A New Petrov–Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems

Abstract

We propose a new variant of smoothed aggregation (SA) suitable for nonsymmetric linear systems. The new algorithm is based on two key generalizations of SA: restriction smoothing and local damping. Restriction smoothing refers to the smoothing of a tentative restriction operator via a damped Jacobi-like iteration. Restriction smoothing is analogous to prolongator smoothing in standard SA and in fact has the same form as the transpose of prolongator smoothing when the matrix is symmetric. Local damping refers to damping parameters used in the Jacobi-like iteration. In standard SA, a single damping parameter is computed via an eigenvalue computation. Here, local damping parameters are computed by considering the minimization of an energy-like quantity for each individual grid transfer basis function. Numerical results are given showing how this method performs on highly nonsymmetric systems.