Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
maximize
next up previous
Next: Test example : Circular Up: Numerical examples Previous: Test example : Diagonally

Test example $ T^{\gamma }$: Dependence on the angle of convection


Table 6: Example $ T^{\gamma }$: Standard multiscale W(1,1)-cycle, average error reduction $ \varrho _{it,it-4}$.
$ \omega\backslash\;b$      $ 0.0$          $ 10^{1}$          $ 10^{2}$          $ 10^{3}$          $ 10^{4}$          $ 10^{5}$          $ 10^{6}$          $ 10^{8}$     
    $ h_0=1/256$                                                  
0      0.09          0.09          0.09          0.09          0.09          0.04          0.01          0.01     
$ \pi/8$      0.09          0.09          0.09          0.13          0.24          0.26          0.26          0.26     
$ \pi/4$      0.09          0.09          0.09          0.15          0.30          0.33          0.33          0.33     
$ 3\pi/8$      0.09          0.09          0.09          0.13          0.23          0.25          0.25          0.25     
$ \pi/2$      0.09          0.09          0.09          0.08          0.08          0.03          0.01          0.01     
    $ h_0=1/512$                                                  
0      0.09          0.09          0.09          0.09          0.09          0.07          0.02          0.01     
$ \pi/8$      0.09          0.09          0.09          0.11          0.19          0.22          0.23          0.23     
$ \pi/4$      0.09          0.09          0.09          0.11          0.28          0.33          0.33          0.33     
$ 3\pi/8$      0.09          0.09          0.09          0.10          0.21          0.24          0.25          0.25     
$ \pi/2$      0.09          0.09          0.09          0.08          0.09          0.06          0.02          0.01     
     $ h_0=1/1024$                                                  
0      0.09          0.09          0.09          0.09          0.09          0.11          0.05          0.01     
$ \pi/8$      0.09          0.09          0.09          0.09          0.18          0.19          0.22          0.21     
$ \pi/4$      0.09          0.09          0.09          0.10          0.23          0.31          0.32          0.32     
$ 3\pi/8$      0.09          0.09          0.09          0.09          0.18          0.23          0.24          0.24     
$ \pi/2$      0.09          0.09          0.09          0.08          0.09          0.11          0.05          0.01     
$ P^{k}_{k+1,{\cal V}}$ matrix-dependent $ Q^{k}_{k+1,{\cal V}}$ pre-prewavelet filter
$ P^{k}_{k+1,{\cal S}}$ bilinear $ Q^{k}_{k+1,{\cal S}}$ hierarchical filter

To check the dependence on the angle of convection of the convergence behavior of the above multiscale W(1,1)-cycle we apply exactly the same multiscale setting to $ T^{\gamma }$ as for the last example. We use the classical hierarchical basis transformation for the decomposition of the test spaces and matrix-dependent prolongations together with pre-prewavelet-based decompositions on the trial sides. Here, averages $ \varrho _{it,it-4}$ have been derived from the last five iterations to get a better impression of the asymptotic behavior of the iterations. From Table 6 we conclude that one obtains robust solvers also for angles different from $ \omega=0$, $ \omega=\frac{\pi}{4}$ and $ \omega=\frac{\pi}{2}$ with the multiscale W(1,1)-cycles. The rates for the corresponding axis-oriented cases are of course significantly better compared to other angles.
next up previous
Next: Test example : Circular Up: Numerical examples Previous: Test example : Diagonally
Frank Kiefer
2001-10-25