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Hierzu b \psfrag{tmu 1t1n0 A}{\tiny $\tilde \mu$ (uniform)} \psfrag{eta 1t1n0 A}{\tiny $\eta$ (uniform)} \psfrag{fehler 1t1n0 A}{\tiny Fehler (uniform)} -\psfrag{Zeit 1t1n0 A}{\tiny Zeit (uniform)} -\psfrag{cond 1t1n0 A}{\tiny Kondition (uniform)} +\psfrag{Zeit 1t1n0 A}{\tiny Rechenzeit (uniform)} +\psfrag{cond 1t1n0 A}{\tiny Konditionszahl (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)} @@ -2158,8 +2158,8 @@ Zunächst wollen wir drei Verfeinerungs-Strategien genauer untersuchen. Hierzu b \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{Zeit 1t05n0 A}{\tiny Rechenzeit (isotrop)} +\psfrag{cond 1t05n0 A}{\tiny Konditionszahl (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)} @@ -2167,8 +2167,8 @@ Zunächst wollen wir drei Verfeinerungs-Strategien genauer untersuchen. Hierzu b \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{Zeit 1t05n05 A}{\tiny Rechenzeit (anisotrop)} +\psfrag{cond 1t05n05 A}{\tiny Konditionszahl (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)} @@ -2176,7 +2176,7 @@ Zunächst wollen wir drei Verfeinerungs-Strategien genauer untersuchen. Hierzu b \centering \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 $\hat V_{\ell}$ Matrix für das Quadrat \label{fig:2DQuad:verfeinern:cond}]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_cond}} +\subfloat[Konditionszahl der $\hat V_{\ell}$ 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} @@ -2210,7 +2210,7 @@ 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 Matrix $\hat V_{\ell}$ in Abhängigkeit der Elementanzahl. Für die "`uniforme"' Strategie in \figLineA[] erkennen wir sehr gute Konditionszahlen, wissen aber, dass auch der Fehler der Galerkin-Lösung nur langsam gegen 0 konvergiert. Die Konditionszahlen der "`adaptiv anisotropen"' Strategie in \figLineC[] wachsen hingegen am schnellsten und steigen bei etwa 3000 Elementen sogar sprunghaft an. \noindent -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. +Weiterhin können wir in Abbildung \ref{fig:2DQuad:verfeinern:time} die benötigte Rechenzeit 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 Rechenzeit 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 Form 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. @@ -2226,46 +2226,46 @@ Bei der folgenden Berechnung verwenden wir wieder den Algorithmus \ref{alg:adapt \centering -\psfrag{tmu 2222t05n05 0 QA}{\tiny $\tilde \mu$ (Grad $2^ 0 $)} -\psfrag{eta 2222t05n05 0 QA}{\tiny $\eta$ (Grad $2^ 0 $)} -\psfrag{fehler 2222t05n05 0 QA}{\tiny Fehler (Grad $2^ 0 $)} -\psfrag{Zeit 2222t05n05 0 QA}{\tiny Zeit (Grad $2^ 0 $)} -\psfrag{cond 2222t05n05 0 QA}{\tiny Kondition (Grad $2^ 0 $)} -\psfrag{min hmin/max hmax 2222t05n05 0 QA}{\tiny $\min (h_{\min})/\max (h_{\max})$ (Grad $2^ 0 $)} -\psfrag{min hmax/max hmax 2222t05n05 0 QA}{\tiny $\min (h_{\max})/\max (h_{\max})$ (Grad $2^ 0 $)} -\psfrag{min hmin/hmax 2222t05n05 QA}{\tiny $\min (h_{\min}/h_{\max})$ (Grad $2^ 0 $)} - -\psfrag{tmu 2222t05n05 1 QA}{\tiny $\tilde \mu$ (Grad $2^ 1 $)} -\psfrag{eta 2222t05n05 1 QA}{\tiny $\eta$ (Grad $2^ 1 $)} -\psfrag{fehler 2222t05n05 1 QA}{\tiny Fehler (Grad $2^ 1 $)} -\psfrag{Zeit 2222t05n05 1 QA}{\tiny Zeit (Grad $2^ 1 $)} -\psfrag{cond 2222t05n05 1 QA}{\tiny Kondition (Grad $2^ 1 $)} -\psfrag{min hmin/max hmax 2222t05n05 1 QA}{\tiny $\min (h_{\min})/\max (h_{\max})$ (Grad $2^ 1 $)} -\psfrag{min