From e0b8883142819449c04320d021f1448f4e8cf053 Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Sun, 14 Apr 2013 00:23:48 +0200 Subject: [PATCH] =?utf8?q?[doc]=202D=20Verfeinern=20Strategie=20vergleich?= =?utf8?q?=20Seitenverh=C3=A4ltniss=20fertig?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- doc/code.pdf | Bin 378012 -> 395858 bytes doc/code.tex | 35 +- doc/doc.pdf | Bin 359387 -> 368211 bytes doc/doc.tex | 84 ++-- doc/fig/1t05n05_2DLShape_cond.eps | 283 ++++++++++--- doc/fig/1t05n05_2DLShape_error.eps | 149 ++++--- doc/fig/1t05n05_2DLShape_hminmax.eps | 602 +++++++++++++++------------ doc/fig/1t05n05_2DLShape_time.eps | 414 +++++++++--------- src/A_plots.m | 9 +- src/export_exmpl.m | 3 +- 10 files changed, 963 insertions(+), 616 deletions(-) diff --git a/doc/code.pdf b/doc/code.pdf index 86968124d25cd99c624937fdcc57b19fa744583d..9de000538709595184165b4c81cb915ce687f993 100644 GIT binary patch delta 223932 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zusDFmPyjT3olp4x<2NQg3;>{sG*U(3sR_d7M4&bq*eTIy8eDifpw)gL9Gu|yQjv;e>XhlW7_PdjcDiP|Do>5-tW~g#Qs>2`HM$0)XZF#{j|g<=rYl8W)9^8Z^-v4nP}2smn&!@v^-K-~!ej?&a^M0m+W%^(p^BEeGlcNs+lNEm8c zCK72^DMS(+sHhl(jf}?TNwBDCTo_CEkBT>zQ6xN?W_}=%Xd)!MW`Ql_-;S|xNT6Lt z5mB_!4URHYxEKH=VQ3u-{(z(wKLMTqs3im28=e-9NE5VRU!q=oVc;-LT=j@m`Tq&E z1Pr{u{&#e+j$sG*w+%~xcSimb$xeIOI=R|=vC7K}%j!G%+Y4jRC}CMWGWoPH%(Aea zug6;QV!Z diff --git a/doc/code.tex b/doc/code.tex index 34fd77c..3e1ed51 100644 --- a/doc/code.tex +++ b/doc/code.tex @@ -42,9 +42,9 @@ \psfrag{fehler 1t05n05 A}{\tiny Fehler} \psfrag{Zeit 1t05n05 A}{\tiny Zeit} \psfrag{cond 1t05n05 A}{\tiny Kondition} -\psfrag{min hmin/max hmax 1t05n05 A}{\tiny $(\min h_{min})/(\max h_{max})$} -\psfrag{min hmax/max hmax 1t05n05 A}{\tiny $(\min h_{max})/(\max h_{max})$} -\psfrag{min hmin/hmax 1t05n05 A}{\tiny $(\min h_{min})/h_{max}$} +\psfrag{min hmin/max hmax 1t05n05 A}{\tiny $\min (h_{\min})/(\max h_{\max})$} +\psfrag{min hmax/max hmax 1t05n05 A}{\tiny $\min (h_{\max})/(\max h_{\max})$} +\psfrag{min hmin/hmax 1t05n05 A}{\tiny $\min (h_{\min}/h_{\max})$} \psfrag{tmu 132t05n05 A}{\tiny $\tilde \mu$ analytisch} @@ -118,27 +118,27 @@ \psfrag{fehler 1t1n0 A}{\tiny Fehler (uniform)} \psfrag{Zeit 1t1n0 A}{\tiny Zeit (uniform)} \psfrag{cond 1t1n0 A}{\tiny Kondition (uniform)} -\psfrag{min hmin/max hmax 1t1n0 A}{\tiny $(\min h_{min})/(\max h_{max})$ (uniform)} -\psfrag{min hmax/max hmax 1t1n0 A}{\tiny $(\min h_{max})/(\max h_{max})$ (uniform)} -\psfrag{min hmin/hmax 1t1n0 A}{\tiny $(\min h_{min})/h_{max}$ (uniform)} +\psfrag{min hmin/max hmax 1t1n0 A}{\tiny $\min (h_{\min})/(\max h_{\max})$ (uniform)} +\psfrag{min hmax/max hmax 1t1n0 A}{\tiny $\min (h_{\max})/(\max h_{\max})$ (uniform)} +\psfrag{min hmin/hmax 1t1n0 A}{\tiny $\min (h_{\min}/h_{\max})$ (uniform)} \psfrag{tmu 1t05n0 A}{\tiny $\tilde \mu$ (isotrop)} \psfrag{eta 1t05n0 A}{\tiny $\eta$ (isotrop)} \psfrag{fehler 1t05n0 A}{\tiny Fehler (isotrop)} \psfrag{Zeit 1t05n0 A}{\tiny Zeit (isotrop)} \psfrag{cond 1t05n0 A}{\tiny Kondition (isotrop)} -\psfrag{min hmin/max hmax 1t05n0 A}{\tiny $(\min h_{min})/(\max h_{max})$ (isotrop)} -\psfrag{min hmax/max hmax 1t05n0 A}{\tiny $(\min h_{max})/(\max h_{max})$ (isotrop)} -\psfrag{min hmin/hmax 1t05n0 A}{\tiny $(\min h_{min})/h_{max}$ (isotrop)} +\psfrag{min hmin/max hmax 1t05n0 A}{\tiny $\min (h_{\min})/(\max h_{\max})$ (isotrop)} +\psfrag{min hmax/max hmax 1t05n0 A}{\tiny $\min (h_{\max})/(\max h_{\max})$ (isotrop)} +\psfrag{min hmin/hmax 1t05n0 A}{\tiny $\min (h_{\min}/h_{\max})$ (isotrop)} \psfrag{tmu 1t05n05 A}{\tiny $\tilde \mu$ (anisotrop)} \psfrag{eta 1t05n05 A}{\tiny $\eta$ (anisotrop)} \psfrag{fehler 1t05n05 A}{\tiny Fehler (anisotrop)} \psfrag{Zeit 1t05n05 A}{\tiny Zeit (anisotrop)} \psfrag{cond 1t05n05 A}{\tiny Kondition (anisotrop)} -\psfrag{min hmin/max hmax 1t05n05 A}{\tiny $(\min h_{min})/(\max h_{max})$ (anisotrop)} -\psfrag{min hmax/max hmax 1t05n05 A}{\tiny $(\min h_{max})/(\max h_{max})$ (anisotrop)} -\psfrag{min hmin/hmax 1t05n05 A}{\tiny $(\min h_{min})/h_{max}$ (anisotrop)} +\psfrag{min hmin/max hmax 1t05n05 A}{\tiny $\min (h_{\min})/(\max h_{\max})$ (anisotrop)} +\psfrag{min hmax/max hmax 1t05n05 A}{\tiny $\min (h_{\max})/(\max h_{\max})$ (anisotrop)} +\psfrag{min hmin/hmax 1t05n05 A}{\tiny $\min (h_{\min}/h_{\max})$ (anisotrop)} \caption{2D Quad im Vergleich vollanalytisch $V\phi = 1$} \centering @@ -183,12 +183,21 @@ \begin{figure}[ht] \caption{3D FichCube adaptiv anisotrop vollanalytisch $V\phi = 1$} \centering -\label{fig:exmplAA_3DQuad_A} +\label{fig:exmplAA_3DFichCube_A} \subfloat[Fehler]{\includegraphics[width=0.7\textwidth]{fig/1t05n05_3DFichCube_error}}\\ \subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_hminmax}} \subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_cond}} \end{figure} +\begin{figure}[ht] +\caption{3D Cube adaptiv anisotrop vollanalytisch $V\phi = 1$} +\centering +\label{fig:exmplAA_3DCube_A} +\subfloat[Fehler]{\includegraphics[width=0.7\textwidth]{fig/1t05n05_3DCube_error}}\\ +\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DCube_hminmax}} +\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DCube_cond}} +\end{figure} + \begin{figure}[ht] diff --git a/doc/doc.pdf b/doc/doc.pdf index 763f4c485be6e2eb4c115c4d1921478a3de13207..ad78e3111c5326ba879f6240f5409fa9baffb62b 100644 GIT binary patch delta 253181 zcmZU)Q*bU^ux=gOwr$(C?PSHalQ*_)+qUhj*iKe#`~S{g`<$D7)w8<#Zr=1WYV_7C z`ce-@f+Rp!)&;K}x$m}q{Yb6~0^)D%nzHk8ifSaSidJ8VbAojCCJXiCGHD#c&*i(f z!N81(^g`3{EKO*^a>L_;3%@b_yyRy!38p4BkP8@L1m+cOC6)CXYvz5sI?2EX>Y#JYFI@>WBu3 zcOyV{`@?Dj!YuAYeGId0_G@iy6iWF&A20ma@iKeU2zLm|d*Li*F3w($z!S1C>UrJE z)K!W(**`_fL~;VJ)38&*a6W7_+!ki>8N(9rC*Ni+FOUl=;jz>&jb4lHH6o5W|AKg^ z*Y_I&Mr-CwY#!(~kBRZg;sQIb-Za8|;wAwcw`I(a`?B1k#jP(2Atn-`Bg=1x%ZdfDVWLPNR+=?r{F(HsVF?^w>@w zf~_lLOGj4YNB8z+X!wWz!OVGMf;oV}e&LXxmIvI&R4%KGa~K`BvDSDBfY4gNyHWux 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zS_*iOaOq$tit8P0^f0mj8i3;kbfCWz#M<>2($!TI)G~JucL7*v3hQ|)qOik2AKv1F zbA1gs_|_BG8LWW?