Robustness of multi-grid applied to anisotropic equations on convex domains and on domains with re-entrant corners
We analyse multigrid applied to anisotropic equations within the framework of smoothing and approximation properties developed by \textit{W. Hackbusch} [Multi-grid methods and applications (1985; Zbl 0595.65106)]. For a model anisotropic equation on a square, we give an up-till-now missing proof of an estimate concerning the approximation property which is essential to show robustness. Furthermore, we show a corresponding estimate for a model anisotropic equation on an \(L\)-shaped domain. The existing estimates for the smoothing property are not suitable to prove robustness for either 2-cyclic Gauss-Seidel smoothers or for less regular problems such as our second model equation. For both cases, we give sharper estimates. By combination of our results concerning smoothing and approximation properties, robustness of \(W\)-cycle multigrid applied to both our model equations will follow for a number of smoothers.
scientific article; zbMATH DE number 569197
scientific article