 
Summary: A Distributed Memory Unstructured GaussSeidel Algorithm for
Multigrid Smoothers
Mark F. Adams
November 19, 2001
Abstract
GaussSeidel is a popular multigrid smoother as it is provably optimal on structured grids and exhibits
superior performance on unstructured grids. GaussSeidel is not used to our knowledge on distributed
memory machines as it is not obvious how to parallelize it eectively. We, among others, have found that
Krylov solvers preconditioned with Jacobi, block Jacobi or overlapped Schwarz are eective on unstruc
tured problems. GaussSeidel 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 GaussSeidel 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.
Key words: unstructured multigrid, algebraic multigrid, parallel graph algorithms, parallel GaussSeidel
1 Introduction
The availability of large high performance computers is providing scientists and engineers with the oppor
tunity to simulate a variety of complex physical systems with ever more accuracy and thereby exploit the
advantages of computer simulations over laboratory experiments. The nite element method is widely used
for these simulations. The nite element method requires that one or several linearized systems of sparse
