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z=GHU?@i|mbMuCiAko6XjO|42lW{Ka}z|^0okCD!TNl^k>x&Qt3>5DbqJo)+k`;WZi zvOZr2K6&--?fcuU(YpQj{Px#xpG_}b!sp`;)O9Z)zxCO3;3LPU(^DF)PoLl2{{N8Q hZtw1H{`~m*!^giqym@=WXJ26nH;+$`U;Om)=08!>MYR9` diff --git a/doc/doc.tex b/doc/doc.tex index 9e06832..0372a1e 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -77,14 +77,14 @@ \section{Einleitung} \subsection{Allgemein} -In dieser Arbeit wollen wir uns mit numerischen Lösungsverfahren für die homogene Laplace-Gleichung +In dieser Arbeit beschäftigen wir uns mit numerischen Lösungsverfahren für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen, \begin{align} - \varDelta u &= 0 &\text{ in } \Omega& \subset \R^3\\ u &= f &\text{ auf } \Gamma&:=\partial\Omega \nonumber \end{align} -beschäftigen, wobei + wobei $\Omega \subset \R^3$ beschränkte Teilmenge von $\R^3$ -mit Rand $\Gamma := \partial \Omega$\\ +mit Dirichlet-Rand $\Gamma := \partial \Omega$\\ $\abs{u(x)} = O(\abs{x}^{-1}) \nonumber$ Daraus folgt: @@ -191,22 +191,40 @@ Damit ist $\phi_{\ell}$ die Galerkinapproximation an $\phi$ % \end{align} \subsection{Netz} +Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. \begin{defi} - Sei $T \subseteq \R^3$ ein achsenorientiertes Rechteck, wenn sich aus dem Eckpunkt $C \in \R^3$, alle weiteren Knoten schreiben lassen als $\{ C + a\cdot A, C + a \cdot A + b \cdot B , C + b \cdot B\}$, wobei Vektoren $A,B \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ und Skalare $a,b\neq0$ sind und der Flächeninhalt $|T| > 0$. + Sei $T \subseteq \R^3$ ein achsenorientiertes Rechteck, wenn sich aus dem Eckpunkt $C \in \R^3$, alle weiteren Knoten schreiben lassen als $\{ C + a\cdot A, C + a \cdot A + b \cdot B , C + b \cdot B\}$, wobei Vektoren $A,B \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ und Skalare $a,b\neq0$ sind und der Flächeninhalt $|T| > 0$. Weiterhin sei + \begin{align} + C_T &:= \{ c_i:1\leq i \leq 4\} + \end{align} +die Menge der Eckpunkte des Rechtecks. \end{defi} + \begin{defi} - Sei $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine Triangulierung von $\Gamma$. Dann gilt: + Sei $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine \todo{Triangulierung ($\to$ Dreiecke?)} von $\Gamma$. Dann gilt: \begin{itemize} \item $\overline{\Gamma} = \bigcup_{j=1}^NT_j$ \item $\forall T_j \in T_{\ell}$ abgeschlossen und nicht null $\abs{T_j}>0$ \item $\abs{T_j \cap T_k} = 0$ mit $T_j,T_k\in\T_{\ell}$ und $T_j\neq T_k$ \end{itemize} \end{defi} +Den Begriff der Triangulierung wollen wir hier noch stärker definieren, da wir pro Seite eines Rechtecks nur einen hängenden Knoten zulassen wollen. +\begin{defi} + Eine Triangulierung $\T$ von $\Gamma$ sei gültig, wenn auf jeder Kante eines Rechtecks maximal ein Knoten liegt. Daraus folgt, dass pro Seite eines Rechtecks + \begin{itemize} + \item entweder kein Nachbarelement + \item oder genau ein Nachbarelement + \item oder genau zwei Nachbarelemente + \end{itemize} +anliegen können. Weiterhin wollen wir auch verlangen, dass der auf der Kante liegende Knoten die Seite halbiert, die beiden Nachbarlemente also die gleiche Seitenlänge an der anliegenden Kante haben. +\end{defi} +\todo{Reguläres Netz auch definieren?} +\todo{+Bild} \subsubsection{Verfeinern} \begin{defi} - Sei $\T_{\ell/2}$ das aus der Verfeinerung von $\T_{\ell}$ resultierende Netz + Sei $\T_{\ell/2}$ die aus der Verfeinerung von $\T_{\ell}$ resultierende gültige Triangulierung. \end{defi} \begin{alg}[Verfeinern] \label{alg:refine} Sei $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine gegebenes Netz und $marked$ eine gegebene Markierung wobei $marked_j \hat =$ Markierung von $T_j$. Nun sei $j = 1$ und gehe so vor: \begin{enumerate} -- 2.47.3