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zE&|sMj-y?&2cX@N&$KNTd14vs;3Ry;|K6@gp}tLBm^*Tkak{uN7iDAfaXxrfRv}sH z!t+<2ea@QmOUE2Dy!3r%e{#)mtJ7@`FFkeb+6#xFZ}`;VrSCZRLu-d2VffV2(s!IV z{I|zse{~|1p4dF8dQ$u3taU=CSv#rTKH;S7L^j@&PpBXJ|3jQnI`-Jo1sAS7|H4n4 Vzve?FqMxv`bok*Xz4zqO{{bw4m^1(Y diff --git a/doc/doc.tex b/doc/doc.tex index 360a635..0bf468b 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -307,7 +307,7 @@ mit $V := \gamma_0\tilde V$ gilt. Ziel ist es nun, aus \eqref{math:slp:gls} eine Da wir das Problem \eqref{math:slp:gls} im allgemeinen nicht Lösen können, werden wir es mithilfe des Galerkin-Verfahrens näherungsweise lösen. Die Idee dabei ist $H^{-1/2}(\Gamma)$ durch einen endlich dimensionalen Unterraum zu ersetzen. \noindent -Bezeichne nun $\langle \cdot, \cdot \rangle$ das erweiterte $L^2$-Skalarprodukt, so existiert, da $V$ ein symmetrischer, und mit geeigneter Skalierung ein elliptischer Isomorphismus ist, auf $\widetilde H^{-1/2}$ ein äquivalentes Skalarprodukt $\llangle \cdot, \cdot \rrangle$ mit $\llangle \phi, \psi \rrangle := \langle V\phi,\psi\rangle$ und der induzierten Norm $\enorm{\cdot}$. +Bezeichne nun $\langle \cdot, \cdot \rangle$ das erweiterte $L^2$-Skalarprodukt, so existiert, da das Einfachschichtpotential $V$ ein symmetrischer, und mit geeigneter Skalierung elliptischer Isomorphismus ist, auf $\widetilde H^{-1/2}$ ein äquivalentes Skalarprodukt $\llangle \cdot, \cdot \rrangle$ mit $\llangle \phi, \psi \rrangle := \langle V\phi,\psi\rangle$ und der induzierten Norm $\enorm{\cdot}$. Sei nun $\phi$ die eindeutige Lösung von \eqref{math:slp:gls} und bezeichne $g$ die Dirichlet-Daten am Rand. Dann gilt \begin{align} \llangle \phi,\psi\rrangle = \langle g,\psi\rangle \quad \text{für alle }\psi \in \widetilde H^{-1/2}. @@ -686,7 +686,7 @@ gilt, wobei $p$ das Interpolationspolynom von $f$ vom Grad $n$ zu den Knoten $t_ % \int_{T_j} \int_{T_k} &\approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2}\\ % \int_{T_j} \int_{T_k} &\approx \q_{T_j} \q_{J_1} \int_{J_2} % \end{align}} -\subsection{Volle Quadratur} +\subsection{Volle Quadratur} \label{sec:semi:Q} % \subsubsection{Glatter Kern} % Das Integral über $T_j$ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über $T_k$ wird nun mittels einer doppelten Gauss-Quadratur berechnet. @@ -998,7 +998,7 @@ Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{beweis} -\subsection{Quadratur über ein Element} +\subsection{Quadratur über ein Element}\label{sec:semi:QE} \begin{defi} Seien $T_j,T_k \in \R^3$ achsenorientierte Rechtecke und sei $\zeta_E > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_E$-zulässig, genau dann wenn @@ -1862,10 +1862,12 @@ Relevant zum Verfeinern eines Netzes sind also die Koordinaten $COO$, Elemente $ \noindent Da wir später den 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 $F2S$ an, in der die maximal vier Elementindizes gespeichert werden. 