A Distributed Memory Unstructured Gauss-Seidel Algorithm for Multigrid Smoothers

Gauss-Seidel is a popular multigrid smoother as it is provably optimal on structured grids and exhibits superior performance on unstructured grids. Gauss-Seidel is not used to our knowledge on distributed memory machines as it is not obvious how to parallelize it effectively. We, among others, have found that Krylov solvers preconditioned with Jacobi, block Jacobi or overlapped Schwarz are effective on unstructured problems. Gauss-Seidel does however have some attractive properties, namely: fast convergence, no global communication (ie, no dot products) and fewer .ops per iteration as one can incorporate an initial guess naturally. This paper discusses an algorithm for parallelizing Gauss-Seidel for distributed memory computers for use as a multigrid smoother and compares its performance with preconditioned conjugate gradients on unstructured linear elasticity problems with up to 76 million degrees of freedom.