\subfloat[Zeit]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_time}}
\end{figure}
+\begin{figure}[ht]
+
+\caption{2D Quad im Vergleich Quadratur $V\phi = 1$}
+\centering
+\label{fig:exmpl_2DQuad_Q}
+\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_error}}
+\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_hminmax}}\\
+\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_cond}}
+\subfloat[Zeit]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_time}}
+\end{figure}
+
\begin{figure}[ht]
\caption{2D Quad adaptiv anisotrop vollanalytisch $V\phi = 1$}
\centering
% \subfloat[Zeit]{\includegraphics[width=0.33\textwidth]{fig/1t05n05_2DQuad_time}}
\end{figure}
+\begin{figure}[ht]
+\caption{2D LShape adaptiv anisotrop vollanalytisch $V\phi = 1$}
+\centering
+\label{fig:exmplAA_2DLShape_A}
+\subfloat[Fehler]{\includegraphics[width=0.7\textwidth]{fig/1t05n05_2DLShape_error}}\\
+\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_2DLShape_hminmax}}
+\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_2DLShape_cond}}
+% \subfloat[Zeit]{\includegraphics[width=0.33\textwidth]{fig/1t05n05_2DLShape_time}}
+\end{figure}
+
\begin{figure}[ht]
\caption{3D FichCube adaptiv anisotrop vollanalytisch $V\phi = 1$}
\centering
\numberwithin{defi}{section}
% \numberwithin{bew}{section}
% \numberwithin{sat}{section}
+\numberwithin{equation}{section}
\psfrag{T}{\scriptsize $T$}
\noindent
In dieser Arbeit beschäftigen wir uns mit der Randelementmethode für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen
-\begin{align*}
+\begin{align}
+\begin{aligned}
- \varDelta u &= 0 \quad\text{ in } \Omega \subset \R^3,\\
u &= g \quad \text{ auf }\Gamma,
-\end{align*}
+\end{aligned}
+\end{align}
wobei $\varDelta u := \partial_x^2u+\partial_y^2u+\partial_z^2u$ den Laplace-Operator bezeichnet und $\Omega \subset \R^3$ eine beschränkte Teilmenge von $\R^3$ mit Lipschitz-Rand $\Gamma := \partial \Omega$ ist.\\
In Abschnitt 2 stellen wir zunächst die Randelementmethode für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen vor. Dabei verwenden wir den indirekten Ansatz, um anschließend mithilfe des Galerkin-Verfahrens die Gleichung zu lösen. An dieser Stelle werden wir auch kurz die Parametrisierung des Randes vorstellen. Denn wir werden im Folgenden den Rand in affine achsenorientierte Rechtecke $T$ zerlegen, das heißt die Punkte in einem Rechteck liegen in einer zu den Achsen des Koordinatensystem parallelen Ebene.\\
In Abschnitt 3 werden wir uns mit der approximativen Berechnung des Doppelintegrals
\noindent
An dieser Stelle betrachten wir die homogene Laplace-Gleichung mit Dirichlet-\-Rand\-bedingungen
-\begin{align*} \label{math:num:lapGLS}
+\begin{align}\label{math:slp:lapGLS}
+\begin{aligned}
- \varDelta u &= 0 \quad\text{ in } \Omega \subset \R^3, \\
u &= g \quad \text{ auf }\Gamma,
-\end{align*}
-wobei $\varDelta u := \partial_x^2u+\partial_y^2u+\partial_z^2u$ den Laplace-Operator bezeichnet und $\Omega \subset \R^3$ eine beschränkte Teilmenge von $\R^3$ mit Lipschitz-Rand $\Gamma := \partial \Omega$ ist.\\
-
-\begin{align}
- V \phi := - \frac 1 {4\pi} \int_{T_j} \int_{T_k} \frac {1}{\abs{\bs x - \bs y}} \phi(\bs y) ds_{\bs y} ds_{\bs x}
+\end{aligned}
\end{align}
+wobei $\varDelta u := \partial_x^2u+\partial_y^2u+\partial_z^2u$ den Laplace-Operator bezeichnet und $\Omega$ eine beschränkte Teilmenge von $\R^3$ mit Lipschitz-Rand $\Gamma := \partial \Omega$ ist.
+% \begin{align}
+% V \phi := \frac 1 {4\pi} \int_{T_j} \int_{T_k} \frac {1}{\abs{\bs x - \bs y}} \phi(\bs y) ds_{\bs y} ds_{\bs x}
+% \end{align}
+% \subsection{MAYR}
+% Sei $\tilde V$ gegeben durch
+% \begin{align}
+% \tilde V \phi = \frac 1 {4\pi} \int_{T_j}\int_{T_k} \frac 1 {\abs{\bs x -\bs y}} \phi(\bs y) ds_{\bs x}ds_{\bs y}
+% \end{align}
+% und bezeichne $\gamma_0 \in L(H^1(\Omega)),H^{1/2}(\Gamma))$ den Spuroperator, der einer $C^{\infty}(\Omega)$-Funktion eine Funktion zuordnet sodass
+% \begin{align}
+% \gamma_0 u = u|_{\Gamma}.
+% \end{align}
+% Gemäß \todo{ref} gilt mit diesen Bezeichnungen
+% \begin{align}
+% -\varDelta \tilde V \phi = 0 \quad \text{für alle }\phi \in H^{1/2}(\Gamma).
