*i)] ...
- ,sym{i+1});
+ G_D(:,2+1+rows*i)...*G_D(1,2)/G_D(1,2+1+rows*i) ...
+ sqrt(abs(sol - G_D(:,2+2+rows*i)))...*G_D(1,2)/sqrt(abs(sol - G_D(1,2+2+rows*i)))...
+ G_D(:,2+3+rows*i)...*G_D(1,2)/G_D(1,2+3+rows*i) ...
+ ],sym{i+1});
end
loglog(X(:,1),[7*X(:,1).^(-1/2),3*X(:,1).^(-1/4),2*X(:,1).^(-3/4)],'-.')
%Energienorm^2 Berechnen
xe_fine = x_fine'*A_fine*x_fine;
+ xe = x'*A*x;
% xe_fine = x'*A*x;
% sqrt(sum(ind))
- data = [data type(i) sqrt(sum(ind)) sqrt(ind2) xe_fine sqrt(sum(ind3))];
+ data = [data type(i) sqrt(sum(ind)) sqrt(ind2) xe_fine sqrt(sum(ind3)) sqrt(abs(xe-xe_fine))];
end
time(2) = toc;
%!PS-Adobe-2.0 EPSF-1.2
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+%%Creator: MATLAB, The MathWorks, Inc. Version 7.13.0.564 (R2011b). Operating System: Linux 3.0.0-16-generic #29-Ubuntu SMP Tue Feb 14 12:48:51 UTC 2012 x86_64.
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DD
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DD
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DO
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0 sg
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+3850 270 mt 3850 288 L
+3970 3205 mt 3970 3186 L
+3970 270 mt 3970 288 L
+4068 3205 mt 4068 3186 L
+4068 270 mt 4068 288 L
+4151 3205 mt 4151 3186 L
+4151 270 mt 4151 288 L
+4223 3205 mt 4223 3186 L
+4223 270 mt 4223 288 L
+4287 3205 mt 4287 3186 L
+4287 270 mt 4287 288 L
+4344 3205 mt 4344 3186 L
+4344 270 mt 4344 288 L
+4344 3205 mt 4344 3167 L
+4344 270 mt 4344 307 L
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-3354 3333 mt
+4283 3331 mt
(10) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 55.5556 FMSR
-3446 3282 mt
+4375 3280 mt
(3) s
-3695 3207 mt 3695 3189 L
-3695 269 mt 3695 288 L
-3859 3207 mt 3859 3189 L
-3859 269 mt 3859 288 L
-3975 3207 mt 3975 3189 L
-3975 269 mt 3975 288 L
-4065 3207 mt 4065 3189 L
-4065 269 mt 4065 288 L
-4139 3207 mt 4139 3189 L
-4139 269 mt 4139 288 L
-4201 3207 mt 4201 3189 L
-4201 269 mt 4201 288 L
-4255 3207 mt 4255 3189 L
-4255 269 mt 4255 288 L
-4303 3207 mt 4303 3189 L
-4303 269 mt 4303 288 L
-4346 3207 mt 4346 3189 L
-4346 269 mt 4346 288 L
-4346 3207 mt 4346 3170 L
-4346 269 mt 4346 307 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
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+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 2325 mt 661 2325 L
+4344 2325 mt 4306 2325 L
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-4285 3333 mt
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(10) s
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-4377 3282 mt
-(4) s
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 2328 mt 661 2328 L
-4346 2328 mt 4308 2328 L
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-
- 400 2358 mt
-(10) s
-%%IncludeResource: font Helvetica
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-
- 492 2307 mt
+ 492 2304 mt
(0.65) s
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 1024 mt 661 1024 L
-4346 1024 mt 4308 1024 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
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+ 624 3205 mt 624 3205 L
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+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
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+ 624 1022 mt 661 1022 L
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- 400 1054 mt
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(10) s
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- 492 1003 mt
+ 492 1001 mt
(0.66) s
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 624 3207 L
- 624 3207 mt 4346 3207 L
- 624 269 mt 4346 269 L
- 624 3207 mt 624 269 L
-4346 3207 mt 4346 269 L
-gs 624 269 3723 2939 MR c np
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 624 3205 L
+ 624 3205 mt 4344 3205 L
+ 624 270 mt 4344 270 L
+ 624 3205 mt 624 270 L