$3UC7#(=6%Ceqf>R_Xq~DeQ#+urwgh$v-gE-^Eo3*uc;x2q`PC J+p=Eh{{VR%+%5nB diff --git a/doc/doc.tex b/doc/doc.tex index e8e80a2..e332123 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -34,12 +34,12 @@ \definecolor{fig_help}{rgb}{.9,.6,0} \definecolor{fig_err}{rgb}{.9,0,0} -\newcommand{\figLineA}[1][]{\textcolor{fig_line1}{dunkelgrün#1}} -\newcommand{\figLineB}[1][]{\textcolor{fig_line2}{zyan#1}} -\newcommand{\figLineC}[1][]{\textcolor{fig_line3}{magenta#1}} -\newcommand{\figLineD}[1][]{\textcolor{fig_line4}{blau#1}} -\newcommand{\figHelp}[1][]{\textcolor{fig_help}{orange#1}} -\newcommand{\figErr}[1][]{\textcolor{fig_err}{rot#1}} +\newcommand{\figLineA}[1][]{{\color{fig_line1}dunkelgrün#1}} +\newcommand{\figLineB}[1][]{{\color{fig_line2}zyan#1}} +\newcommand{\figLineC}[1][]{{\color{fig_line3}magenta#1}} +\newcommand{\figLineD}[1][]{{\color{fig_line4}blau#1}} +\newcommand{\figHelp}[1][]{{\color{fig_help}orange#1}} +\newcommand{\figErr}[1][]{{\color{fig_err}rot#1}} \usepackage{fancyhdr} @@ -97,8 +97,8 @@ \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\Abs}[1]{ \left\lvert#1\right\rvert} -\newcommand{\todo}[1]{\textcolor{dred}{#1}} -\newcommand{\why}[1]{\textcolor{dblue}{#1}} +\newcommand{\todo}[1]{{\color{dred}#1}} +\newcommand{\why}[1]{{\color{dblue}#1}} \newcommand{\Matlab}{{\sc Matlab}} \newcommand{\Cpp}{{\sc C++}} @@ -1825,6 +1825,7 @@ Da wir später den Fehlerschätzer berechnen wollen, ist es wichtig sich zu jede \label{exmpl1:f2s:part} \end{figure} +\subsection{Berechnung der $V$ Matrix} \clearpage @@ -1893,7 +1894,7 @@ Dadurch gilt auf isotropen Netzen: % %\mu_{\ell}^2 &= \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ % %&=\sum_{T\in \T_{\ell}}\mu_{\ell}(T)^2\\ % \mu_{\ell}(T)^2 &= \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ -% &= h_{min}(T)\norm{\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ +% &= h_{\min}(T)\norm{\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ % && T_j \in \tau_{\ell}, t_j^{(1)},\dots,t_j^{(4)} \in \tau_{\frac l 2} \\ % \phi_{\frac l 2}|_{T_j} &=x_j^{(1)}\chi_{t_j^{(1)}}+\dots+x_j^{(4)}\chi_{t_j^{(4)}}\\ % \Pi_{\ell}\phi_{\frac l 2}|_{T_j}&=\frac 1 {\abs{T_j}}\int_{T_j}\phi_{\frac l 2}d\Gamma\\ @@ -1907,7 +1908,7 @@ Dadurch gilt auf isotropen Netzen: % &=\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_{\ell}(T_j)^2 &= \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} +% \mu_{\ell}(T_j)^2 &= \frac{ h_{\min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} % \end{align*} % @@ -2006,8 +2007,16 @@ mit dem Startnetz aus Abbildung \ref{fig:mesh:2DQuad}, einem Quadrat in der $z=0 \begin{align*} \{ (0,0,0), (1,0,0), (1,1,0), (0,1,0) \}. \end{align*} +\todo{Des Weiteren werden wir als exakte Lösung $\enorm{\phi}^2 = 4.609193$ annehmen.} +\begin{figure}[ht] + \centering + \includegraphics[width=0.5\textwidth]{fig/exmpl_2DQuad_ref} + \caption{Quadrat in der $z=0$ Ebene} + \label{fig:mesh:2DQuad} +\end{figure} +\subsubsection{Vergleich verschiedener Verfeinerungsstrategien} \noindent Zunächst wollen wir drei Verfeinerungs-Strategien genauer untersuchen. Hierzu betrachten wir zum einen die Strategie "`uniform"' ($\theta=1,\nu=0$) bei der das verfeinerte Netz $\T_{\ell+1}$ durch isotrope Verfeinerung