Basic smoothing procedures for the multigrid treatment of elliptic 3D operators

摘要

From 2D multigrid it is a fundamental insight that pointwise relaxation combined with standard coarsening gives no reasonable smoothing effect if the given elliptic operator shows appreciably 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 highly efficient even 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.