Using Symmetries and Antisymmetries to Analyze a Parallel Multigrid Algorithm: the Elliptic Boundary Value Problem Case

Abstract. Symmetry and antisymmetry properties of a class of elliptic partial di erential equations are exploited to prove when a particular parallel multilevel algorithm is a direct method rather than the usual iterative method. No smoothing is required for this result. Examples are presented, including variable coe cient ones. A connection between the algorithm in this article and domain decomposition is established, even though this algorithm is more general and di erent. The parallel algorithm is also analyzed when it is iterative and it is shown how to increase processor utilization. Hackbusch's robust multigrid algorithm is analyzed for some model problems and it is shown that the parallel algorithm in this article uses much less computer time with at most the same amount of storage.