hmax/max hmax 2222t05n05 1 QA}{\tiny $\min (h_{\max})/\max (h_{\max})$ (Grad $2^ 1 $)} -\psfrag{min hmin/hmax 2222t05n05 QA}{\tiny $\min (h_{\min}/h_{\max})$ (Grad $2^ 1 $)} - -\psfrag{tmu 2222t05n05 2 QA}{\tiny $\tilde \mu$ (Grad $2^ 2 $)} -\psfrag{eta 2222t05n05 2 QA}{\tiny $\eta$ (Grad $2^ 2 $)} -\psfrag{fehler 2222t05n05 2 QA}{\tiny Fehler (Grad $2^ 2 $)} -\psfrag{Zeit 2222t05n05 2 QA}{\tiny Zeit (Grad $2^ 2 $)} -\psfrag{cond 2222t05n05 2 QA}{\tiny Kondition (Grad $2^ 2 $)} -\psfrag{min hmin/max hmax 2222t05n05 2 QA}{\tiny $\min (h_{\min})/\max (h_{\max})$ (Grad $2^ 2 $)} -\psfrag{min hmax/max hmax 2222t05n05 2 QA}{\tiny $\min (h_{\max})/\max (h_{\max})$ (Grad $2^ 2 $)} -\psfrag{min hmin/hmax 2222t05n05 QA}{\tiny $\min (h_{\min}/h_{\max})$ (Grad $2^ 2 $)} - -\psfrag{tmu 2222t05n05 3 QA}{\tiny $\tilde \mu$ (Grad $2^ 3 $)} -\psfrag{eta 2222t05n05 3 QA}{\tiny $\eta$ (Grad $2^ 3 $)} -\psfrag{fehler 2222t05n05 3 QA}{\tiny Fehler (Grad $2^ 3 $)} -\psfrag{Zeit 2222t05n05 3 QA}{\tiny Zeit (Grad $2^ 3 $)} -\psfrag{cond 2222t05n05 3 QA}{\tiny Kondition (Grad $2^ 3 $)} -\psfrag{min hmin/max hmax 2222t05n05 3 QA}{\tiny $\min (h_{\min})/\max (h_{\max})$ (Grad $2^ 3 $)} -\psfrag{min hmax/max hmax 2222t05n05 3 QA}{\tiny $\min (h_{\max})/\max (h_{\max})$ (Grad $2^ 3 $)} -\psfrag{min hmin/hmax 2222t05n05 QA}{\tiny $\min (h_{\min}/h_{\max})$ (Grad $2^ 3 $)} +\psfrag{tmu 2222t05n05 0 QA}{\tiny $\tilde \mu$ (Grad $1 $)} +\psfrag{eta 2222t05n05 0 QA}{\tiny $\eta$ (Grad $1 $)} +\psfrag{fehler 2222t05n05 0 QA}{\tiny Fehler (Grad $1 $)} +\psfrag{Zeit 2222t05n05 0 QA}{\tiny Rechenzeit (Grad $1 $)} +\psfrag{cond 2222t05n05 0 QA}{\tiny Konditionszahl (Grad $1 $)} +\psfrag{min hmin/max hmax 2222t05n05 0 QA}{\tiny $\min (h_{\min})/\max (h_{\max})$ (Grad $1 $)} +\psfrag{min hmax/max hmax 2222t05n05 0 QA}{\tiny $\min (h_{\max})/\max (h_{\max})$ (Grad $1 $)} +\psfrag{min hmin/hmax 2222t05n05 QA}{\tiny $\min (h_{\min}/h_{\max})$ (Grad $1 $)} + +\psfrag{tmu 2222t05n05 1 QA}{\tiny $\tilde \mu$ (Grad $2 $)} +\psfrag{eta 2222t05n05 1 QA}{\tiny $\eta$ (Grad $2 $)} +\psfrag{fehler 2222t05n05 1 QA}{\tiny Fehler (Grad $2 $)} +\psfrag{Zeit 2222t05n05 1 QA}{\tiny Rechenzeit (Grad $2 $)} +\psfrag{cond 2222t05n05 1 QA}{\tiny Konditionszahl (Grad $2 $)} +\psfrag{min hmin/max hmax 2222t05n05 1 QA}{\tiny $\min (h_{\min})/\max (h_{\max})$ (Grad $2 $)} +\psfrag{min hmax/max hmax 2222t05n05 1 QA}{\tiny $\min (h_{\max})/\max (h_{\max})$ (Grad $2 $)} +\psfrag{min hmin/hmax 2222t05n05 QA}{\tiny $\min (h_{\min}/h_{\max})$ (Grad $2 $)} + +\psfrag{tmu 2222t05n05 2 QA}{\tiny $\tilde \mu$ (Grad $4 $)} +\psfrag{eta 2222t05n05 2 QA}{\tiny $\eta$ (Grad $4 $)} +\psfrag{fehler 2222t05n05 2 QA}{\tiny Fehler (Grad $4 $)} +\psfrag{Zeit 2222t05n05 2 QA}{\tiny Rechenzeit (Grad $4 $)} +\psfrag{cond 2222t05n05 2 QA}{\tiny Konditionszahl (Grad $4 $)} +\psfrag{min hmin/max hmax 2222t05n05 2 QA}{\tiny $\min (h_{\min})/\max (h_{\max})$ (Grad $4 $)} +\psfrag{min hmax/max hmax 2222t05n05 2 QA}{\tiny $\min (h_{\max})/\max (h_{\max})$ (Grad $4 $)} +\psfrag{min hmin/hmax 2222t05n05 QA}{\tiny $\min (h_{\min}/h_{\max})$ (Grad $4 $)} + +\psfrag{tmu 2222t05n05 3 QA}{\tiny $\tilde \mu$ (Grad $8 $)} +\psfrag{eta 2222t05n05 3 QA}{\tiny $\eta$ (Grad $8 $)} +\psfrag{fehler 2222t05n05 3 QA}{\tiny Fehler (Grad $8 $)} +\psfrag{Zeit 2222t05n05 3 QA}{\tiny Rechenzeit (Grad $8 $)} +\psfrag{cond 2222t05n05 3 QA}{\tiny Konditionszahl (Grad $8 $)} +\psfrag{min hmin/max hmax 2222t05n05 3 QA}{\tiny $\min (h_{\min})/\max (h_{\max})$ (Grad $8 $)} +\psfrag{min hmax/max hmax 2222t05n05 3 QA}{\tiny $\min (h_{\max})/\max (h_{\max})$ (Grad $8 $)} +\psfrag{min hmin/hmax 2222t05n05 QA}{\tiny $\min (h_{\min}/h_{\max})$ (Grad $8 $)} \subfloat[Fehler und Fehlerschätzer für das Quadrat \label{fig:2DQuad:quad:err}]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_error}} % \subfloat[Seitenverhältnisse auf dem Quadrat \label{fig:2DQuad:quad:hminmax}]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_hminmax}} % \\ -% \subfloat[Kondition der $\hat V_{\ell}$ Matrix \label{fig:2DQuad:quad:cond}]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_cond}} +% \subfloat[Konditionszahl der $\hat V_{\ell}$ Matrix \label{fig:2DQuad:quad:cond}]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_cond}} \subfloat[Berechnungszeit für das Quadrat \label{fig:2DQuad:quad:time}]{\includegraphics[width=0.5\textwidth]{fig/2222t05n05_2DQuad_time}} \caption{Vergleich der Quadraturgrade auf dem Quadrat} \label{fig:2DQuad:quad} @@ -2275,7 +2275,7 @@ Bei der folgenden Berechnung verwenden wir wieder den Algorithmus \ref{alg:adapt 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 dass 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 für die Matrix $\hat V_{\ell}$ in Abbildung \ref{fig:2DQuad:quad:time} untersucht. Wie sich leicht erkennen lässt, benötigt die Berechnung mit Quadraturgrad 8, dargestellt durch die Linie in \figLineD[], für 3000 Elemente etwa $1000$ Sekunden. Die Berechnungszeiten für die Quadraturgrade 1,2,4 benötigen hingegen wesentlich weniger Zeit und sind fast äquivalent. Sie berechnen die Matrix $\hat V_{\ell}$ mit 3000 Elementen in etwa $100$ Sekunden und sind damit etwa 16 Mal schneller als die Quadratur vom Grad 8. +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. Wie sich leicht erkennen lässt, benötigt die Berechnung mit Quadraturgrad 8, dargestellt durch die Linie in \figLineD[], für 3000 Elemente etwa $1000$ Sekunden. Die Berechnungszeiten für die Quadraturgrade 1,2,4 benötigen hingegen wesentlich weniger Rechenzeit und sind fast äquivalent. Sie berechnen die Matrix $\hat V_{\ell}$ mit 3000 Elementen in etwa $100$ Sekunden und sind damit etwa 16 Mal schneller als die Quadratur vom Grad 8. \noindent Aufgrund dieser Ergebnisse werden wir für die folgenden Berechnungen einen Quadraturgrad von 4 wählen. Da wir zum einen die Berechnungszeiten gering halten und zum anderen die Stabilität der Berechnungen sicherstellen wollen. @@ -2292,7 +2292,7 @@ In der letzten Strategie "`volle Quadratur"' berechnen wir alle $\zeta_Q$-zuläs \subfloat[Fehler und Fehlerschätzer für das Quadrat \label{fig:2DQuad:sem:err}]{\includegraphics[width=0.5\textwidth]{fig/142t05n05_2DQuad_error}} % \subfloat[Seitenverhältnisse auf dem Quadrat \label{fig:2DQuad:quad:hminmax}]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_hminmax}} \\ -\subfloat[Kondition der $\hat V_{\ell}$ Matrix \label{fig:2DQuad:sem:cond}]{\includegraphics[width=0.5\textwidth]{fig/142t05n05_2DQuad_cond}} +\subfloat[Konditionszahlen der $\hat V_{\ell}$ Matrix \label{fig:2DQuad:sem:cond}]{\includegraphics[width=0.5\textwidth]{fig/142t05n05_2DQuad_cond}} \subfloat[Berechnungszeit für das Quadrat \label{fig:2DQuad:sem:time}]{\includegraphics[width=0.5\textwidth]{fig/142t05n05_2DQuad_time}} \caption{Vergleich der Berechnungsarten von $\hat V_{\ell}$ auf dem Quadrat} \label{fig:2DQuad:sem} @@ -2306,7 +2306,7 @@ In Abbildung \ref{fig:2DQuad:sem:err} haben wir die Ergebnisse des Fehlers gegen In Abbildung \ref{fig:2DQuad:sem:cond} vergleichen wir noch einmal die Konditionszahlen der $\hat V_{\ell}$ Matrix für die verschiedenen Strategien. Auch hier wurde die \figErr[e] Grenzlinie bei etwa 2000 Elementen eingezeichnet. Anhand der Linie in \figLineA[] erkennen wir deutlich, dass ab der 2000 Elemente Grenze die Konditionszahlen für die analytische Berechnung sprunghaft gegenüber den Konditionszahlen der beiden Quadraturen in \figLineB[] und \figLineC[] ansteigen. Wir erkennen an dieser Stelle aber auch, dass die Kondtionszahlen der "`analytischen"' Berechnung schon ab 1000 Elementen von den Quadraturen geringfügig abweichen und damit auf eine ungünstige Berechnung hindeuten. \noindent -Weiterhin wollen wir noch einmal die Laufzeit der drei Strategien in Abbildung \ref{fig:2DQuad:sem:time} betrachten. Gemessen wurde die Zeit eines Berechnungsschritts, welches wieder das Aufstellen der Matritzen und Lösen des Gleichungssystems beinhaltet. Hier erkennen wir, dass sich die Laufzeiten der drei Strategien nur minimal voneinander unterscheiden. Für fast $10^4$ Elemente benötigen alle drei Berechnungsarten etwa $4*10^4$ Sekunden, was etwa elf Stunden entspricht. Da die Zeit zum Aufstellen im Vergleich der Quadraturgrade wesentlich kürzer ist, können wir daraus schließen, dass die meiste Zeit beim Lösen und Berechnen der Konditionszahlen verloren geht. +Weiterhin wollen wir noch einmal die Laufzeit der drei Strategien in Abbildung \ref{fig:2DQuad:sem:time} betrachten. Gemessen wurde die Zeit eines Berechnungsschritts, welches wieder das Aufstellen der Matritzen und Lösen des Gleichungssystems beinhaltet. Hier erkennen wir, dass sich die Laufzeiten der drei Strategien nur minimal voneinander unterscheiden. Für fast $10^4$ Elemente benötigen alle drei Berechnungsarten etwa $4*10^4$ Sekunden, was etwa elf Stunden entspricht. Da die Rechenzeit zum Aufstellen im Vergleich der Quadraturgrade wesentlich kürzer ist, können wir daraus schließen, dass die meiste Rechenzeit beim Lösen und Berechnen der Konditionszahlen verloren geht. \noindent Die Ergebnisse zeigen, dass wir durch Gauss-Quadratur über das kleinere Element auf $\zeta_E$-zulässigen Elementen bis an die Speichergrenze stabil rechnen können und es nicht nötig ist, alle Integrale durch Quadratur zu ersetzen. Weiterhin nimmt die Art der Berechnung von $\hat V_{\ell}$ kaum Einfluss auf die Laufzeit. @@ -2350,7 +2350,7 @@ Weiterhin werden wir in der Strategie "`volle Quadratur"' alle auftretenden Inte \\ \subfloat[Seitenverhältnisse auf dem Fischer Würfel \label{fig:3DFichCube:quad:hminmax}]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_3DFichCube_hminmax}} % \\ -\subfloat[Kondition der $\hat V_{\ell}$ Matrix \label{fig:3DFichCube:sem:cond}]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_3DFichCube_cond}} +\subfloat[Konditionszahlen der $\hat V_{\ell}$ Matrix \label{fig:3DFichCube:sem:cond}]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_3DFichCube_cond}} % \subfloat[Berechnungszeit für das Fischer Würfel \label{fig:3DFichCube:sem:time}]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_3DFichCube_time}} \caption{Vergleich der Berechnungsarten von $\hat V_{\ell}$ auf dem Fischer Würfel} \label{fig:3DFichCube:sem} -- 2.47.3