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


Table 1: Example $ T^{\alpha }$: 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     0.29         0.38         0.58         0.56         0.58         0.63         0.64         0.64     
16      0.36          0.51          0.71          0.78          0.79          0.80          0.78          0.79     
32      0.52          0.57          0.81          0.87          0.87          0.87          0.88          0.88     
64      0.60          0.67          0.85          0.91          0.93          0.93          0.93          0.94     
128      0.64          0.69          0.89          0.93          0.93          0.95          0.96          0.96     
256      0.69          0.69          0.90          0.93          0.94          0.96          0.97          0.97     
512      0.72          0.75          0.86          0.93          0.94          0.96          0.98          0.98     
1024      0.75          0.76          0.88          0.92          0.96          0.95          0.98          0.99     
$ P^{k}_{k+1,{\cal V}}$ matrix-dependent $ Q^{k}_{k+1,{\cal V}}$ hierarchical filter
$ P^{k}_{k+1,{\cal S}}$ bilinear $ Q^{k}_{k+1,{\cal S}}$ hierarchical filter


Table 2: Example $ T^{\alpha }$: 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     0.17         0.13         0.08         0.07         0.07         0.07         0.07         0.07     
16      0.20          0.16          0.11          0.08          0.08          0.08          0.08          0.08     
32      0.24          0.19          0.14          0.11          0.08          0.07          0.07          0.08     
64      0.25          0.21          0.17          0.13          0.09          0.08          0.07          0.07     
128      0.26          0.23          0.19          0.17          0.11          0.08          0.08          0.08     
256      0.27          0.24          0.20          0.19          0.16          0.10          0.08          0.08     
512      0.27          0.25          0.21          0.21          0.21          0.13          0.09          0.08     
1024      0.28          0.26          0.21          0.22          0.24          0.19          0.11          0.08     
$ P^{k}_{k+1,{\cal V}}$ matrix-dependent $ Q^{k}_{k+1,{\cal V}}$ linear prewavelet filter
$ P^{k}_{k+1,{\cal S}}$ bilinear $ Q^{k}_{k+1,{\cal S}}$ hierarchical filter

If we choose for both decompositions on the trial and test side the classical hierarchical basis transformation [41], the Galerkin-like multiscale V(1,1)-cycle will soon break down when $ b$ increases. A reason for this is that the resulting coarse-grid operators become more and more unstable (w.r.t. convection) and physically meaningless. The hereby computed ``corrections'' finally destroy the convergence of the method [18,43]. Therefore, we apply a matrix-dependent coarsening strategy (c.f. report, Section 3). Table 1 shows the average error reduction rates for a standard multiscale V(1,1)-cycle. Here, we use a matrix-dependent hierarchical decomposition of the trial space together with the classical hierarchical decomposition of the test space. The rates increase slightly with increasing problem size, which is already clear from the behavior of the classical HBMG method with respect to the non-perturbed equation (c.f. the first column, which is the Poisson problem with $ b=0$). However, $ \varrho _{it,1}$ grows also with increasing size of $ b$. This is somewhat surprising since the coarse-grid operators by our method are identical to those of robust multigrid iterations. The same behavior can even be observed if the blocks $ A_k$ are inverted exactly. We realized this by applying an iterative solver to invert these blocks. We omit the corresponding results. Hence, we see from the multigrid interpretation of the hierarchical multiscale method that the loss of robustness must be due to a principal defect of the hierarchical smoother.

However, we will obtain a fairly robust method, if we use a wavelet-like decomposition on the trial side instead of the hierarchical one. The wavelet-like decomposition is based on the classical linear prewavelet filter $ [1\,-6\quad \underline{10}\,-6\quad 1]/10$ and its modifications $ (Q_{k+1,{\cal V},w}^{k})_{*}=[\underline{9} \quad\!\! -6 \quad 1]/10$ and $ (Q_{k+1,{\cal V},e}^{k})_{*}=[1 \quad\!\! -6 \quad \underline{9}]/10$ at the western and eastern boundaries. The robustness of the method is clear from the numbers displayed in Table 2. One observes that the rates even decrease with increasing size of $ b$. Note that we obtained similar results in further experiments when using tensor product $ L$-spline prewavelets.


next up previous
Next: Test example : Diagonally Up: Numerical examples Previous: Computational costs
Frank Kiefer
2001-10-25