\subsection{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 $i \in \N$:
+$\theta \in (0,1),i =0$
\begin{enumerate}
- \item $(\T^{(i)}_{\frac l 2},f2s^{(i)}_{\frac l 2}) = refineQuad(\T^{(i)},2)$
- \item $V^{(i)}_{\frac l 2} = mex\_build\_AU(\T^{(i)}_{\frac l 2},$ vollAnalytisch $)$
- \item $b^{(i)}_{\frac l 2} = \left\{b_k=size(T_k) | T_k\in\T^{(i)}_{\frac l 2}, k \in \{1,2 \dots\#\T^{(i)}_{\frac l 2}\} \right\}$
- \item Löse: $V^{(i)}_{\frac l 2} \cdot x^{(i)}_{\frac l 2} = b^{(i)}_{\frac l 2}$
- \item $\mu(\T^{(i)})^2 = \left\{\frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_{\frac l 2,j}^{(i)(k)}-\frac 1 4\sum_{l=1}^4x_{\frac l 2,j}^{(i)(l)})^2}, j \in \{1,2 \dots\#\T^{(i)}\} \right\}$\\$ = computeTstSlpMuTilde(\phi^{(i)}_{\frac l 2},\T^{(i)},f2s^{(i)}_{\frac l 2})$
- \item $marked^{(i)} = mark(x^{(i)}_{\frac l 2}(f2s^{(i)}_{\frac l 2}),\mu^{(i)}(\T^{(i)})^2$,theta, nu$)$
- \item $\T^{(i+1)} = refineQuad(\T^{(i)},marked)$
+ \renewcommand{\theenumi}{(\roman{enumi})}
+ \item Verfeinere $T_l^{(i)}$ um $T_{l/2}^{(i)}$ zu erhalten
+ \item Berechne die Galerkinlösung $\phi_{l/2}^{(i)} \in P^0(\T_{l/2}^{(i)})$
+ \item Berechne Fehlerschätzer $\tilde \mu_{l,i} := \norm{\varrho^{1/2}(\phi_{l/2}^{(i)} - \Pi_l \phi_{l/2}^{(i)} )}$
+ \item Wähle $M_l^{(i)} \subseteq T_l^{(i)}$ mit minimaler Kardinalität, so dass
+\begin{align}
+ \theta \sum_{T\in \T^{(i)}_l} \tilde\mu_{l,i}^2 & \leq \sum_{T\in M^{(i)}_l} \tilde\mu_{l,i}^2
+\end{align}
+ \item Verfeinere die Markierten Elemente $M_l^{(i)}$ um $\T_l^{(i+1)}$ zu erhalten
+ \item $i \mapsto i+1$, gehe zu $(i)$
\end{enumerate}
-Zum Plotten (\ref{exmplAA_2DQuad}) wird zusätzlich auch folgendes berechnet:
+
+Zum Plotten (\ref{exmplAA_2DQuad})werden noch folgende Schritte ausgeführt
\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: $V^{(i)} \cdot x^{(i)} = b^{(i)}$
- \item $\enorm{\phi^{(i)}_{\frac l 2}}^2 = x^{(i)\prime}_{\frac l 2}\cdot V^{(i)}_{\frac l 2} \cdot x^{(i)}_{\frac l 2}$
- \item $\enorm{\phi^{(i)}}^2 = x^{(i)\prime}\cdot V^{(i)} \cdot x^{(i)}$
- \item $error = \sqrt{\enorm{\phi}^2 - \enorm{\phi^{(i)}}^2}$
-% \item $error_2 = \sqrt{\enorm{\phi} - \enorm{\phi^{(i)}_{\frac l 2}}}$
- \item $\eta = \sqrt{\enorm{x^{(i)}_{\frac l 2}(f2s^{(i)}_{\frac l 2})}^2 - \enorm{\phi^{(i)}}^2}$
-% \item $error_2 = \enorm{\phi^{(i)}_{\frac l 2}(f2s^{(i)}_{\frac l 2}) - \phi^{(i)}}$
- \item $\mu_2(\T^{(i)})^2 = \left\{\frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_{\frac l 2,j}^{(i)(k)}-x_{j}^{(i)})^2}, j \in \{1,2 \dots\#\T^{(i)}\} \right\}$
+ \item Berechne Galerkinlösung $\phi_{l}^{(i)} \in P^0(\T_l^{(i)})$
+% \item $\enorm{\phi^{(i)}_{l/2}}$
+% \item $\enorm{\phi^{(i)}_l}$
+ \item $error = \sqrt{\enorm{\phi}^2 - \enorm{\phi_l^{(i)}}^2}$
+ \item $\mu_{l,i} := \norm{\varrho^{1/2}(\phi_{l/2}^{(i)} - \phi_{l}^{(i)} )}$
+ \item $\eta_{l,i} = \enorm{\phi_{l/2}^{(i)} - \phi_{l}^{(i)}}$
+
\end{itemize}
\begin{figure}[ht]
-\caption{2D Quad adaptiv anisotrop vollanalytisch}
+\caption{2D Quad adaptiv anisotrop vollanalytisch $V\phi = 1$}
\centering
\label{exmplAA_2DQuad}
\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/exmplAA_2DQuad_error}}
%!PS-Adobe-2.0 EPSF-1.2
%%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.