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 sichergestellt, dass das arithmetische Mittel über die Elemente immer gültig auszuführen ist. -(Siehe Abbildung \ref{exmpl13}) +Vergleiche hierzu Abbildungen \ref{exmpl12} und \ref{exmpl13}. +Durch die Funktion \ref{code:refineQuad} \begin{lstlisting}[language=M, numbers=none] [COO_fine, ELE_fine, NEI_fine, F2S] = refineQuad(COO, ELE, NEI, marked); \end{lstlisting} +kann die Verfeinerung in \Matlab~durchgeführt werden. % $[\T_{\ell}, F2S ] = refineQuad(\T_{\ell}, marked);$\\ @@ -1964,6 +1966,20 @@ Die Implementierung der Berechnung eines Eintrags $A_{jk}$ befindet sich in der \end{itemize} } +\noindent +In diesem Abschnitt werden wir anhand von zwei Beispielgeometrien die gezeigten Zusammenhänge genauer untersuchen. Hierzu werden wir zunächst einige Fehlerschätzer definieren, welche auch lokale Aussagen auf den Netzen zulassen. Weiterhin werden wir eine Strategie zum Markieren von Elementen vorstellen, die die Definition eines adaptiven Algorithmus ermöglicht. Da wir durch verschiedene Strategien die Stabilität der Berechnungen sicherstellen wollen, werden wir unsere Berechnungen auf das Lösen der Gleichung +\begin{align*} + V\phi = 1 +\end{align*} +mit konstanter rechter Seite beschränken. + +\noindent +Alle Experimente wurden auf einem Rechner mit acht Intel(R) Xeon(R) Prozessoren mit jeweils 2.5GHz durchgeführt. Auf dem Betriebsystem Ubuntu 3.0.0-32-server mit 32GB verfügbaren Arbeitsspeicher wurde die gcc Version 4.6.1 und \Matlab~R2011a (7.12.0) 64bit verwendet. Während den Berechnungen konnte eine Systemauslastung von etwa 700\% beobachtet werden, welches der Benutzung von sieben Kernen entspricht. + + + + + \subsection{Fehlerschätzer} Mithilfe des Galerkin-Verfahrens berechnen wir nur eine approximative Lösung von \eqref{math:slp:gls}, für die wir eine Aussage über die Genauigkeit der Lösung treffen wollen. Da wir für den Fehler $\enorm{\phi - \phi_{\ell}}$ nur die Galerkin-Lösung $\phi_{\ell}$ kennen und die exakte Lösung $\phi$ unbekannt ist, werden wir im Folgenden verschiedene Fehlerschätzer betrachten. % In diesem Abschnitt definieren wir die a-posteriori Fehlerschätzer, die wir zur Steuerung des adaptiven Algorithmus einsetzen werden. @@ -1983,35 +1999,29 @@ und unter der Saturationsannahme \end{align*} zuverlässig \begin{align*} -\enorm{\phi - \phi_{\ell}}&\leq \frac 1{\sqrt{1-C_{sat}}}\eta_{\ell}. +\enorm{\phi - \phi_{\ell}}&\leq \frac 1{\sqrt{1-C_{sat}^2}}\eta_{\ell}. \end{align*} \end{defi} -Der Schätzer ist berechenbar, liefert aber keine lokalen Beiträge und kann daher nicht unmittelbar verwendet werden, um einen adaptiven Algorithmus zu steuern. Hierzu definieren wir den $\tilde \mu$ Schätzer. +Der Schätzer ist berechenbar, liefert aber keine lokalen Beiträge und kann daher nicht unmittelbar verwendet werden, um einen adaptiven Algorithmus zu steuern. Hierzu definieren wir weitere Schätzer. Für jedes $T_j\in\T_{\ell}$ bezeichnen wir mit $\varrho_j>0$ den Durchmesser des größten eingeschriebenen Innenkreises in $T_j$ und mit $h_j:=\diam(T_j)>0$ den maximalen