Basic Smoothing Procedures for the Multigrid Treatment of Elliptic 3D-Operators
From 2D-multigrid it is a fundamental insight that pointwise relaxation combined with standard coarsening gives no reasonable smoothing effect as soon as the given elliptic operator has a considerably anisotropic behavior. Corresponding questions and practical consequences are discussed in this paper for the 3D-case: In the general anisotropic 3D-case, even line relaxation is not sufficient if standard coarsening is maintained. Instead, “plane relaxation” is necessary in certain cases. If plane relaxation is applied correctly and performed by use of appropriate 2D-multigrid, the resulting 3D-multigrid method has an asymptotic complexity of O(N) (where N = number of 3D-grid points) and is — indeed — highly efficient already for moderate values of N. In contrast to the common opinion, plane relaxation turns out to be a simple and general smoothing concept for standard elliptic 3D-problems.
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