Volume 10, pp. 92-114, 2000.

Behavior of plane relaxation methods as multigrid smoothers

Ignacio M. Llorente and N. Duane Melson

Abstract

This paper contains the first published numerical results and analysis of the behavior of alternating plane relaxation methods as multigrid smoothers for cell-centered grids. The results are very satisfactory: plane smoothers work well in general and their performance improves considerably for strong anisotropies in the right direction because they effectively become exact solvers. In fact, the convergence rate decreases (improves) linearly with increasing anisotropy strength. The methods compared are plane Jacobi with damping, plane Jacobi with partial damping, plane Gauss-Seidel, plane zebra Gauss-Seidel, and line Gauss-Seidel. Based on numerical experiments and local mode analysis, the smoothing factor and cost per cycle of the different methods in the presence of strong anisotropies for Dirichlet boundary conditions are compared. A four-color Gauss-Seidel method is found to have the best numerical and architectural properties of the methods considered in the present work. Although alternating direction plane relaxation schemes are simpler and more robust than other approaches, they are not currently used in industrial and production codes because they require the solution of a two-dimensional problem for each plane in each direction. We verify the theoretical predictions of Thole and Trottenberg that an exact solution of each plane is not necessary; in fact, a single two-dimensional multigrid cycle gives the same result as an exact solution, in much less execution time. As a result, alternating-plane smoothers are found to be highly efficient multigrid smoothers for anisotropic elliptic problems.

Full Text (PDF) [234 KB]

Key words

multigrid methods, anisotropic discrete operators, plane implicit methods, robust multigrid.

AMS subject classifications

65M55.

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