Durchmesser von $T_j$. Weiterhin definieren wir die lokalen Netzweiten $\varrho,h\in L^\infty$ durch $\varrho_{|T_j} = \varrho_j$ und $h_{|T_j} = h_j$. -\begin{sat}[A-posteriori Fehlerschätzer] Seien aus \cite[Theorem 3.2 und 3.4]{fer:errbem}: +\begin{defi}[A-posteriori Fehlerschätzer] Es bezeichne $\phi$ die exakte Lösung von \eqref{math:slp:gls}, $\phi_{\ell}$ die Galerkin-Lösung von \eqref{math:slp:gls:galerkin} auf dem Gitter $\T_{\ell}$ und $\hat \phi_{\ell}$ die Galerkin-Lösung auf dem uniform verfeinerten Gitter $\hat \T_{\ell}$. Dann gilt nach \cite[Theorem 3.2 und 3.4]{fer:errbem}, die Fehlerschätzer \begin{align} -\eta_{\ell} &= \enorm{\hat \phi_{\ell} - \phi_{\ell}}\\ +\mu_{\ell} &= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \phi_{\ell})}_{L^2(\Gamma)}\\ \tilde\eta_{\ell} &= \enorm{\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell}}\\ -\mu_{\ell} &= \norm{\varrho^{\ell/2}(\hat \phi_{\ell} - \phi_{\ell})}_{L^2(\Gamma)}\\ -\tilde\mu_{\ell} &= \norm{\varrho^{\ell/2}(\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell})}_{L^2(\Gamma)} +\tilde\mu_{\ell} &= \norm{\varrho^{1/2}(\hat \phi_{\ell} - \Pi_{\ell}\hat \phi_{\ell})}_{L^2(\Gamma)} \end{align} -wobei $\Pi_{\ell}$ die $L_2$ Projektion auf $P^0(\T_{\ell})$ ist. -\end{sat} -Weiterhin ist $\mu_{\ell}$ noch immer zuverlässig und effizient, da $\mu_{\ell} \approx \eta_{\ell}$ gilt.\\ -Dadurch gilt auf isotropen Netzen: +wobei $\Pi_{\ell}$ die $L_2$ Projektion auf $P^0(\T_{\ell})$ ist, + \begin{itemize} -\item Schätzer sind equivalent\\ $\tilde \mu_{\ell} \leq \mu_{\ell} \leq \sqrt{2}C_3 \eta_{\ell}$ +\item sind mit Konstanten $C_3,C_4>0$ äquivalent\\ +$\eta_{\ell} \leq \tilde \eta_{\ell} \leq C_4\norm{(h/\varrho)^{1/2}}_{L^\infty(\Gamma)} \tilde \mu_{\ell}$ und $\tilde \mu_{\ell} \leq \mu_{\ell} \leq \sqrt{2}C_3 \eta_{\ell}$ \item sie sind effizient -\item sie sind unter Saturationsannahme auch zuverlässig +\item sie sind unter Saturationsannahme auch zuverlässig. \end{itemize} +\end{defi} + -% -% % Siehe \cite[Theorem 3.2 \& 3.4]{fer:errbem}. -% -% -% -% % \begin{align*} % %\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\\ @@ -2032,7 +2042,7 @@ Dadurch gilt auf isotropen Netzen: % &=\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} % \end{align*} -% + @@ -2041,14 +2051,16 @@ Im Adaptiven Algorithmus werden wir die Elemente abhängig vom Fehlerschätzer $ \begin{defi}[Dörfler-Markierung]\label{thm:def:mark} Bestimme für feste Konstante $\theta \in (0,1]$ die Menge $M_{\ell} \subseteq T_{\ell}$ mit minimaler Kardinalität \begin{align*} - \theta \sum_{T\in T_{\ell}} \tilde \mu_{\ell}(T)^2 &\leq \sum_{ T\in M_{\ell}} \tilde \mu_{\ell}(T)^2. + \theta \sum_{T\in T_{\ell}} \tilde \mu_{\ell}(T)^2 &\leq \sum_{ T\in M_{\ell}} \tilde \mu_{\ell}(T)^2. \end{align*} - Um die Symmetrie des Netzes zu erhalten bestimme weiterhin die kleinste Teilmenge $\tilde M_{\ell} \subseteq T_{\ell}$ für die gilt + Um die Symmetrie des Netzes zu erhalten bestimme weiterhin für feste