%%Title: ./exmplAA_2DQuad_error.eps
-%%CreationDate: 03/26/2012 08:17:51
+%%CreationDate: 03/26/2012 10:22:43
%%DocumentNeededFonts: Helvetica
%%DocumentProcessColors: Cyan Magenta Yellow Black
%%Extensions: CMYK
( ) s
SO
1 sg
-0 854 1199 0 0 -854 665 3165 4 MP
+0 854 1238 0 0 -854 665 3165 4 MP
PP
--1199 0 0 854 1199 0 0 -854 665 3165 5 MP stroke
+-1238 0 0 854 1238 0 0 -854 665 3165 5 MP stroke
2.77778 w
DO
SO
4.16667 w
0 sg
- 665 3165 mt 1864 3165 L
+ 665 3165 mt 1903 3165 L
665 3165 mt 665 2311 L
+ 960 2406 mt
+(tilde ) s
%%IncludeResource: font Symbol
/Symbol /ISOLatin1Encoding 83.3333 FMSR
- 969 2406 mt
+1136 2406 mt
(m) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 83.3333 FMSR
-1017 2406 mt
+1184 2406 mt
( testAA Analytisch) s
-gs 665 2311 1200 855 MR c np
+gs 665 2311 1239 855 MR c np
c8
-234 0 711 2375 2 MP stroke
-gs 777 2324 103 103 MR c np
- 25 25 828 2375 FO
+227 0 710 2375 2 MP stroke
+gs 772 2324 103 103 MR c np
+ 25 25 823 2375 FO
gr
gr
%%IncludeResource: font Symbol
/Symbol /ISOLatin1Encoding 83.3333 FMSR
- 969 2507 mt
+ 960 2507 mt
(h) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 83.3333 FMSR
-1019 2507 mt
+1010 2507 mt
( testAA Analytisch) s
-gs 665 2311 1200 855 MR c np
+gs 665 2311 1239 855 MR c np
c9
-234 0 711 2477 2 MP stroke
-gs 777 2426 103 103 MR c np
- 25 25 828 2477 FO
+227 0 710 2477 2 MP stroke
+gs 772 2426 103 103 MR c np
+ 25 25 823 2477 FO
gr
gr
c9
0 sg
- 969 2609 mt
+ 960 2609 mt
(error testAA Analytisch) s
-gs 665 2311 1200 855 MR c np
+gs 665 2311 1239 855 MR c np
c10
-234 0 711 2578 2 MP stroke
-gs 777 2527 103 103 MR c np
- 25 25 828 2578 FO
+227 0 710 2578 2 MP stroke
+gs 772 2527 103 103 MR c np
+ 25 25 823 2578 FO
gr
gr
%%IncludeResource: font Symbol
/Symbol /ISOLatin1Encoding 83.3333 FMSR
- 969 2710 mt
+ 960 2710 mt
(m) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 83.3333 FMSR
-1017 2710 mt
-(2 testAA Analytisch) s
-gs 665 2311 1200 855 MR c np
+1008 2710 mt
+( testAA Analytisch) s
+gs 665 2311 1239 855 MR c np
c11
-234 0 711 2679 2 MP stroke
-gs 777 2628 103 103 MR c np
- 25 25 828 2679 FO
+227 0 710 2679 2 MP stroke
+gs 772 2628 103 103 MR c np
+ 25 25 823 2679 FO
gr
gr
c11
0 sg
- 969 2841 mt
+ 960 2841 mt
(N) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 66.6667 FMSR
-1029 2800 mt
+1020 2800 mt
(-1/2) s
-gs 665 2311 1200 855 MR c np
+gs 665 2311 1239 855 MR c np
DD
c8
-234 0 711 2800 2 MP stroke
+227 0 710 2800 2 MP stroke
SO
gr
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 83.3333 FMSR
- 969 2981 mt
+ 960 2981 mt
(N) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 66.6667 FMSR
-1029 2940 mt
+1020 2940 mt
(-1/4) s
-gs 665 2311 1200 855 MR c np
+gs 665 2311 1239 855 MR c np
DD
c9
-234 0 711 2940 2 MP stroke
+227 0 710 2940 2 MP stroke
SO
gr
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 83.3333 FMSR
- 969 3122 mt
+ 960 3122 mt
(N) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 66.6667 FMSR
-1029 3081 mt
+1020 3081 mt
(-3/4) s
-gs 665 2311 1200 855 MR c np
+gs 665 2311 1239 855 MR c np
DD
c10
-234 0 711 3080 2 MP stroke
+227 0 710 3080 2 MP stroke
SO
gr
%!PS-Adobe-2.0 EPSF-1.2
%%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.
%%Title: ./exmplAA_2DQuad_norm.eps
-%%CreationDate: 03/26/2012 08:17:52
+%%CreationDate: 03/26/2012 10:22:43
%%DocumentNeededFonts: Helvetica
%%DocumentProcessColors: Cyan Magenta Yellow Black
%%Extensions: CMYK
l0 = [files{i}(p1:p1+p2-2) ' '];
l1 = {type2str{data(1,[2+(0:step-1)*rows])}};
for i = 1:step
- leg0 = {leg0{:} ['\mu ' l0 l1{i}] ['\eta ' l0 l1{i}]...
- ['error ' l0 l1{i}] ['\mu2 ' l0 l1{i}]}';
+ leg0 = {leg0{:} ['tilde \mu ' l0 l1{i}] ['\eta ' l0 l1{i}]...
+ ['error ' l0 l1{i}] ['\mu ' l0 l1{i}]}';
leg1 = {leg1{:} [ l0 l1{i}]}';
sym = {sym{:} type2sym{data(1,[2+(i-1)*rows])}}'
end
i=0;
-[shift k] = min(G_D(:,2+1+rows*i)*G_D(1,2)/G_D(1,3)-G_D(:,2+0+rows*i));
+[shift k] = min(G_D(1:end-5,2+1+rows*i)*G_D(1,2)/G_D(1,3)-G_D(1:end-5,2+0+rows*i));
shift = shift+shift/10;
% eta = G_D(:,2+1+rows*i)*(G_D(k,2)-shift)/G_D(k,3)
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