+% \end{align}
+% Wir machen nun den sogenannten indirekten Ansat $u = \tilde V\phi$. Wegen \todo{$(4)$} ist die Laplace-Gleichung erfüllt, und es gilt
+% \begin{align}
+% V\phi=g,
+% \end{align}
+% wobei $V := \gamma_0\tilde V \in L(H^{-1/2}(\Gamma),H^{1/2}(\Gamma))$. Ziel ist es nun aus \todo{$(5)$} eine Funktion $\phi$ zu bestimmen, die die obige Gleichung erfüllt, denn dann ist $\tilde V\phi$ die Lösung des Problems.
+\subsection{Galerkin-Verfahren}
+Die Fundamentallösung des Laplaceoperators \todo{cite Steinbach} ist für $\Omega \subset \R^3$ gegeben durch
+\begin{align*} % \label{math:slp:fundamental}
+ G(\bs z) := \frac 1 {4 \pi \abs{\bs z}}.
+\end{align*}
+Weiterhin sei $\tilde V$ gegeben durch
+\begin{align*} % \label{math:slp:tildeV}
+ \tilde V\phi(\bs x) := \int_{\Gamma} G(\bs x -\bs y)\phi(\bs y) ds_{\bs y} \quad \text{für } \bs x \in \Omega
+\end{align*}
+und bezeichne $\gamma_0 \in L(H^1(\Omega)),H^{1/2}(\Gamma))$ den Spuroperator, der einer $C^{\infty}(\Omega)$-Funktion eine Funktion zuordnet sodass
+\begin{align*} % \label{math:slp:spurOp}
+ \gamma_0 u = u|_{\Gamma}.
+\end{align*}
+In der schwachen Formulierung lautet die Laplace-Gleichung
+\begin{align*} % \label{math:slp:lapGLS:weak}
+- \varDelta u &= 0 \quad \text{ in } H^{-1}(\Omega), \\
+\gamma_0 u &= g \quad \text{ auf }H^{1/2}(\Gamma),
+\end{align*}
+Gemäß \todo{cite Steinbach} kann $\tilde V$ auch als Operator $\tilde V \in L(H^{-1/2}(\Gamma),H^{1}(\Omega))$ aufgefasst werden, und es gilt
+\begin{align*}
+ -\varDelta\tilde V\phi = 0 \in H^{-1}(\Omega) \quad \text{für alle }\phi \in H^{-1/2}(\Gamma)
+\end{align*}
+Insbesondere ist auch der Spuroperator
+\begin{align*}
+ \gamma_0 \tilde V \in L(H^{-1/2}(\Gamma),H^{1/2}(\Gamma))
+\end{align*}
+wohldefiniert.
+Wir machen nun den sogenannten indirekten Ansatz $u = \tilde V\phi$, wodurch
+\begin{align}\label{math:slp:gls}
+ V \phi = g
+\end{align}
+mit $V := \gamma_0\tilde V$ gilt. Ziel ist es nun, aus \eqref{math:slp:gls} eine Funktion $\phi$ zu bestimmen, die die obige Gleichung erfüllt, denn dann ist $\tilde V\phi$ die Lösung des Problems \eqref{math:slp:lapGLS}.
-% \noindent
+\subsection{old Galerkin}
% Wir wissen, dass die Laplace-Gleichung erfüllt wird durch:
% \begin{align}
-% V\phi &= f \label{Formel}
+% V\phi &= g \label{Formel}
% \end{align}
% % $\abs{u(x)} = O(\abs{x}^{-1}) \nonumber$
% Sei nun die $G$ Fundamentallösung,
% \begin{align}
% V : H^{-1/2+s}(\Gamma) &\rightarrow H^{1/2+s}(\Gamma)& \text{mit } s\in [-1/2,1/2]
% \end{align}
-% \begin{lem}[Lax-Milgram]
-% Sei eine Abbildung $a: X\times X \rightarrow \R$ wobei $X$ ein reflexiver Banachraum. Und gilt:
-% \begin{itemize}
-% \item $a$ stetig, d.h. : $\abs{a(x,y)} \leq C \cdot \norm{x} \cdot \norm{y}$
-% \item $a$ eliptisch, d.h. : $a(x,x) \geq X \cdot \norm{x}^2$
-% \end{itemize}
-% So folgt daraus
-% $\forall f \in X'$ $\exists $ eindeutiges $x \in X$ mit $a(x,\cdot) = f$.
-% \end{lem}
-% \begin{defi}
-% Sei $\langle \cdot, \cdot \rangle$ das erweiterte $L_2$ - Skalarprodukt
-% \end{defi}
-% Wendet man nun das Lax-Milgram Lemma auf die schwache Formulierung an,
-% \begin{align}
-% \langle V \phi, \psi\rangle &= \langle f,\psi\rangle & \psi \in H^{-1/2}\\
-% a(\phi,\psi) &:= \langle V\phi,\psi\rangle&:H^{-1/2}(\Gamma)\times H^{-1/2}(\Gamma) \rightarrow \R
-% \end{align}
-% zeigen wir noch, dass:
-% \begin{itemize}
-% \item $H^{-1/2}(\Gamma)$ ist reflexiver Banachraum, welches aus der Definition von $H^{-1/2}$ folgt
-% \item $ \abs{\langle V\phi,\psi \rangle} \leq C \cdot \norm{\phi}_{H^{-1/2}} \cdot \norm{\psi}_{H^{-1/2}}$
-% \item $ \langle V\phi,\phi \rangle \geq C \cdot \norm{\phi}_{H^{-1/2}}^2$
-% \end{itemize}
-% Daraus folgt nun dass, $\forall f \in H^{-1/2}$ $\exists$ eindeutige Lösung $\phi \in H^{-1/2}(\Gamma)$ von
-% \begin{align}
-% \langle V \phi, \psi \rangle &=\langle f, \psi \rangle & \forall \psi \in H^{-1/2}(\Gamma)
-% \end{align}
-% Wollen wir nun das Galerkin-Verfahren anwenden benötigen wir die schwache Formulierung:
-% \begin{align}
-% \int_{\Gamma} V \phi(x) \cdot \psi(x) dx &= \int_{\Gamma} f(x)\cdot\psi(x) dx
-% \end{align}
-% Nun wählen wir einen endlich-dimensionalen Teilraum $P^0(\T_n) \subseteq H^{-1/2}$ und betrachten
-%
-% \begin{defi}\label{1}
-% \begin{align}
-% \langle V\phi_{\ell},\psi_{\ell} \rangle & = \langle f,\psi_{\ell} \rangle& \forall \psi_{\ell} \in P^0(\T_{\ell})
-% \end{align}
-% \end{defi}
-% Gesucht ist jetzt also $\phi_{\ell} \in P^0(\T_{\ell})$
-%
-% \noindent
-% Aus dem Max-Milgram Lemma und $X = P^0(\T_{\ell})$ folgt wiederum, es $\exists$ eindeutige Lösung $\phi_{\ell} \in P^0(\T_{\ell})$, da $\psi_{\ell} \in P^0(\T_{\ell}),\phi_{\ell} \in P^0(\T_{\ell})$.
