A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations

Abstract

Standard multigrid algorithms have proven ine ective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's e ectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as two hundred. For xed wave numbers, it displays grid-independent convergence rates and has costs proportional to number of unknowns.Abstract. Standard multigrid algorithms have proven ine ective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's e ectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as two hundred. For xed wave numbers, it displays grid-independent convergence rates and has costs proportional to number of unknowns.

A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

“…The Helmholtz equation, however, does not belong to the class of PDEs for which off-the-shelf multigrid methods perform efficiently. Convergence degradation and, consequently, loss of O(N ) complexity are caused by difficulties encountered in the smoothing and coarse-grid correction components; see [7,33] for a discussion.We present an efficient numerical solution technique for the heterogeneous highwavenumber Helmholtz equation, discretized by fourth-order finite differences. Recently,in [10], a robust preconditioned Bi-CGSTAB method has been proposed for solving these problems, in which the preconditioner is based on a second Helmholtz equation with an imaginary shift.…”

“…Many authors, e.g. [5,7,13,19], have contributed to the development of appropriate multigrid methods for the Helmholtz equation, but an efficient multigrid treatment of heterogeneous problems with high wavenumers arising in engineering settings has not yet been proposed in the literature. The multigrid method [4,14] is known to be a highly efficient iterative method, for example, for discrete Poisson-type equations, even with fourth-order accurate discretizations [6,33].…”

“…The Helmholtz equation, however, does not belong to the class of PDEs for which off-the-shelf multigrid methods perform efficiently. Convergence degradation and, consequently, loss of O(N ) complexity are caused by difficulties encountered in the smoothing and coarse-grid correction components; see [7,33] for a discussion.…”

Smoothed aggregation for Helmholtz problems

“…An alternative relaxation strategy is to use a non-stationary Krylov or Krylov-like method, such as GMRES [17] or USYMQR [10,38]. In particular, USYMQR forms a favorable Krylov-like subspace in which the residual is reduced.…”

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

“…Controllability methods have been proposed for both Helmholtz and Navier problems in [17,18]. Multigrid methods have been considered for acoustic and elastic problems in [19,20,21,22]. With multigrid methods, it is difficult to define a stable and sufficiently accurate coarse grid problem and smoother for it.…”