aller Elemente entsteht, also jedes Element wird in vier gleich Große Elemente geteilt. In der zweiten Strategie "`isotrop"' ($\theta=0.5,\nu=0$) werden wir zulassen, dass nicht alle Elemente verfeinert werden, also nur eine Teilmenge wird jeweils in vier gleich große Elemente geteilt. Und in der letzten Strategie "`anisotrop"' ($\theta=0.5,\nu=0.5$) werden wir außerdem anisotrope Teilungen zulassen, also ein Teil der Elemente wird geeignet in zwei oder vier gleich Große Elemente geteilt. Für alle drei Strategien werden wir den Algorithmus \ref{alg:adapt} mit entsprechenden Parametern $\theta,\nu$ verwenden. @@ -2018,45 +2027,66 @@ Zunächst wollen wir drei Verfeinerungs-Strategien genauer untersuchen. Hierzu b \psfrag{fehler 1t1n0 A}{\tiny Fehler (uniform)} \psfrag{Zeit 1t1n0 A}{\tiny Zeit (uniform)} \psfrag{cond 1t1n0 A}{\tiny Kondition (uniform)} -\psfrag{min hmin/max hmax 1t1n0 A}{\tiny $(\min h_{min})/(\max h_{max})$ (uniform)} -\psfrag{min hmax/max hmax 1t1n0 A}{\tiny $(\min h_{max})/(\max h_{max})$ (uniform)} -\psfrag{min hmin/hmax 1t1n0 A}{\tiny $(\min h_{min})/h_{max}$ (uniform)} +\psfrag{min hmin/max hmax 1t1n0 A}{\tiny $\min (h_{\min})/\max ( h_{\max})$ (uniform)} +\psfrag{min hmax/max hmax 1t1n0 A}{\tiny $\min ( h_{\max})/\max ( h_{\max})$ (uniform)} +\psfrag{min hmin/hmax 1t1n0 A}{\tiny $\min (h_{\min}/h_{\max})$ (uniform)} \psfrag{tmu 1t05n0 A}{\tiny $\tilde \mu$ (isotrop)} \psfrag{eta 1t05n0 A}{\tiny $\eta$ (isotrop)} \psfrag{fehler 1t05n0 A}{\tiny Fehler (isotrop)} \psfrag{Zeit 1t05n0 A}{\tiny Zeit (isotrop)} \psfrag{cond 1t05n0 A}{\tiny Kondition (isotrop)} -\psfrag{min hmin/max hmax 1t05n0 A}{\tiny $(\min h_{min})/(\max h_{max})$ (isotrop)} -\psfrag{min hmax/max hmax 1t05n0 A}{\tiny $(\min h_{max})/(\max h_{max})$ (isotrop)} -\psfrag{min hmin/hmax 1t05n0 A}{\tiny $(\min h_{min})/h_{max}$ (isotrop)} +\psfrag{min hmin/max hmax 1t05n0 A}{\tiny $\min (h_{\min})/\max ( h_{\max})$ (isotrop)} +\psfrag{min hmax/max hmax 1t05n0 A}{\tiny $\min ( h_{\max})/\max ( h_{\max})$ (isotrop)} +\psfrag{min hmin/hmax 1t05n0 A}{\tiny $\min (h_{\min}/h_{\max})$ (isotrop)} \psfrag{tmu 1t05n05 A}{\tiny $\tilde \mu$ (anisotrop)} \psfrag{eta 1t05n05 A}{\tiny $\eta$ (anisotrop)} \psfrag{fehler 1t05n05 A}{\tiny Fehler (anisotrop)} \psfrag{Zeit 1t05n05 A}{\tiny Zeit (anisotrop)} \psfrag{cond 1t05n05 A}{\tiny Kondition (anisotrop)} -\psfrag{min hmin/max hmax 1t05n05 A}{\tiny $(\min h_{min})/(\max h_{max})$ (anisotrop)} -\psfrag{min hmax/max hmax 1t05n05 A}{\tiny $(\min h_{max})/(\max h_{max})$ (anisotrop)} -\psfrag{min hmin/hmax 1t05n05 A}{\tiny $(\min h_{min})/h_{max}$ (anisotrop)} +\psfrag{min hmin/max hmax 1t05n05 A}{\tiny $\min (h_{\min})/\max (h_{\max})$ (anisotrop)} +\psfrag{min hmax/max hmax 1t05n05 A}{\tiny $\min (h_{\max})/\max (h_{\max})$ (anisotrop)} +\psfrag{min hmin/hmax 1t05n05 A}{\tiny $\min (h_{\min}/h_{\max})$ (anisotrop)} \centering -\subfloat[Fehler \label{fig:2DQuad:verfeinern:err}]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_error}} -\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_hminmax}}\\ -\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_cond}} -\subfloat[Zeit]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_time}} +\subfloat[Fehler und Fehlerschätzer für das Quadrat \label{fig:2DQuad:verfeinern:err}]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_error}} +\subfloat[Seitenverhältnisse für das Quadrat \label{fig:2DQuad:verfeinern:hminmax}]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_hminmax}}\\ +\subfloat[Kondition der $V_{\ell}$ Matrix für das Quadrat \label{fig:2DQuad:verfeinern:cond}]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_cond}} +\subfloat[Berechnungszeit für das Quadrat \label{fig:2DQuad:verfeinern:time}]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_time}} \caption{Vergleich der Verfeinerungsstrategien auf dem Quadrat} \label{fig:2DQuad:verfeinern} \end{figure} \noindent In Abbildung \ref{fig:2DQuad:verfeinern:err} betrachten wir für die jeweilige Strategie jeweils die Fehlerschätzer $\tilde \mu$ und $\eta$, sowie den Fehler gegenüber der tatsächlichen Lösung. Die Werte wurden jeweils über die Elementanzahl des $\T_{\ell}$ Netzes aufgetragen. Anhand der \figLineA[en] Linien erkennen wir, dass der Fehler bei der "`uniformen"' Strategie mit etwa $\O(N^{-1/4})$ gegen $0$ strebt. Gerechnet wurde hier bis zu einer Elementanzahl des $\T_{\ell}$ Netzes von etwa 3000, für das der Fehler der Energienorm im Bereich von 1 liegt. -Anhand der Linien in \figLineB, beobachten wir eine schnellere Konvergenz der "`isotropen"' Strategie. Denn hier lässt sich eine Konvergenzrate von $\O(N^{-1/2})$ gegen 0 erkennen. Bei dieser Strategie erreichen wir für die gleiche Elementanzahl des $\T_{\ell}$ Gitters schon einen Fehler im Bereich von 0.1. -Betrachten wir nun die Strategie "`anisotrop"' in \figLineC, so beobachten wir eine kurzzeitig sehr starke Konvergenz, welche dann von unten gegen eine sehr gute Konvergenzrate von $\O(N^{-3/4})$ strebt. Hierbei erkennen wir, dass der Fehler der Energienorm schon im Bereich von 0.01 für die gleiche Elementanzahl liegt. Jedoch sehen wir auch, dass der Fehlerschätzer $\tilde \mu$ ab dieser Elementanzahl seine Konvergenzrate verliert und damit unzuverlässig wird. \noindent -Weiterhin lässt sich für jede Strategie anhand der Parallelität der Fehlerschätzer zum tatsächlichen Fehler, die Effektivität und Zuverlässigkeit der Fehlerschätzer erkennen. Sowie auch die Äquivalents des $h-h/2$ Schätzers zum lokalen $\tilde \mu$ Schätzers aufgrund der Parallelität zu beobachten ist. Außerdem beschreiben die Fehlerschätzer auch in der Größenordnung den tatsächlichen Fehler sehr gut. +Anhand der Linien in \figLineB[], beobachten wir eine schnellere Konvergenz der "`isotropen"' Strategie. Denn hier lässt sich eine Konvergenzrate von $\O(N^{-1/2})$ gegen 0 erkennen. Bei dieser Strategie erreichen wir für die gleiche Elementanzahl des $\T_{\ell}$ Gitters schon einen Fehler im Bereich von 0.1. + +\noindent +Betrachten wir nun die Strategie "`anisotrop"' in \figLineC[], so beobachten wir eine kurzzeitig sehr starke Konvergenz, welche