-% \begin{defi}
-% Sei nun die Basis von $P^0(\T_{\ell})$ die charakteristischen Funktionen
-% \begin{align}
-% \{\chi_T | T\in\T_{\ell}\} &= \{\chi_{T_1},\chi_{T_2},\dots\}
-% \end{align}
-% \end{defi}
-% So können wir mit $N = \dim P^0(\T_{\ell})$ und $\psi_{\ell},\phi_{\ell}\in\R$ wobei $l\in \{1\dots N\}$ schreiben
-% \begin{align}
-% \psi_{\ell} &= \sum_{l=1}^N \psi_{\ell} \cdot \chi_{T_{\ell}} \\
-% \phi_{\ell} &= \sum_{l=1}^N \phi_{\ell} \cdot \chi_{T_{\ell}}
-% \end{align}
-% Dadurch können wir Definition \ref{1} nun einfacher Lösen durch:
-% \begin{align}
-% \langle V \phi_{\ell},\chi_k\rangle & = \langle f, \chi_k \rangle & k = 1\dots N
-% \end{align}
-% Aufgrund der Linearität von $V$ und dem Skalarprodukt schreiben wir:
-% \begin{align}
-% \sum_{l=1}^N\langle V\phi_{\ell}\chi_{\ell},\chi_k\rangle & = \langle f,\chi_k\rangle
-% \end{align}
-% welches sich wiederum so schreiben lässt
-% \begin{defi}[Galerkinapproximation]
-% \begin{align}
-% \ul{\ul{V}} \cdot \ul{\phi} = \ul{f}
-% \end{align}
-% wobei $\ul{\ul{V}}\in R^{N \times N},\ul{\phi}\in\R^{N \times 1},\ul{f}\in\R^{N \times 1}$
-% \begin{align}
-% \ul{\ul{V}}_{\ell,k} &= \langle V \chi_{\ell}, \chi_k \rangle\\
-% \ul{\phi}_{\ell} &= \phi_{\ell} \nonumber\\
-% \ul{f}_k &= \langle f, \chi_k\rangle \nonumber
-% \end{align}
-% Damit ist $\phi_{\ell}$ die Galerkinapproximation an $\phi$
-% \end{defi}
+\begin{lem}[Lax-Milgram]
+Sei eine Abbildung $a: X\times X \rightarrow \R$ wobei $X$ ein reflexiver Banachraum. Und gilt:
+\begin{itemize}
+ \item $a$ stetig, d.h. : $\abs{a(x,y)} \leq C \cdot \norm{x} \cdot \norm{y}$
+ \item $a$ eliptisch, d.h. : $a(x,x) \geq X \cdot \norm{x}^2$
+\end{itemize}
+So folgt daraus
+$\forall f \in X'$ $\exists $ eindeutiges $x \in X$ mit $a(x,\cdot) = f$.
+\end{lem}
+\begin{defi}
+Sei $\langle \cdot, \cdot \rangle$ das erweiterte $L_2$ - Skalarprodukt
+\end{defi}
+Wendet man nun das Lax-Milgram Lemma auf die schwache Formulierung an,
+\begin{align}
+ \langle V \phi, \psi\rangle &= \langle f,\psi\rangle & \psi \in H^{-1/2}\\
+a(\phi,\psi) &:= \langle V\phi,\psi\rangle&:H^{-1/2}(\Gamma)\times H^{-1/2}(\Gamma) \rightarrow \R
+\end{align}
+zeigen wir noch, dass:
+\begin{itemize}
+ \item $H^{-1/2}(\Gamma)$ ist reflexiver Banachraum, welches aus der Definition von $H^{-1/2}$ folgt
+ \item $ \abs{\langle V\phi,\psi \rangle} \leq C \cdot \norm{\phi}_{H^{-1/2}} \cdot \norm{\psi}_{H^{-1/2}}$
+ \item $ \langle V\phi,\phi \rangle \geq C \cdot \norm{\phi}_{H^{-1/2}}^2$
+\end{itemize}
+Daraus folgt nun dass, $\forall f \in H^{-1/2}$ $\exists$ eindeutige Lösung $\phi \in H^{-1/2}(\Gamma)$ von
+\begin{align}
+ \langle V \phi, \psi \rangle &=\langle f, \psi \rangle & \forall \psi \in H^{-1/2}(\Gamma)
+\end{align}
+Wollen wir nun das Galerkin-Verfahren anwenden benötigen wir die schwache Formulierung:
+\begin{align}
+ \int_{\Gamma} V \phi(x) \cdot \psi(x) dx &= \int_{\Gamma} f(x)\cdot\psi(x) dx
+\end{align}
+Nun wählen wir einen endlich-dimensionalen Teilraum $P^0(\T_n) \subseteq H^{-1/2}$ und betrachten
+
+\begin{defi}\label{1}
+\begin{align}
+ \langle V\phi_{\ell},\psi_{\ell} \rangle & = \langle f,\psi_{\ell} \rangle& \forall \psi_{\ell} \in P^0(\T_{\ell})
+\end{align}
+\end{defi}
+Gesucht ist jetzt also $\phi_{\ell} \in P^0(\T_{\ell})$
+
+\noindent
+Aus dem Max-Milgram Lemma und $X = P^0(\T_{\ell})$ folgt wiederum, es $\exists$ eindeutige Lösung $\phi_{\ell} \in P^0(\T_{\ell})$, da $\psi_{\ell} \in P^0(\T_{\ell}),\phi_{\ell} \in P^0(\T_{\ell})$.