Konstante $\vartheta \in (0,1)$ die kleinste Teilmenge $\tilde M_{\ell} \subseteq T_{\ell}\backslash M_{\ell}$ für die gilt \begin{align*} - \sum_{T\in M_{\ell}} \tilde \mu_{\ell}(T)^2 - \sum_{T\in \tilde M_{\ell}} \tilde \mu_{\ell}(T)^2 / \sum_{T\in M_{\ell}} \tilde \mu_{\ell}(T)^2 &> 10^{-2}. + \vartheta \sum_{T\in M_{\ell}} \tilde \mu_{\ell}(T)^2 &< \sum_{T\in \tilde M_{\ell}} \tilde \mu_{\ell}(T)^2 \end{align*} -\end{defi}9 -Da wir im Adaptiven Algorithmus auch anisotrope Verfeinerungen zulassen werden definieren wir an dieser Stelle eine Auswahlstrategie zum bestimmen der Verfeinerungsart für jedes Element. + Die Menge der Markierten Elemente ist dann gegeben durch $M_{\ell} \cup \tilde M_{\ell}$. +\end{defi} +Im Folgenden werden wir $\vartheta = 10^{-2}$ wählen. +Da wir im Adaptiven Algorithmus auch anisotrope Verfeinerungen zulassen wollen, definieren wir an dieser Stelle eine Auswahlstrategie zum bestimmen der Verfeinerungsart für jedes Element. \begin{defi} Seien $\phi_j^{(1)},\ldots, \phi_j^{(4)}$ die Lösungen der isotropen Verfeinerung $T_j^{(1)},\ldots, T_j^{(4)} \in \hat \T_{\ell}$ von $T_j \in T_{\ell}$, % das heißt $T_j = \sum_{k=1,\ldots,4} T_j^{(k)}$, @@ -2079,7 +2091,7 @@ marked_j := \end{align*} Weiterhin sei $marked_j = 1$ für alle $T_j \in \T_{\ell} \backslash \tilde M_{\ell}$. \end{defi} -Die Funktion +Die Funktion \ref{code:mark} \begin{lstlisting}[language=M,numbers=none] marked = mark(xF2S, tmu, theta, nu); \end{lstlisting} @@ -2107,7 +2119,7 @@ Mithilfe der oben Definierten Funktionen ist es uns nun möglich den Ablauf der \renewcommand{\theenumi}{(\roman{enumi})} \item Verfeinere $T_{\ell}$ um $\hat T_{\ell}$ zu erhalten \label{alg:adapt:begin} \item Berechne die Galerkin-Lösung $\hat \phi_{\ell} \in P^0(\hat T_{\ell})$ von \eqref{math:slp:gls:galerkin} -\item Berechne Fehlerschätzer $\tilde \mu_{i} := \norm{\varrho^{\ell/2}(\hat \phi_{\ell} - \Pi_{\ell} \hat \phi_{\ell} )}$ +\item Berechne Fehlerschätzer $\tilde \mu_{i} := \norm{\varrho^{1/2}(\hat \phi_{\ell} - \Pi_{\ell} \hat \phi_{\ell} )}$ \item Wähle Teilmenge $M_{\ell} \subseteq T_{\ell}$ wie in Definition \ref{thm:def:mark} \item Verfeinere mindestens die Markierten Elemente $M_{\ell}$ durch Algorithmus \ref{alg:refine} um $\T_{\ell+1}$ zu erhalten \item $\ell \mapsto \ell+1$, gehe zu \ref{alg:adapt:begin} @@ -2206,7 +2218,7 @@ Anhand der \figLineC[] farbenen Linien, also der "`adaptiv anisotropen"' Strateg 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. \noindent -Weiterhin können wir in Abbildung \ref{fig:2DQuad:verfeinern:time} die benötigte Zeit pro Berechnungsschritt ablesen. Zu einem Berechnungsschritt gehört das Aufstellen der Matrix $\hat{\bs V_{\ell}}$, berechnen der Kondition von $\hat{\bs V_{\ell}}$, berechnen der Galerkin-Lösung und berechnen der Fehlerschätzer. Hierbei fällt auf, dass die Wahl der Strategie keinen Einfluss auf die benötigte Zeit hat, sondern nur die Anzahl der Elemente. Für die Berechnung mit 3000 Elementen benötigen alle drei