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

Introduction We propose a new variant of smoothed aggregation (SA) suitable for nonsymmetric linear systems. SA is a highly successful and popular algebraic multigrid method for symmetric positive-definite systems [3, 2]. A relatively large number of significant parallel smoothed aggregation codes have been developed at universities, companies, and laboratories. Many of these codes are quite sophisticated and represent a significant investment in time and effort. Despite the large body of work on multigrid methods for fluid dynamics and the significant successes of smoothed aggregation, there have been surprisingly few attempts at generalizing the smoothed aggregation idea to nonsymmetric systems. Most smoothed aggregation variants for nonsymmetric systems either sacrifice performance on diffusion-dominated problems or do not perform well on highly convective problems. In this talk, a new variant is proposed that performs well in both the highly diffusive and highly convective regimes. 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. Restrictor Smoothing and Local Damping Let A refer to a discretized partial differential equation, P be an interpolation operator, and R be a restriction operator. The key to any algebraic multigrid scheme is the precise definition of P and R. In standard smoothed aggregation, P is normally defined by