+\begin{defi}
+ Sei nun die Basis von $P^0(\T_{\ell})$ die charakteristischen Funktionen
+\begin{align}
+ \{\chi_T | T\in\T_{\ell}\} &= \{\chi_{T_1},\chi_{T_2},\dots\}
+\end{align}
+\end{defi}
+So können wir mit $N = \dim P^0(\T_{\ell})$ und $\psi_{\ell},\phi_{\ell}\in\R$ wobei $l\in \{1\dots N\}$ schreiben
+\begin{align}
+ \psi_{\ell} &= \sum_{l=1}^N \psi_{\ell} \cdot \chi_{T_{\ell}} \\
+ \phi_{\ell} &= \sum_{l=1}^N \phi_{\ell} \cdot \chi_{T_{\ell}}
+\end{align}
+Dadurch können wir Definition \ref{1} nun einfacher Lösen durch:
+\begin{align}
+ \langle V \phi_{\ell},\chi_k\rangle & = \langle f, \chi_k \rangle & k = 1\dots N
+\end{align}
+Aufgrund der Linearität von $V$ und dem Skalarprodukt schreiben wir:
+\begin{align}
+\sum_{l=1}^N\langle V\phi_{\ell}\chi_{\ell},\chi_k\rangle & = \langle f,\chi_k\rangle
+\end{align}
+welches sich wiederum so schreiben lässt
+\begin{defi}[Galerkinapproximation]
+\begin{align}
+\ul{\ul{V}} \cdot \ul{\phi} = \ul{f}
+\end{align}
+wobei $\ul{\ul{V}}\in R^{N \times N},\ul{\phi}\in\R^{N \times 1},\ul{f}\in\R^{N \times 1}$
+\begin{align}
+ \ul{\ul{V}}_{\ell,k} &= \langle V \chi_{\ell}, \chi_k \rangle\\
+ \ul{\phi}_{\ell} &= \phi_{\ell} \nonumber\\
+ \ul{f}_k &= \langle f, \chi_k\rangle \nonumber
+\end{align}
+Damit ist $\phi_{\ell}$ die Galerkinapproximation an $\phi$
+\end{defi}
% \subsection{Vorkonditionieren}
\begin{beweis} Die Ableitung der Funktion $\kappa \circ g : \R^4 \to \R^6 \to \R$ ist
\begin{align*}
- \partial(\kappa \circ g)(\lambda) = \partial \kappa(g(\lambda)) \circ \partial g(\lambda) \qquad \in \R^{1\times 4}.
+ \partial(\kappa \circ g)(\lambda) = \partial \kappa(g(\lambda)) \circ \partial g(\lambda) \quad \in \R^{1\times 4}.
\end{align*}
Mithilfe der Jacobimatrizen $A := \partial \kappa(g(\lambda)) \in \R^{1\times 6}$, $B := \partial g(\lambda) \in\R^{6\times 4}$ untersuchen wir zunächst die partiellen Ableitungen
\begin{align}
\end{align*}
Die Stammfunktion ist trotz Parametrisierung stetig. Daher gilt für $T_k$ mit geringem Durchmesser $\diam(T_k)$ aber $\tilde F_{\bs x}(\lambda) \approx \tilde F_{\bs x}(\tilde \lambda)$ für alle $\lambda,\tilde \lambda \in [0,1]^2$, wodurch für den Ausdruck
\begin{align*}
- \tilde F_{\bs x}(0,0) - \tilde F_{\bs x}(1,0) -\tilde F_{\bs x}(0,1) + \tilde F_{\bs x}(1,1) = \int_{[0,1]^2} \frac 1 {\abs{\bs x -\gamma_k(\bs \lambda)}} d\bs \lambda = \int_{T_k} \frac 1 {\abs{\bs x -\bs y}} ds_{\bs y}
+ \tilde F_{\bs x}(0,0) - \tilde F_{\bs x}(1,0) -\tilde F_{\bs x}(0,1) + \tilde F_{\bs x}(1,1)
+ = \int_{[0,1]^2} \frac 1 {\abs{\bs x -\gamma_k(\bs \lambda)}} d\bs \lambda = \int_{T_k} \frac 1 {\abs{\bs x -\bs y}} ds_{\bs y}
\end{align*}
starke Auslöschungseffekte auftreten.\\
- Deshalb werden wir für zulässige Randelemente die Integrale geeignet durch Gauss-Quadratur ersetzen. Dabei wird der Integrand mehrfach ausgewehrtet und mit stets positiven Quadraturgewichten multipliziert. Da der Integrand in der Praxis immer das gleiche Vorzeichen hat, werden während der Gauss-Quadratur lediglich Werte gleichen Vorzeichens addiert, wodurch keine Auslöschung auftritt. Deshalb lässt sich beobachten, dass die Auswertung mittels Gauss-Quadratur für zulässige Randstücke genauer ist als die Berechnung mithilfe der Stammfunktion.\\