dann von unten gegen eine sehr gute Konvergenzrate von $\O(N^{-3/4})$ strebt. Hierbei erkennen wir, dass der Fehler der Energienorm schon im Bereich von 0.01 für die gleiche Elementanzahl liegt. Jedoch sehen wir auch, dass der Fehlerschätzer $\tilde \mu$ ab dieser Elementanzahl seine Konvergenzrate verliert und damit unzuverlässig wird. + +\noindent +Weiterhin lässt sich für jede Strategie anhand der Parallelität der Fehlerschätzer zum tatsächlichen Fehler, die Effektivität und Zuverlässigkeit der Fehlerschätzer erkennen. Sowie auch die Äquivalents des $h-h/2$ Schätzers zum lokalen $\tilde \mu$ Schätzers aufgrund der Parallelität zu beobachten ist. Außerdem beschreiben die Fehlerschätzer den tatsächlichen Fehler auch in der Größenordnung sehr gut. + +\noindent +In Abbildung \ref{fig:2DQuad:verfeinern:hminmax} betrachten wir bestimmte Eigenschaften zwischen den Seitenverhältnissen der Elemente aus dem $\T_{\ell}$ Netz für die jeweilige Strategie. $h_{\min}$ steht hierbei für die kürzere Seite eines Rechtecks $T \in \T_{\ell}$ und $h_{\max}$ für die längere der beiden Seiten. Wir werden nun das Verhältnis der kleinsten kurzen Seite gegenüber der größten langen Seite $\min(h_{\min}) / \max (h_{\max})$, das Verhältnis der kleinsten langen gegenüber der größten langen Seite $\min(h_{\max}) / \max (h_{\max})$, sowie das kleinste Verhältnis der kurzen gegenüber der langen Seiten $\min(h_{\min} /h_{\max})$ für die drei Strategien genauer betrachten. + +\noindent +Bei der "`uniformen"' Strategie, dargestellt durch die \figLineA[e] Linien, sind wie erwartet alle Verhältnisse gleich 1 da alle Elemente deckungsgleich sind. + +\noindent +Bei der "`isotropen"' Strategie in \figLineB[] ist am Verhältnis $\min(h_{\min} /h_{\max}) = 1$ gut zu erkennen, dass alle Elemente Quadrate sind. Die beiden anderen Verhältnisse zeigen, dass für zunehmende Elementanzahl die Differenz der Elementgrößen zunimmt. + +\noindent +Anhand der \figLineC[] farbenen Linien, also der "`anisotropen"' Strategie beobachten wir, am kleiner werdenden Verhältnis $\min(h_{\min} /h_{\max})$, dass mit zunehmender Elementanzahl lange schmale Elemente entstehen. Am kleiner werden der anderen beiden Verhältnisse erkennen wir auch hier, dass die Differenz der Elementgrößen zunimmt. + +\noindent +Um auch die Stabilität der drei Strategien untersuchen zu können, sehen wir in der Abbildung \ref{fig:2DQuad:verfeinern:cond} die Konditionszahlen der $V_{\ell}$ Matrix in Abhängigkeit der Elementanzahl. +\subsubsection{Vergleich verschiedener Quadraturgrade} +\subsubsection{Vergleich verschiedener Berechnungsarten} % Zum Plotten (Abb.