Guidance for Choosing Multigrid Preconditioners for Systems of Elliptic Partial Differential Equations

Abstract

This paper presents some guidance for choosing multigrid preconditioners for systems of partial differential equations (PDEs). Rather than developing a black-box algorithm for arbitrary systems of PDEs, the strategy in this paper is to classify systems with multigrid preconditioners. From this classification, a new preconditioner is developed, and it is shown to be suitable for a particular class of strongly cross-coupled systems, solving the whole cross-coupled system almost as effectively as solving just the diagonal system. Also, this paper gives an explanation of why block diagonal preconditioners are somewhat effective for certain strongly cross-coupled systems that at first glance appear to be intractable for multigrid preconditioning.