+ Deshalb werden wir für zulässige Randelemente Integrale geeignet durch Gauss-Quadratur ersetzen. Dabei wird der Integrand mehrfach ausgewehrtet und mit stets positiven Quadraturgewichten multipliziert. Da der Integrand in der Praxis immer das gleiche Vorzeichen hat, werden während der Gauss-Quadratur lediglich Werte gleichen Vorzeichens addiert, wodurch keine Auslöschung auftritt. Deshalb lässt sich beobachten, dass die Auswertung mittels Gauss-Quadratur für zulässige Randstücke genauer ist als die Berechnung mithilfe der Stammfunktion.\\
Da wir die Integrationsreihenfolge laut Satz \ref{thm:sem:switch} beliebig vertauschen dürfen, ist es sinnvoll über die großen Integrationsbereiche zu erst zu integrieren und die Integration über die kleinen nach außen zu stellen. Denn dadurch wird jene Operation, durch die starke Auslöschungseffekte auftreten, ans Ende der Berechnung gestellt.\\
Weiterhin kann die Berechnung durch Quadratur abhängig vom gewählten Quadraturgrad sehr aufwändig werden, welches auf die hohe Anzahl der Auswertungsstellen zurückzuführen ist. Für die Quadratur über beispielsweise zwei Integrale mit einem Quadraturgrad von 8 werden dann schon $8^2 = 64$ Auswertungen benötigt, über vier Integrale hingegen schon $8^4 = 4096$. Deshalb werden wir nicht nur die Strategie betrachten in der für alle zulässigen Randelemente alle Integrale durch Quadratur ersetzt werden, sondern auch eine in der nur ein Teil der auftretenden Integrale geeignet durch Gauss-Quadratur ersetzt wird.
\end{bem}
\end{align*}
gilt nach \cite[Seite 13]{mai:3dbem}
\begin{align}
- \begin{split}
+ \begin{aligned}
F_{p}(x_1&,x_2,y_1,y_2,\bs \delta)\\
=& (x_1-y_1-\delta_1)(x_2-y_2-\delta_2)G(-1/2;x_1,x_2;y_1+\delta_1,y_2+\delta_2,\delta_3)\\
&-(x_1-y_1-\delta_1) g(1/2;x_1;y_1+\delta_1;\{(x_2-y_2-\delta_2)^2 + \delta_3^2\}^{1/2})\\
&-(x_2-y_2-\delta_2) g(1/2;x_2;y_2+\delta_2;\{(x_1-y_1-\delta_1)^2 + \delta_3^2\}^{1/2})\\
&+\frac 1 3 \{(x_1-y_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+\delta_3^2\}^{3/2}.
- \end{split}
+ \end{aligned}
\end{align}
\end{lem}
\end{align*}
gilt nach \cite{mai:3dbem}
\begin{align}
- \begin{split}
+ \begin{aligned}
2F_{o}(x_1&,x_2,y_2,y_3,\bs \delta)\\
=&-G(1/2;y_3,x_1;-\delta_3,\delta_1,x_2-y_2-\delta_2)\\
&-(x_1-\delta_1)(x_2-y_2-\delta_2)G(-1/2;x_2,y_3;y_2+\delta_2,-\delta_3,x_1-\delta_1)\\
&+(x_1-\delta_1)g(1/2);y_3;-\delta_3,\{(x_1-\delta_1)^2+(x_2-y_2-\delta_2)^2\}^{1/2})\\
&-(y_3-\delta_3)(x_2-y_2-\delta_2)G(-1/2;x_1,x_2;\delta_1,y_2+\delta_2,-y_3-\delta_3)\\
&+(y_3-\delta_3)g(1/2);x_1;\delta_1,\{(x_2-y_2-\delta_2)^2+(y_3+\delta_3)^2\}^{1/2}).
- \end{split}
+ \end{aligned}
\end{align}
\end{lem}
\end{align*}
und unter der Saturationsannahme
\begin{align*}
-\norm{\phi -\hat \phi_{\ell}} &\leq C_{sat} \cdot \norm{\phi - \phi_{\ell}} \qquad \text{mit }0 < C_{sat} < 1
+\norm{\phi -\hat \phi_{\ell}} &\leq C_{sat} \cdot \norm{\phi - \phi_{\ell}} \quad \text{mit }0 < C_{sat} < 1
\end{align*}
zuverlässig
\begin{align*}
\begin{align*}
marked_j :=
\begin{cases} %
- 3 \qquad (\text{vertikal}) &\text{falls } \nu \abs{ C_j^{(3)}} \geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(4)}}^2}\\ %
- 4 \qquad (\text{horizontal}) &\text{falls } \nu \abs{ C_j^{(4)}} \geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(3)}}^2}\\ %
- 2 \qquad (\text{isotrop}) & \text{sonst.}
+ 3 \quad (\text{vertikal}) &\text{falls } \nu \abs{ C_j^{(3)}} \geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(4)}}^2}\\ %
+ 4 \quad (\text{horizontal}) &\text{falls } \nu \abs{ C_j^{(4)}} \geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(3)}}^2}\\ %
+ 2 \quad (\text{isotrop}) & \text{sonst.}
\end{cases}
\end{align*}
Weiterhin sei $marked_j = 1$ für alle $T_j \in \T_{\ell} \backslash \tilde M_{\ell}$.