\ref{fig:exmplAA_2DQuad})werden noch folgende Schritte ausgeführt % \begin{itemize} @@ -2102,7 +2132,7 @@ Weiterhin lässt sich für jede Strategie anhand der Parallelität der Fehlersch \centering \label{fig:objects} \subfloat[2D L Shape\label{fig:mesh:2DLShape}]{\includegraphics[width=0.5\textwidth]{fig/exmpl_2DLShape_ref}} -\subfloat[2D Quad\label{fig:mesh:2DQuad}]{\includegraphics[width=0.5\textwidth]{fig/exmpl_2DQuad_ref}}\\ +% \subfloat[2D Quad\label{fig:mesh:2DQuad}]{\includegraphics[width=0.5\textwidth]{fig/exmpl_2DQuad_ref}}\\ \subfloat[3D Cube\label{fig:mesh:3DCube}]{\includegraphics[width=0.5\textwidth]{fig/exmpl_3DCube_ref}} \subfloat[3D FichCube\label{fig:mesh:3DFichCube}]{\includegraphics[width=0.5\textwidth]{fig/exmpl_3DFichCube_ref}} \end{figure} diff --git a/doc/fig/1t05n05_2DLShape_cond.eps b/doc/fig/1t05n05_2DLShape_cond.eps index c69e4bd..8b75a27 100644 --- a/doc/fig/1t05n05_2DLShape_cond.eps +++ b/doc/fig/1t05n05_2DLShape_cond.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The MathWorks, Inc. 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665 3165 mt 665 2529 L - 977 2624 mt + 665 3165 mt 665 2428 L + 977 2523 mt (tmu 1t05n05 A) s -gs 665 2529 962 637 MR c np +gs 665 2428 962 738 MR c np c8 -240 0 712 2593 2 MP stroke -gs 781 2542 103 103 MR c np - 25 25 832 2593 FO +240 0 712 2492 2 MP stroke +gs 781 2441 103 103 MR c np + 25 25 832 2492 FO gr gr c8 0 sg - 977 2725 mt + 977 2624 mt (eta 1t05n05 A) s -gs 665 2529 962 637 MR c np +gs 665 2428 962 738 MR c np c8 -240 0 712 2695 2 MP stroke -gs 781 2644 103 103 MR c np +240 0 712 2594 2 MP stroke +gs 781 2543 103 103 MR c np 0 j -29 50 -58 0 29 -50 832 2728 4 MP +29 50 -58 0 29 -50 832 2627 4 MP DP gr @@ -699,52 +721,63 @@ gr c8 0 sg - 977 2827 mt + 977 2726 mt (fehler 1t05n05 A) s -gs 665 2529 962 637 MR c np +gs 665 2428 962 738 MR c np c8 -240 0 712 2796 2 MP stroke -gs 781 2745 103 103 MR c np - 815 2779 mt 849 2813 L - 849 2779 mt 815 2813 L +240 0 712 2695 2 MP stroke +gs 781 2644 103 103 MR c np + 815 2678 mt 849 2712 L + 849 2678 mt 815 2712 L gr gr c8 0 sg - 977 2928 mt + 977 2827 mt (N-12) s -gs 665 2529 962 637 MR c np +gs 665 2428 962 738 MR c np DD c9 -240 0 712 2897 2 MP stroke +240 0 712 2796 2 MP stroke SO gr c9 0 sg - 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624 270 mt 661 270 L + 623 270 mt 661 270 L 4344 270 mt 4306 270 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 83.3333 FMSR @@ -525,45 +555,49 @@ SO /Helvetica /ISOLatin1Encoding 55.5556 FMSR 537 249 mt -(4) s - 624 3205 mt 4344 3205 L - 624 270 mt 4344 270 L - 624 3205 mt 624 270 L +(5) s + 623 3205 mt 4344 3205 L + 623 270 mt 4344 270 L + 623 3205 mt 623 270 L 4344 3205 mt 4344 270 L -gs 624 270 3721 2936 MR c np +gs 624 269 3721 2937 MR c np /c8 { 0.000000 0.300000 0.300000 sr} bdef c8 -100 -155 91 -144 92 -143 84 -124 80 -127 70 -114 73 -140 76 -134 -71 -141 74 -120 65 -46 67 -73 66 -68 77 -102 62 -31 74 -84 -95 -123 95 -70 99 -11 133 -134 147 -177 141 369 190 -593 254 18 -396 456 1067 2653 26 MP stroke -gs 1016 591 2875 2588 MR c np - 25 25 1067 2653 FO - 25 25 1463 3109 FO - 25 25 1717 3127 FO - 25 25 1907 2534 FO - 25 25 2048 2903 FO - 25 25 2195 2726 FO - 25 25 2328 2592 FO - 25 25 2427 2581 FO - 25 25 2522 2511 FO - 25 25 2617 2388 FO - 25 25 2691 2304 FO - 25 25 2753 2273 FO - 25 25 2830 2171 FO - 25 25 2896 2103 FO - 25 25 2963 2030 FO - 25 25 3028 1984 FO - 25 25 3102 1864 FO - 25 25 3173 1723 FO - 25 25 3249 1589 FO - 25 25 3322 1449 FO - 25 25 3392 1335 