Coarse-Grained Parallel Matrix-Free Solution of a Three-Dimensional Elliptic Prototype Problem

The finite difference discretization of the Poisson equation in three dimensions results in a large, sparse, and highly structured system of linear equations. This prototype problem is used to analyze the performance of the parallel linear solver on coarse-grained clusters of workstations. The conjugate gradient method with a matrix-free implementation of the matrix-vector product with the system matrix is shown to be optimal with respect to memory usage and runtime performance. Parallel performance studies confirm that speedup can be obtained. When only an ethernet interconnect is available, best performance is limited to up to 4 processors, since the conjugate gradient method involves several communications per iteration. Using a high performance Myrinet interconnect, excellent speedup is possible for at least up to 32 processors. These results justify the use of this linear solver as the computational kernel for the time-stepping in a system of reaction-diffusion equations.