\end{defi}
Die Funktion
\begin{align*}
-marked = mark(xF2S, tmu, theta, nu); \qquad \text{mit } xF2S := x_{fine}[F2S]
+marked = mark(xF2S, tmu, theta, nu); \quad \text{mit } xF2S := x_{fine}[F2S]
\end{align*}
implementiert die Definitionen zum Bestimmen der Markierung.
\bibliographystyle{gerabbrv}
\bibliography{doc}
-
\end{document}
\ No newline at end of file
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gr
/c9 { 0.000000 0.600000 0.600000 sr} bdef
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/c10 { 0.600000 0.000000 0.600000 sr} bdef
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gr
/c9 { 0.000000 0.600000 0.600000 sr} bdef
c9
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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gr
/c10 { 0.600000 0.000000 0.600000 sr} bdef
c10
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
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0 j
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
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0 j
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
DD
/c11 { 0.900000 0.600000 0.000000 sr} bdef
c11
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-99 29 92 28 97 28 131 39 114 34 152 45 120 35 219 65
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DO
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-99 14 92 14 97 14 131 20 114 17 152 22 120 18 219 32
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DA
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c2
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gr
c2
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
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gr
/c10 { 0.600000 0.000000 0.600000 sr} bdef
c10
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
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DP
gr
DA
c2
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+
+end %%Color Dict
+
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+29 50 -58 0 29 -50 3897 889 4 MP
+DP
+29 50 -58 0 29 -50 4019 749 4 MP
+DP
+29 50 -58 0 29 -50 4139 610 4 MP
+DP
+gr
+
+/c10 { 0.600000 0.000000 0.600000 sr} bdef
+c10
+120 -138 122 -139 107 -124 99 -113 97 -111 94 -105 87 -105 72 -78
+55 -75 84 -114 54 -66 64 -97 60 -62 61 -77 61 -54 45 -35
+62 -50 48 -32 52 -25 70 -72 51 -20 90 -71 71 -36 103 -29
+123 -84 190 -86 254 -219 560 -139 1183 2931 29 MP stroke
+gs 1132 524 3059 2459 MR c np
+1166 2914 mt 1200 2948 L
+1200 2914 mt 1166 2948 L
+1726 2775 mt 1760 2809 L
+1760 2775 mt 1726 2809 L
+1980 2556 mt 2014 2590 L
+2014 2556 mt 1980 2590 L
+2170 2470 mt 2204 2504 L
+2204 2470 mt 2170 2504 L
+2293 2386 mt 2327 2420 L
+2327 2386 mt 2293 2420 L
+2396 2357 mt 2430 2391 L
+2430 2357 mt 2396 2391 L
+2467 2321 mt 2501 2355 L
+2501 2321 mt 2467 2355 L
+2557 2250 mt 2591 2284 L
+2591 2250 mt 2557 2284 L
+2608 2230 mt 2642 2264 L
+2642 2230 mt 2608 2264 L
+2678 2158 mt 2712 2192 L
+2712 2158 mt 2678 2192 L
+2730 2133 mt 2764 2167 L
+2764 2133 mt 2730 2167 L
+2778 2101 mt 2812 2135 L
+2812 2101 mt 2778 2135 L
+2840 2051 mt 2874 2085 L
+2874 2051 mt 2840 2085 L
+2885 2016 mt 2919 2050 L
+2919 2016 mt 2885 2050 L
+2946 1962 mt 2980 1996 L
+2980 1962 mt 2946 1996 L
+3007 1885 mt 3041 1919 L
+3041 1885 mt 3007 1919 L
+3067 1823 mt 3101 1857 L
+3101 1823 mt 3067 1857 L
+3131 1726 mt 3165 1760 L
+3165 1726 mt 3131 1760 L
+3185 1660 mt 3219 1694 L
+3219 1660 mt 3185 1694 L
+3269 1546 mt 3303 1580 L
+3303 1546 mt 3269 1580 L
+3324 1471 mt 3358 1505 L
+3358 1471 mt 3324 1505 L
+3396 1393 mt 3430 1427 L
+3430 1393 mt 3396 1427 L
+3483 1288 mt 3517 1322 L
+3517 1288 mt 3483 1322 L
+3577 1183 mt 3611 1217 L
+3611 1183 mt 3577 1217 L
+3674 1072 mt 3708 1106 L
+3708 1072 mt 3674 1106 L
+3773 959 mt 3807 993 L
