Parallel support graph preconditioners

Abstract

Support graph preconditioning is a relatively new technique that has gained attention in recent years. Unlike incomplete factorization-based preconditioning, this is a robust technique whose performance is not affected significantly by domain characteristics such as anisotropy and inhomogeneity. A major limitation of this technique is that it is applicable to symmetric diagonally dominant M-matrices only. In this paper, we outline an extension of the technique to symmetric positive definite matrices arising from finite element discretization of elliptic problems. An added advantage of our approach is the inherent parallelism that can be exploited to develop efficient parallel preconditioners. Our method allows trade-off between the preconditioner's parallelism and the rate of convergence of the iterative solver. In contrast, efforts to parallelize incomplete factorization-based preconditioners often result in much slower convergence. Numerical results show that our preconditioner achieves good parallel speedup on distributed memory multiprocessors such as Beowulf workstation clusters. © 2006 Springer-Verlag.