}
\subsection{Datenstruktur}
-Für die Implementierung in \Matlab~und C++ wollen wollen wir eine einheitliche Datenstruktur einführen.
+Für die Implementierung in \Matlab~und C++ werden wir eine einheitliche Datenstruktur einführen.
Die für die Partition $\T_{\ell} = \{T_1\ldots T_N\}$ benötigten Knoten $\K_{\ell} = \{k_1\ldots k_M\}$ stellen wir in einer $M \times 3$ Matrix $COO$ dar. Dabei enthält die $j$-te Zeile die Koordinaten des Knoten $C_j$, d.h. :
\begin{align}
COO[j,1:3] = k_j := (x_j,y_j,z_j)^{T} \text{ wobei } x_j,y_j,z_j \in \R.
\begin{align}
ELE[i,1:4] = T_i := (j,k,l,m).
\end{align}
-Die Knoten wollen wir gegen den Uhrzeigersinn anordnen und der Knoten $C_j$ sei der Punkt $\bs v$ aus Definition \ref{thm:def:T}, also der kleinste Bezüglich des Koordinatensystems.
+Die Knoten ordnen wir gegen den Uhrzeigersinn an und der Knoten $C_j$ sei der Punkt $\bs v$ aus Definition \ref{thm:def:T}, also der kleinste Bezüglich des Koordinatensystems.
\noindent
-Für die bessere Handhabung der Elemente beim Verfeinern der Partition, wollen wir auch die Nachbarschaftsrelationen geeignet abspeichern. Aufgrund der Netzstabilität wollen wir maximal zwei Nachbarn pro Kante eines Elements zulassen.
+Für die bessere Handhabung der Elemente beim Verfeinern der Partition, müssen wir auch die Nachbarschaftsrelationen geeignet abspeichern. Aufgrund der Netzstabilität werden wir maximal zwei Nachbarn pro Kante eines Elements zulassen.
Wir legen also eine $M \times 8$ Matrix für die Indizes der Nachbarelemente an, wobei die $i$-te Zeile die Nachbarelemente $\{T_{n_1},\ldots,T_{n_8}\}$ zum Element $T_i$ enthält.
\begin{align}
NEI[i,1:8] = N_i := (n_1,\ldots,n_8)
\end{align}
-Offensichtlich ist $i \notin N_i$. Wir wollen uns aber noch genauer eine geeignete Anordnung für die Nachbarelemente überlegen. Hierbei bezeichnen wir die Seite $[j,k]$ eines Elements als Seite 1 und gegen den Uhrzeigersinn alle weiteren mit $\{2,3,4\}$. Für den einfacheren Zugriff auf die Elemente einer Seite $s$, wollen wir die Nachbarelemente zur Seite $s$ unter den Indizes $n_s$ und $n_{s+4}$ abspeichern. Die Nachbarelemente $\{T_{n_s}, T_{n_{s+4}}\}$ liegen also an der Seite $s$ des Elements. Für Seiten die nur einen Nachbarn $T_{n_s}$ besitzen setzen wir $n_{s+4}=0$ und für Seiten mit keinem Nachbarn setzen wir $n_s = n_{s+4} = 0$. Daraus folgt unmittelbar, dass für $n_s =0$ auch $n_{s+4} = 0$ gilt, die Seite $s$ also keine Nachbarelemente besitzt und umgekehrt folgt aus $n_{s+4} \neq 0$ $n_s \neq 0$, womit die Seite $s$ genau zwei Nachbarelemente hat.
+Offensichtlich ist $i \notin N_i$. Wir wollen uns aber noch genauer eine geeignete Anordnung für die Nachbarelemente überlegen. Hierbei bezeichnen wir die Seite $[j,k]$ eines Elements als Seite 1 und gegen den Uhrzeigersinn alle weiteren mit $\{2,3,4\}$. Für den einfacheren Zugriff auf die Elemente einer Seite $s$, seien die Nachbarelemente zur Seite $s$ unter den Indizes $n_s$ und $n_{s+4}$ abgespeichert. Die Nachbarelemente $\{T_{n_s}, T_{n_{s+4}}\}$ liegen also an der Seite $s$ des Elements. Für Seiten die nur einen Nachbarn $T_{n_s}$ besitzen setzen wir $n_{s+4}=0$ und für Seiten mit keinem Nachbarn setzen wir $n_s = n_{s+4} = 0$. Daraus folgt unmittelbar, dass für $n_s =0$ auch $n_{s+4} = 0$ gilt, die Seite $s$ also keine Nachbarelemente besitzt und umgekehrt folgt aus $n_{s+4} \neq 0$ $n_s \neq 0$, womit die Seite $s$ genau zwei Nachbarelemente hat.
(Siehe Abb.:\ref{exmpl13:nei:part})
\begin{figure}[ht]
-\caption{Nachbarschaftsrelationen Element 4 aus Abb.\ref{exmpl13}}
\label{exmpl13:nei:part}
\centering
\subfloat[Lage]{\includegraphics{fig/Net_Neigh}}
\subfloat[Nachbarn]{\input{fig/exmpl3_nei_part}}
+\caption{Nachbarschaftsrelationen Element 4 aus Abb.\ref{exmpl13}, wobei nur der Index der Elemente $T_{n_s}$ notiert wurde}
\end{figure}
\noindent
\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/22222t05n05_2DQuad_30_hminmax}}
\end{figure}
+\begin{figure}[ht]
+\caption{3D FichCube adaptiv anisotrop analytisch und volleQuadratur $V\phi = 1$}
+\centering
+\label{fig:12t05n05_3DFichCube_22}
+\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/12t05n05_3DFichCube_22_error}}
+\subfloat[Energie Norm]{\includegraphics[width=0.5\textwidth]{fig/12t05n05_3DFichCube_22_norm}}\\
+\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/12t05n05_3DFichCube_22_cond}}
+\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/12t05n05_3DFichCube_22_hminmax}}
+\end{figure}
+
\section{Anhang Code}
\todo{
Die wichtigsten Funktionen
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