+3807 959 mt 3773 993 L
+3880 835 mt 3914 869 L
+3914 835 mt 3880 869 L
+4002 696 mt 4036 730 L
+4036 696 mt 4002 730 L
+4122 558 mt 4156 592 L
+4156 558 mt 4122 592 L
+gr
+
+/c11 { 0.000000 0.000000 0.900000 sr} bdef
+c11
+120 -126 122 -126 107 -109 99 -99 97 -94 94 -91 87 -80 72 -67
+55 -55 84 -83 54 -44 64 -56 60 -40 61 -67 61 -40 45 -40
+62 -45 48 -48 52 -23 70 -79 51 -14 90 -74 71 -19 103 -96
+123 -100 190 -126 254 -260 560 -269 1183 2902 29 MP stroke
+gs 1132 481 3059 2473 MR c np
+16 W
+1183 2902 PD
+16 W
+1743 2633 PD
+16 W
+1997 2373 PD
+16 W
+2187 2247 PD
+16 W
+2310 2147 PD
+16 W
+2413 2051 PD
+16 W
+2484 2032 PD
+16 W
+2574 1958 PD
+16 W
+2625 1944 PD
+16 W
+2695 1865 PD
+16 W
+2747 1842 PD
+16 W
+2795 1794 PD
+16 W
+2857 1749 PD
+16 W
+2902 1709 PD
+16 W
+2963 1669 PD
+16 W
+3024 1602 PD
+16 W
+3084 1562 PD
+16 W
+3148 1506 PD
+16 W
+3202 1462 PD
+16 W
+3286 1379 PD
+16 W
+3341 1324 PD
+16 W
+3413 1257 PD
+16 W
+3500 1177 PD
+16 W
+3594 1086 PD
+16 W
+3691 992 PD
+16 W
+3790 893 PD
+16 W
+3897 784 PD
+16 W
+4019 658 PD
+16 W
+4139 532 PD
+gr
+
+gr
+
+c11
+0 sg
+%%IncludeResource: font Helvetica
+/Helvetica /ISOLatin1Encoding 83.3333 FMSR
+
+2207 3432 mt
+(Elementanzahl) s
+ 388 1927 mt -90 rotate
+(Sekunden) s
+90 rotate
+ 612 3237 mt
+( ) s
+4333 300 mt
+( ) s
+1 sg
+0 434 1097 0 0 -434 665 746 4 MP
+PP
+-1097 0 0 434 1097 0 0 -434 665 746 5 MP stroke
+2.77778 w
+DO
+SO
+4.16667 w
+0 sg
+ 665 746 mt 1762 746 L
+ 665 746 mt 665 312 L
+ 994 407 mt
+(Zeit 2222t05n05 QA) s
+gs 665 312 1098 435 MR c np
+c8
+253 0 715 376 2 MP stroke
+gs 791 325 103 103 MR c np
+ 25 25 842 376 FO
+gr
+
+gr
+
+c8
+0 sg
+ 994 509 mt
+(Zeit 2222t05n05 QA) s
+gs 665 312 1098 435 MR c np
+c9
+253 0 715 478 2 MP stroke
+gs 791 427 103 103 MR c np
+0 j
+29 50 -58 0 29 -50 842 511 4 MP
+DP
+gr
+
+gr
+
+c9
+0 sg
+ 994 610 mt
+(Zeit 2222t05n05 QA) s
+gs 665 312 1098 435 MR c np
+c10
+253 0 715 579 2 MP stroke
+gs 791 528 103 103 MR c np
+ 825 562 mt 859 596 L
+ 859 562 mt 825 596 L
+gr
+
+gr
+
+c10
+0 sg
+ 994 712 mt
+(Zeit 2222t05n05 QA) s
+gs 665 312 1098 435 MR c np
+c11
+253 0 715 681 2 MP stroke
+gs 791 630 103 103 MR c np
+16 W
+842 681 PD
+gr
+
+gr
+
+c11
+
+end %%Color Dict
+
+eplot
+%%EndObject
+
+epage
+end
+
+showpage
+
+%%Trailer
+%%EOF
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function A_plots(files,printt)
+global type2sy
figure(4)
-figure(5)
+% figure(5)
figure(6)
figure(7)
figure(8)
% type2str = {'Analytisch ' 'Quad Element' 'Quad Achse ' 'Quad Seite '};
type2str = {'A ' 'QA' 'QEQA' 'QA ' 'QS '};
-type2sym = {'o-' 'v-' 'x-' '.-' '^-','+-','p-','s-','d-'};
-type2color = [[0 .3 .3]; [0 .6 .6]; [.6 0 .6 ]; [.9 0 0]];
+type2sy = {'o-' 'v-' 'x-' '.-' '^-','+-','p-','s-','d-'};
+type2color = [[0 .3 .3]; [0 .6 .6]; [.6 0 .6 ]; [0 0 .9]];
set(0,'DefaultAxesColorOrder',type2color,...
'DefaultAxesLineStyleOrder','o-|v-|x-')
leg4 = {leg4{:}...
['Zeit ' l0 l1{i}]...
}';
- sym = {sym{:} type2sym{data(1,[2+(i-1)*rows])}}';
+ sym = {sym{:} type2sym(data(1,[2+(i-1)*rows]))}';
end
end
+
+% [u ia ic] = unique(leg0)
[m n] = size(G_D);
step = round(n/rows);
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}, ...
+loglog(X(:,i+1),G_D(:,2+rows*i),type2sym(i*3+1), ...
... G_D(:,2+8+rows*i)...
- X(:,i+1),G_D(:,2+1+rows*i),type2sym{i*3+2},...*(G_D(k,2)-shift)/G_D(k,3)...*G_D(1,2)/G_D(1,2+1+rows*i) ...
- X(:,i+1),sqrt(abs(sol - G_D(:,2+2+rows*i))),type2sym{i*3+3},...*(G_D(k,2)-shift)/G_D(k,3)...
+ X(:,i+1),G_D(:,2+1+rows*i),type2sym(i*3+2),...*(G_D(k,2)-shift)/G_D(k,3)...*G_D(1,2)/G_D(1,2+1+rows*i) ...
+ X(:,i+1),sqrt(abs(sol - G_D(:,2+2+rows*i))),type2sym(i*3+3),...*(G_D(k,2)-shift)/G_D(k,3)...
... G_D(:,2+3+rows*i)...*G_D(1,2)/G_D(1,2+3+rows*i) ...
... [ 0; sqrt(G_D(2:end,2+9+rows*i)-G_D(1:end-1,2+9+rows*i))]...