Strategien etwa $10^4$ Sekunden, was fast drei Stunden entspricht. +Weiterhin können wir in Abbildung \ref{fig:2DQuad:verfeinern:time} die benötigte Zeit für einen Berechnungsschritt ablesen. In einem Berechnungsschritt wird die Matrix $\hat V_{\ell}$ und $V_{\ell}$ aufgestellt und die Galerkin-Lösung inklusive aller Fehlerschätzer berechnet. Hierbei fällt auf, dass die Wahl der Strategie keinen Einfluss auf die benötigte Zeit hat, sondern nur die Anzahl der Elemente. Für die Berechnung mit 3000 Elementen benötigen alle drei Strategien etwa $10^4$ Sekunden, was fast drei Stunden entspricht. \noindent Diese Ergebnisse Zeigen also, dass die "`adaptiv anisotrope"' Strategie die beste Konvergenzrate aufweist, wir dafür jedoch eine schlechtere Konditionszahl der Matrix in kauf nehmen müssen. Dies ist letztendlich auf die Größe und Forme der Elemente zurückzuführen. An dieser Stelle wollen wir noch zusätzlich Abbildung \ref{fig:mesh:2DQuad:steps} betrachten, welche das "`adaptiv anisotrop"' verfeinerte Netz nach 12 Schritten darstellt. Denn hier erkennen wir sehr gut, dass diese Strategie das Netz insbesondere an den Singularitäten verfeinert. @@ -2215,7 +2227,7 @@ Diese Ergebnisse Zeigen also, dass die "`adaptiv anisotrope"' Strategie die best Ziel wird es nun sein die Instabilitäten, die bei der analytischen Berechnung auftreten durch Quadratur zu vermeiden. Dafür werden wir vorher noch Berechnungen mit verschiedenen Quadraturgraden genauer untersuchen. \subsubsection{Vergleich verschiedener Quadraturgrade} -Bei der folgenden Berechnung werden wir wieder den Algorithmus \ref{alg:adapt} mit Parametern $\theta=0.5,\nu=0.5$ zum Lösen von \eqref{math:bsp:Quad:gls} verwenden, wobei alle Integrale von $\zeta_Q$-zulässigen Elemente durch die vorgestellte Gauss-Quadratur approximiert werden. Hierbei werden wir jeweils 1, 2, 4 oder 8 Auswertungsstellen verwenden. Alle vier Berechnungsarten werden auf den selben Netzen ausgeführt um die Ergebnisse nicht durch die Wahl des Netzes zu beeinflusse. Zum Berechnen des $\tilde \mu_{\ell}$-Schätzers, welcher die Verfeinerung steuert, werden wir in jedem Schritt die Lösung der Quadratur mit 8 Auswertungsstellen verwenden. +Bei der folgenden Berechnung werden wir wieder den Algorithmus \ref{alg:adapt} mit Parametern $\theta=0.5,\nu=0.5$ zum Lösen von \eqref{math:bsp:Quad:gls} verwenden, wobei alle Integrale von $\zeta_Q$-zulässigen Elemente durch die vorgestellte Gauss-Quadratur approximiert werden. Hierbei werden wir jeweils 1, 2, 4 oder 8 Auswertungsstellen verwenden. Alle vier Berechnungsarten werden auf den selben Netzen ausgeführt um die Ergebnisse nicht durch die Wahl des Netzes zu beeinflussen. Zum Berechnen des $\tilde \mu_{\ell}$-Schätzers, welcher die Verfeinerung steuert, werden wir in jedem Schritt die Lösung der Quadratur mit 8 Auswertungsstellen verwenden. \begin{figure}[ht] @@ -2234,16 +2246,19 @@ Bei der folgenden Berechnung werden wir wieder den Algorithmus \ref{alg:adapt} m In Abbildung \ref{fig:2DQuad:quad:err} haben wir die Ergebnisse der Fehler und Fehlerschätzer für die verschiedenen Quadraturgrade dargestellt. Wir beobachten, dass eine sowie zwei Auswertungsstellen, dargestellt durch die Linien in \figLineA[] und \figLineB[] instabil werden. Für die Quadraturgrade vier und acht dargestellt durch die Linien in \figLineC[] und \figLineD[], erkennen wir, dass die Berechnungen stabil bleiben und das auch für Elementanzahlen bei denen die analytische Berechnung (vergleiche Abbildung \ref{fig:2DQuad:verfeinern}) versagt. \noindent -Da wir möglichst ökonomisch arbeiten wollen, haben wir auch die Berechnungszeiten in Abbildung \ref{fig:2DQuad:quad:time} untersucht. Wie sich leicht erkennen lässt sind die Berechnungszeiten für die verschiedenen Quadraturgrade etwa äquivalent. Lediglich die Quadratur mit 8 Auswertungsstellen benötigt für kleine Elementanzahlen etwas länger, wobei sie für große Anzahlen jedoch fast gleich ist. Daraus lässt sich ablesen, dass die Wahl des Quadraturgrades für große Netze kaum einen Einfluss auf die Berechnungszeit nimmt. +Da wir möglichst ökonomisch arbeiten wollen, haben wir auch die Berechnungszeiten für die Matrix $\hat V_{\ell}$ in Abbildung \ref{fig:2DQuad:quad:time} untersucht. \todo{Wie sich leicht erkennen lässt sind die Berechnungszeiten für die verschiedenen Quadraturgrade etwa äquivalent. Lediglich die Quadratur mit 8 Auswertungsstellen benötigt für kleine Elementanzahlen etwas länger, wobei sie für große Anzahlen jedoch fast gleich ist. Daraus lässt sich ablesen, dass die Wahl des Quadraturgrades für große Netze kaum einen Einfluss auf die Berechnungszeit nimmt.} \noindent Aufgrund dieser Ergebnisse werden wir für die Folgenden Berechnungen einen Quadraturgrad von 4 wählen. Denn wir wollen zum einen die Berechnungszeiten auch für kleine Netze gering halten und zum anderen die Stabilität der Berechnungen sicher stellen. \subsubsection{Vergleich verschiedener Berechnungsarten} -Siehe Abbildung \ref{fig:2DQuad:sem}. +Wie wir für die analytische Berechnung mit adaptiv anisotroper Netzverfeinerung in Abbildung \ref{fig:2DQuad:verfeinern:err} gesehen haben, wird der Fehlerschätzer $\tilde \mu_{\ell}$ ab etwa 3000 Elementen instabil. Deswegen wollen wir an dieser Stelle verschiedene Strategien zur Approximation der Matrix $\hat V_{\ell}$ vorstellen.\\ +Bei den folgenden Berechnungen werden wir wieder den Algorithmus \ref{alg:adapt} mit Parametern $\theta=0.5,\nu=0.5$ zum Lösen von \eqref{math:bsp:Quad:gls} verwenden, wobei für alle auftretenden Gauss-Quadraturen der Grad 4 gewählt wurde. Alle Berechnungsarten werden auf den selben Netzen ausgeführt um die Ergebnisse nicht durch die Wahl des Netzes zu beeinflussen.