A Note on Multi-block Relaxation Schemes for Multigrid Solvers

E cient and robust multigrid solvers for anisotropic problems typically use either semi-coarsened grids or implicit smoothers line relaxation in 2D and plane relaxation in 3D. However, both of these may be di cult to implement in codes using multiblock structured grids where there may be no natural de nition of a global `line' or `plane'. These multi-block structured grids are often used in uid dynamic applications to capture complex geometries and/or to facilitate parallel processing. In this paper, we investigate the performance of multigrid algorithms using implicit smoothers within the blocks of a such a grid. By looking at a model problem, the 2-D anisotropic di usion equation, we show that true multigrid e ciency is achieved only when the block sizes are proportional to the strength of the anisotropy. Further, the blocks must overlap and the size of the overlap must again be proportional to the strength of the anisotropy. This research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the rst author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001