... [ 0; sqrt(G_D(2:end,2+2+rows*i)-G_D(1:end-1,2+2+rows*i))]...
hold on
for i = 1:step-1
-loglog(X(:,i+1),G_D(:,2+rows*i),type2sym{i*3+1}, ...
+loglog(X(:,i+1),G_D(:,2+rows*i),type2sym(i*3+1), ...
... G_D(:,2+8+rows*i)...
- X(:,i+1),G_D(:,2+1+rows*i),type2sym{i*3+2},...*(G_D(k,2)-shift)/G_D(k,3)...*G_D(1,2)/G_D(1,2+1+rows*i) ...
- X(:,i+1),sqrt(abs(sol - G_D(:,2+2+rows*i))),type2sym{i*3+3},...*(G_D(k,2)-shift)/G_D(k,3)...
+ X(:,i+1),G_D(:,2+1+rows*i),type2sym(i*3+2),...*(G_D(k,2)-shift)/G_D(k,3)...*G_D(1,2)/G_D(1,2+1+rows*i) ...
+ X(:,i+1),sqrt(abs(sol - G_D(:,2+2+rows*i))),type2sym(i*3+3),...*(G_D(k,2)-shift)/G_D(k,3)...
... G_D(:,2+3+rows*i)...*G_D(1,2)/G_D(1,2+3+rows*i) ...
... [ 0; sqrt(G_D(2:end,2+9+rows*i)-G_D(1:end-1,2+9+rows*i))]...
... [ 0; sqrt(G_D(2:end,2+2+rows*i)-G_D(1:end-1,2+2+rows*i))]...
figure(6)
i=0;
loglog(...
- X(:,i+1),G_D(:,2+4+rows*i),type2sym{i*3+1},...
- X(:,i+1),G_D(:,2+5+rows*i),type2sym{i*3+2},...
- X(:,i+1),G_D(:,2+6+rows*i),type2sym{i*3+3},...
+ X(:,i+1),G_D(:,2+4+rows*i),type2sym(i*3+1),...
+ X(:,i+1),G_D(:,2+5+rows*i),type2sym(i*3+2),...
+ X(:,i+1),G_D(:,2+6+rows*i),type2sym(i*3+3),...
'color', type2color(i+1,:));
hold on
for i = 1:step-1
loglog(...
- X(:,i+1),G_D(:,2+4+rows*i),type2sym{i*3+1},...
- X(:,i+1),G_D(:,2+5+rows*i),type2sym{i*3+2},...
- X(:,i+1),G_D(:,2+6+rows*i),type2sym{i*3+3},...
+ X(:,i+1),G_D(:,2+4+rows*i),type2sym(i*3+1),...
+ X(:,i+1),G_D(:,2+5+rows*i),type2sym(i*3+2),...
+ X(:,i+1),G_D(:,2+6+rows*i),type2sym(i*3+3),...
'color', type2color(i+1,:));
end
% loglog(X(:,1),[7*X(:,1).^(-1/2),3*X(:,1).^(-1/4),2*X(:,1).^(-3/4)],'-.')
i=0;
loglog(repmat(X(:,i+1),1,1),...
G_D(:,2+7+rows*i)...
- ,type2sym{i+1},'color', type2color(i+1,:));
+ ,type2sym(i+1),'color', type2color(i+1,:));
hold on
for i = 1:step-1
loglog(repmat(X(:,i+1),1,1),...
G_D(:,2+7+rows*i)...
- ,type2sym{i+1},'color', type2color(i+1,:));
+ ,type2sym(i+1),'color', type2color(i+1,:));
end
% loglog(X(:,1),[7*X(:,1).^(-1/2),3*X(:,1).^(-1/4),2*X(:,1).^(-3/4)],'-.')
i=0;
loglog(repmat(X(:,i+1),1,1),[...
G_D(:,2+11+rows*i)...
- ],type2sym{i+1},'color', type2color(i+1,:));
+ ],type2sym(i+1),'color', type2color(i+1,:));
hold on
for i = 1:step-1
loglog(repmat(X(:,i+1),1,1),[...
G_D(:,2+11+rows*i)...
- ],type2sym{i+1},'color', type2color(i+1,:));
+ ],type2sym(i+1),'color', type2color(i+1,:));
end
% loglog(X(:,1),[7*X(:,1).^(-1/2),3*X(:,1).^(-1/4),2*X(:,1).^(-3/4)],'-.')
hold off
print('-r600','-depsc',[printt '_time.eps'])
+end
+
+end
+
+
+function str = type2sym(num)
+global type2sy
+ str = type2sy{mod(num-1,length(type2sy))+1};
end
\ No newline at end of file
%% voll Analytisch
A_plots({'meshSave/1t05n05_3DFichCube_23'},'../doc/fig/1t05n05_3DFichCube')
A_plots({'meshSave/1t05n05_2DQuad_32'},'../doc/fig/1t05n05_2DQuad')
+A_plots({'meshSave/1t05n05_2DLShape_26'},'../doc/fig/1t05n05_2DLShape')
%% Isotrop Uniform
A_plots({'meshSave/1t1n0_2DQuad_6'},'../doc/fig/1t1n0_2DQuad')
%% Semianalytisch
A_plots({'meshSave/132t05n05_2DQuad_30'},'../doc/fig/132t05n05_2DQuad')
-A_plots({'meshSave/132t05n05_3DFichCube_21'},'../doc/fig/132t05n05_3DFichCube')
+A_plots({'meshSave/132t05n05_3DFichCube_23'},'../doc/fig/132t05n05_3DFichCube')
close all
\ No newline at end of file
projects / bacc.git / commitdiff