Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
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Test example $ T^{\beta }$: Diagonally oriented convection


Table 3: Example $ T^{\beta }$: Standard multiscale V(1,1)-cycle, average error reduction $ \varrho _{it,1}$.
$ h_0^{-1}\backslash\;b$      0.0          $ 10^1$          $ 10^2$          $ 10^3$          $ 10^4$          $ 10^5$          $ 10^6$          $ 10^8$     
8      0.13          0.07          0.09          0.14          0.15          0.15          0.16          0.15     
16      0.15          0.09          0.13          0.22          0.24          0.25          0.24          0.24     
32      0.16          0.10          0.15          0.27          0.29          0.29          0.29          0.29     
64      0.17          0.12          0.17          0.29          0.32          0.32          0.32          0.32     
128      0.17          0.13          0.18          0.32          0.36          0.36          0.37          0.37     
256      0.17          0.14          0.19          0.35          0.42          0.42          0.42          0.43     
512      0.17          0.14          0.20          0.38          0.50          0.53          0.53          0.53     
1024      0.17          0.14          0.20          0.40          0.62          0.67          0.68          0.68     
$ 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


Table 4: Example $ T^{\beta }$: Improved multiscale V(1,1)-cycle, average error reduction $ \varrho _{it,1}$.
$ h_0^{-1}\backslash\;b$      0.0          $ 10^1$          $ 10^2$          $ 10^3$          $ 10^4$          $ 10^5$          $ 10^6$          $ 10^8$     
8      0.13          0.06          0.08          0.10          0.10          0.10          0.10          0.10     
16      0.14          0.07          0.10          0.13          0.13          0.13          0.13          0.13     
32      0.15          0.09          0.12          0.15          0.16          0.16          0.16          0.16     
64      0.15          0.10          0.12          0.17          0.18          0.18          0.18          0.18     
128      0.15          0.11          0.13          0.19          0.20          0.20          0.20          0.20     
256      0.15          0.11          0.13          0.20          0.24          0.25          0.25          0.25     
512      0.15          0.12          0.14          0.21          0.32          0.33          0.33          0.33     
1024      0.15          0.12          0.14          0.24          0.41          0.48          0.49          0.49     
$ 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


Table 5: Example $ T^{\beta }$: Standard multiscale W(1,1)-cycle, average error reduction $ \varrho _{it,1}$.
$ h_0^{-1}\backslash\;b$      0.0          $ 10^1$          $ 10^2$          $ 10^3$          $ 10^4$          $ 10^5$          $ 10^6$          $ 10^8$     
8      0.07          0.06          0.08          0.13          0.15          0.15          0.15          0.15     
16      0.08          0.07          0.11          0.23          0.24          0.25          0.24          0.24     
32      0.08          0.08          0.11          0.25          0.29          0.29          0.30          0.28     
64      0.08          0.08          0.09          0.24          0.30          0.31          0.30          0.31     
128      0.08          0.08          0.08          0.20          0.30          0.31          0.31          0.31     
256      0.08          0.08          0.08          0.14          0.28          0.31          0.31          0.31     
512      0.08          0.08          0.08          0.10          0.26          0.30          0.30          0.30     
1024      0.08          0.08          0.08          0.09          0.20          0.28          0.29          0.29     
$ 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

For the case of the diagonally oriented convective vector field we directly present our results for the classical hierarchical basis decomposition on the test and the matrix-dependent prolongation together with a wavelet-like decomposition on the trial side. Here, the decomposition of the trial space is based on the smaller pre-prewavelet filter $ [-1\quad \underline{2}\quad\!\! -1]/2$ and its boundary modifications $ (Q_{k+1,{\cal V},w}^{k})_{*}=[\underline{2} \quad\!\! -1 \quad 0]/2$ and $ (Q_{k+1,{\cal V},e}^{k})_{*}=[0 \quad\!\! -1 \quad \underline{2}]/2$. Linear prewavelet filters produce similar results, but tensor product $ L$-spline prewavelet filters only work for the lower convective range. We omit the corresponding results. From Table 3 we see clearly the stabilizing effect (w.r.t. mesh size) of the wavelet-like decomposition and the standard multiscale V(1,1)-cycle when $ b$ is small ($ b\leq10^2$). However, the method is not robust. Now, Table 4 shows that one obtains much better results for an improved multiscale V(1,1)-cycle. Here, the residuals and the corrections of the projected residuals are computed with the help of Schur-complement approximations that use incomplete factorizations of the blocks $ A_k$ ( $ k=1,...,l-1$). Fine-scale features are transported more accurately to coarser scales. However, we have to pay for the improved convergence behavior with substantially larger computational costs. Furthermore, the method still does not seem to be overall robust, which can be seen from the comparison of the results for $ h_0=1/512$ and $ h_0=1/1024$ for large $ b$. Therefore, we also applied a standard multiscale W(1,1)-cycle using the same transformations. Then, one obtains a truly robust method which is suggested by Table 5. The repeated recursive calls of coarse-grid corrections produce some extra smoothing which finally implies robustness.
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Next: Test example : Dependence Up: Numerical examples Previous: Test example : Axis
Frank Kiefer
2001-10-25