\\ +Wir werden nun die Matrix $\hat V_{\ell}$ in der "`analytischen"' Strategie durch die in Kapitel \ref{sec:analyt} vorgestellten Stammfunktionen analytisch berechnen. +Dann werden wir für $\zeta_E$-zulässigen Elemente wie in Kapitel \ref{sec:semi:QE} das Integral über das kleinere Element durch Gauss-Quadratur ersetzen. Alle anderen Elemente werden bei dieser "`semianalytischen QE"' Strategie weiterhin analytisch berechnet. +In der letzten Strategie "`volle Quadratur"' werden wir alle $\zeta_Q$-zulässige Elemente wie in Kapitel \ref{sec:semi:Q} durch Gauss-Quadratur ersetzen und unzulässige wieder analytisch. Da wir annehmen, dass die letzte Strategie stabil sein wird, werden wir den zugehörigen $\tilde \mu_{\ell}$ Schätzer zum Steuern der Verfeinerung verwenden. \begin{figure}[ht] - \centering \subfloat[Fehler und Fehlerschätzer für das Quadrat \label{fig:2DQuad:sem:err}]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_error}} % \subfloat[Seitenverhältnisse auf dem Quadrat \label{fig:2DQuad:quad:hminmax}]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_hminmax}} @@ -2254,12 +2269,14 @@ Siehe Abbildung \ref{fig:2DQuad:sem}. \label{fig:2DQuad:sem} \end{figure} + + \subsection{Beispiel Fischer Würfel} Im Folgenden werden wir die Laplace-Gleichung \begin{align}\label{math:bsp:FichCube:gls} \begin{aligned} - \varDelta u &= 0 \quad\text{ in } \Omega \subset \R^3, \\ -u &= g \quad \text{ auf }\Gamma, +u_{|\Gamma} &= 1 \quad \text{ auf }\Gamma, \end{aligned} \end{align} mit dem Startnetz aus Abbildung \ref{fig:mesh:3DFichCube:start}, einem Fischer Würfel genauer betrachten. @@ -2312,7 +2329,7 @@ Siehe Abbildung \ref{fig:3DFichCube:sem}. % % \item $\enorm{\hat \phi_{\ell}}$ % % \item $\enorm{\phi_{\ell}}$ % \item $error_{i} = \sqrt{\enorm{\phi}^2 - \enorm{\phi_{\ell}}^2}$ -% \item $\mu_{i} = \norm{\varrho^{\ell/2}(\hat \phi_{\ell} - \phi_{\ell} )}$ +% \item $\mu_{i} = \norm{\varrho^{1/2}(\hat \phi_{\ell} - \phi_{\ell} )}$ % \item $\eta_{i} = \enorm{\hat \phi_{\ell} - \phi_{\ell}}$ % \item $\kappa_{i} = \enorm{\hat \phi_{\ell}}-\enorm{\hat \phi_{\ell-1}}$ % \item $\kappa2_{i} = \enorm{\phi_{\ell}}-\enorm{\phi_{\ell-1}}$ diff --git a/src/A_plots.m b/src/A_plots.m index 3dd1f07..a3d746e 100644 --- a/src/A_plots.m +++ b/src/A_plots.m @@ -154,7 +154,7 @@ i=0; % shift2 = shift2+shift2/10; % error*(eta(l)-shift2)/error(l) -first = 1 +find(([ (G_D(2:end,2+rows*i)-G_D(1:end-1,2+rows*i))./G_D(2:end,2+rows*i)])>=0,1)-1 +first = 1 +find(([ (G_D(2:end,2+rows*i)-G_D(1:end-1,2+rows*i))./G_D(2:end,2+rows*i)])>=0,1) Ferr = (X(first,i+1)+X(first-1,i+1))/2; loglog(X(:,i+1),G_D(:,2+rows*i),type2sym(i*3+1), ... diff --git a/src/compute.m b/src/compute.m index 845d6a0..2290583 100644 --- a/src/compute.m +++ b/src/compute.m @@ -170,10 +170,13 @@ for j = 1:times eta = abs(xe_fine-xe); end_time = toc - start_time; - disp(['Relative Zeit ' t2str((build_time - start_time)/end_time)... - ' absolute Zeit ' t2str(build_time - start_time)]); + disp(['Relative Zeit ' num2str(100*(build_time - start_time)/end_time)... + '% absolute Zeit ' t2str(build_time - start_time)]); end_time = build_time - start_time; + save ([ fileo int2str(size(data,1)) '_' int2str(typ(i))]... + , 'tmu') + dataS = [dataS ... typ(i) ... berechnet mit Typ sqrt(sum(tmu))... tilde mu -- 2.47.3