+4344 3205 mt 4344 270 L
+gs 624 270 3721 2936 MR c np
/c8 { 0.000000 0.000000 1.000000 sr} bdef
c8
-100 0 65 0 95 0 107 0 100 0 97 0 93 0 87 0
-72 -1 55 0 84 -1 55 -1 64 -1 60 -3 60 -3 62 -5
-45 -7 62 -10 48 -14 51 -19 71 -28 51 -38 90 -53 71 -82
-104 -164 122 -184 190 -333 254 -480 560 -1228 1184 3207 30 MP stroke
+112 0 73 -1 85 -2 80 -2 81 -3 81 -5 61 -8 83 -10
+63 -13 69 -20 94 -27 68 -38 120 -53 94 -82 138 -164 164 -185
+253 -332 338 -480 747 -1228 1370 3205 20 MP stroke
gr
c8
-gs 1133 501 3078 2758 MR c np
- 25 25 1184 3207 FO
- 25 25 1744 1979 FO
- 25 25 1998 1499 FO
- 25 25 2188 1166 FO
- 25 25 2310 982 FO
- 25 25 2414 818 FO
- 25 25 2485 736 FO
- 25 25 2575 683 FO
- 25 25 2626 645 FO
- 25 25 2697 617 FO
- 25 25 2748 598 FO
- 25 25 2796 584 FO
- 25 25 2858 574 FO
- 25 25 2903 567 FO
- 25 25 2965 562 FO
- 25 25 3025 559 FO
- 25 25 3085 556 FO
- 25 25 3149 555 FO
- 25 25 3204 554 FO
- 25 25 3288 553 FO
- 25 25 3343 553 FO
- 25 25 3415 552 FO
- 25 25 3502 552 FO
- 25 25 3595 552 FO
- 25 25 3692 552 FO
- 25 25 3792 552 FO
- 25 25 3899 552 FO
- 25 25 3994 552 FO
- 25 25 4059 552 FO
- 25 25 4159 552 FO
+gs 1319 501 2907 2756 MR c np
+ 25 25 1370 3205 FO
+ 25 25 2117 1977 FO
+ 25 25 2455 1497 FO
+ 25 25 2708 1165 FO
+ 25 25 2872 980 FO
+ 25 25 3010 816 FO
+ 25 25 3104 734 FO
+ 25 25 3224 681 FO
+ 25 25 3292 643 FO
+ 25 25 3386 616 FO
+ 25 25 3455 596 FO
+ 25 25 3518 583 FO
+ 25 25 3601 573 FO
+ 25 25 3662 565 FO
+ 25 25 3743 560 FO
+ 25 25 3824 557 FO
+ 25 25 3904 555 FO
+ 25 25 3989 553 FO
+ 25 25 4062 552 FO
+ 25 25 4174 552 FO
gr
-gs 624 269 3723 2939 MR c np
+gs 624 270 3721 2936 MR c np
DD
/c9 { 1.000000 0.000000 0.000000 sr} bdef
c9
-100 0 65 0 95 0 107 0 100 0 97 0 93 0 87 0
-72 0 55 0 84 0 55 0 64 0 60 0 60 0 62 0
-45 0 62 0 48 0 51 0 71 0 51 0 90 0 71 0
-104 0 122 0 190 0 254 0 560 0 1184 552 30 MP stroke
+112 0 73 0 85 0 80 0 81 0 81 0 61 0 83 0
+63 0 69 0 94 0 68 0 120 0 94 0 138 0 164 0
+253 0 338 0 747 0 1370 550 20 MP stroke
gr
c9
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-2226 205 mt
+2225 206 mt
(Energie Norm) s
-2308 3433 mt
+2307 3431 mt
(Elemente) s
- 343 1882 mt -90 rotate
+ 343 1881 mt -90 rotate
(eNorm) s
90 rotate
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/Helvetica /ISOLatin1Encoding 66.6667 FMSR
- 302 1630 mt -90 rotate
+ 302 1629 mt -90 rotate
(2) s
90 rotate
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- 612 3238 mt
+ 612 3236 mt
( ) s
-4335 299 mt
+4333 300 mt
( ) s
SO
1 sg
-0 232 1000 0 0 -232 3305 3167 4 MP
+0 232 1000 0 0 -232 3303 3165 4 MP
PP
--1000 0 0 232 1000 0 0 -232 3305 3167 5 MP stroke
+-1000 0 0 232 1000 0 0 -232 3303 3165 5 MP stroke
2.77778 w
DO
SO
4.16667 w
0 sg
-3305 3167 mt 4305 3167 L
-3305 2935 mt 4305 2935 L
-3305 3167 mt 3305 2935 L
-4305 3167 mt 4305 2935 L
-3305 3167 mt 4305 3167 L
-3305 3167 mt 3305 2935 L
-3305 3167 mt 4305 3167 L
-3305 2935 mt 4305 2935 L
-3305 3167 mt 3305 2935 L
-4305 3167 mt 4305 2935 L
-3628 3030 mt
+3303 3165 mt 4303 3165 L
+3303 2933 mt 4303 2933 L
+3303 3165 mt 3303 2933 L
+4303 3165 mt 4303 2933 L
+3303 3165 mt 4303 3165 L
+3303 3165 mt 3303 2933 L
+3303 3165 mt 4303 3165 L
+3303 2933 mt 4303 2933 L
+3303 3165 mt 3303 2933 L
+4303 3165 mt 4303 2933 L
+3615 3028 mt
(testAA Analytisch) s
-gs 3305 2935 1001 233 MR c np
+gs 3303 2933 1001 233 MR c np
c8
-248 0 3354 2999 2 MP stroke
-gs 3427 2948 103 103 MR c np
- 25 25 3478 2999 FO
+240 0 3350 2998 2 MP stroke
+gs 3419 2947 103 103 MR c np
+ 25 25 3470 2998 FO
gr
gr
c8
0 sg
-3628 3133 mt
+3615 3131 mt
(extrapoliert) s
-gs 3305 2935 1001 233 MR c np
+gs 3303 2933 1001 233 MR c np
DD
c9
-248 0 3354 3102 2 MP stroke
+240 0 3350 3099 2 MP stroke
SO
gr
% Test ausführen
%Anzahl der Schritte
-steps = 30;
+steps = 35;
%LShape adaptiv anisotrop
A_run('exmpl_2DQuad', steps, 0.7, [ 1], 0.5, 0.5, 'testAA_')
projects / bacc.git / commitdiff