From: Peter Schaefer Date: Mon, 26 Mar 2012 08:25:01 +0000 (+0200) Subject: [doc] Vorgehen X-Git-Url: https://git.leopard-lacewing.eu/?a=commitdiff_plain;h=c104e1042a16bd37ee7d841dc9e6374d31c33aec;p=bacc.git [doc] Vorgehen [src] tilde mu --- diff --git a/doc/doc.pdf b/doc/doc.pdf index fa99b9b..8597ce2 100644 Binary files a/doc/doc.pdf and b/doc/doc.pdf differ diff --git a/doc/doc.tex b/doc/doc.tex index 63f9a8e..5f20f08 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -612,33 +612,34 @@ $V = mex\_build\_AU(COO,ELE,zeta,type)$ \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}} diff --git a/doc/fig/exmplAA_2DQuad_error.eps b/doc/fig/exmplAA_2DQuad_error.eps index 1158b89..6c4df3a 100644 --- a/doc/fig/exmplAA_2DQuad_error.eps +++ b/doc/fig/exmplAA_2DQuad_error.eps @@ -1,7 +1,7 @@ %!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 @@ -713,31 +713,33 @@ DD ( ) 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 @@ -747,31 +749,31 @@ c8 %%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 @@ -781,35 +783,35 @@ c10 %%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 @@ -818,17 +820,17 @@ c8 %%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 @@ -837,17 +839,17 @@ c9 %%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 diff --git a/doc/fig/exmplAA_2DQuad_error.pdf b/doc/fig/exmplAA_2DQuad_error.pdf index c5c1603..e902f5b 100644 Binary files a/doc/fig/exmplAA_2DQuad_error.pdf and b/doc/fig/exmplAA_2DQuad_error.pdf differ diff --git a/doc/fig/exmplAA_2DQuad_norm.eps b/doc/fig/exmplAA_2DQuad_norm.eps index c18b172..cd2cd30 100644 --- a/doc/fig/exmplAA_2DQuad_norm.eps +++ b/doc/fig/exmplAA_2DQuad_norm.eps @@ -1,7 +1,7 @@ %!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 diff --git a/doc/fig/exmplAA_2DQuad_norm.pdf b/doc/fig/exmplAA_2DQuad_norm.pdf index 3437550..f442c9a 100644 Binary files a/doc/fig/exmplAA_2DQuad_norm.pdf and b/doc/fig/exmplAA_2DQuad_norm.pdf differ diff --git a/src/A_plots.m b/src/A_plots.m index 6d78d2a..f4bd440 100644 --- a/src/A_plots.m +++ b/src/A_plots.m @@ -40,8 +40,8 @@ for i = 1:length(files) 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 @@ -66,7 +66,7 @@ figure(4) 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)