Parallel Multigrid for Anisotropic Elliptic Equations

Abstract

In this paper two well-known robust multigrid solvers for anisotropic operators on structured grids are compared: alternating-plane smoothers combined with full coarsening and plane smoothers combined with semi-coarsening. The study has taken into account not only numerical properties but also architectural ones, focusing on cache memory exploitation and parallel characteristics. Experimental results for the sequential algorithms have been obtained on two different systems based on the MIPS R10000 processor, but with different L2 cache sizes and memory bandwidths (an SGI O2 workstation and an SGI Origin 2000 system). Although the alternating-plane approach is the best choice for sequential implementations, experimental estimations show poor parallel efficiencies. For the semicoarsening alternative two different parallel implementations have been considered. The first one has optimal parallel characteristics but due to deterioration of the convergence properties its realistic efficiency is not satisfactory. In the second one, some processors remain idle during a short period of time on every multigrid cycle. However, the second parallel algorithm is more efficient since it preserves the numerical properties of the sequential version. Parallel experiments have also been taken on a Cray T3E system.