FO - 25 25 3472 1208 FO - 25 25 3556 1084 FO - 25 25 3648 941 FO - 25 25 3739 797 FO - 25 25 3839 642 FO +104 -144 41 -67 38 -40 76 -106 100 -133 91 -123 92 -123 84 -106 +80 -109 70 -98 73 -120 76 -115 71 -121 74 -102 65 -39 67 -63 +66 -59 77 -87 62 -27 74 -71 95 -106 95 -60 99 -9 133 -116 +147 -151 141 316 190 -508 254 15 396 392 1067 2731 30 MP stroke +gs 1016 600 3134 2590 MR c np + 25 25 1067 2731 FO + 25 25 1463 3123 FO + 25 25 1717 3138 FO + 25 25 1907 2630 FO + 25 25 2048 2946 FO + 25 25 2195 2795 FO + 25 25 2328 2679 FO + 25 25 2427 2670 FO + 25 25 2522 2610 FO + 25 25 2617 2504 FO + 25 25 2691 2433 FO + 25 25 2753 2406 FO + 25 25 2830 2319 FO + 25 25 2896 2260 FO + 25 25 2963 2197 FO + 25 25 3028 2158 FO + 25 25 3102 2056 FO + 25 25 3173 1935 FO + 25 25 3249 1820 FO + 25 25 3322 1700 FO + 25 25 3392 1602 FO + 25 25 3472 1493 FO + 25 25 3556 1387 FO + 25 25 3648 1264 FO + 25 25 3739 1141 FO + 25 25 3839 1008 FO + 25 25 3915 902 FO + 25 25 3953 862 FO + 25 25 3994 795 FO + 25 25 4098 651 FO gr gr @@ -580,26 +614,26 @@ c8 90 rotate 612 3236 mt ( ) s -4333 300 mt +4333 299 mt ( ) s 1 sg -0 130 894 0 0 -130 665 442 4 MP +0 130 894 0 0 -130 665 441 4 MP PP --894 0 0 130 894 0 0 -130 665 442 5 MP stroke +-894 0 0 130 894 0 0 -130 665 441 5 MP stroke 2.77778 w DO SO 4.16667 w 0 sg - 665 442 mt 1559 442 L - 665 442 mt 665 312 L - 980 408 mt + 665 441 mt 1559 441 L + 665 441 mt 665 311 L + 980 407 mt (Zeit 1t05n05 A) s -gs 665 312 895 131 MR c np +gs 665 311 895 131 MR c np c8 -242 0 713 377 2 MP stroke -gs 783 326 103 103 MR c np - 25 25 834 377 FO +242 0 713 376 2 MP stroke +gs 783 325 103 103 MR c np + 25 25 834 376 FO gr gr diff --git a/src/A_plots.m b/src/A_plots.m index b4415f0..758634d 100644 --- a/src/A_plots.m +++ b/src/A_plots.m @@ -117,7 +117,8 @@ else % sol = interp1(1./X((round(1)):(end),1)',G_D((round(1)):(end),4)',0,'spline') % G_D(end,4) - + + if(strcmp(t0, '(3DFichCube)') || strcmp(t0, '(3DFichCube2)')) sol = 16.2265; % Ferraz-Leite Paper % sol = 50; @@ -127,13 +128,15 @@ elseif(strcmp(t0, '(2DLShape)')) % sol = 8.28457; % Ferraz-Leite Dipl. sol = 8.28466; % Ferraz-Leite Paper elseif(strcmp(t0, '(3DCube)') || strcmp(t0, '(3DCube2)')) - sol = 8.303; + sol = 8.30235; elseif(strcmp(t0, '(3DCube3)')) % sol = 1.0379; sol = 16.604703 % Ferraz-Leite Dipl. end - +% G_D(:,2+2) +% +% abs(log10((sqrt(sol-G_D(:,2+2))-G_D(:,2+1)))) % G_D diff --git a/src/export_exmpl.m b/src/export_exmpl.m index 6ca94e6..3f57427 100644 --- a/src/export_exmpl.m +++ b/src/export_exmpl.m @@ -142,8 +142,9 @@ end %% voll Analytisch A_plots({'meshSave/1t05n05_3DFichCube_23'},'../doc/fig/1t05n05_3DFichCube') +A_plots({'meshSave/1t05n05_3DCube_25'},'../doc/fig/1t05n05_3DCube') A_plots({'meshSave/1t05n05_2DQuad_32'},'../doc/fig/1t05n05_2DQuad') -A_plots({'meshSave/1t05n05_2DLShape_26'},'../doc/fig/1t05n05_2DLShape') +A_plots({'meshSave/1t05n05_2DLShape_30'},'../doc/fig/1t05n05_2DLShape') %% Isotrop Uniform A_plots({'meshSave/1t1n0_2DQuad_6'},'../doc/fig/1t1n0_2DQuad') -- 2.47.3
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