Efficient Multilevel Preconditioners for Three-Dimensional Plane Wave Helmholtz Systems with Large Wave Numbers

Abstract

In this paper we are concerned with solvers for the systems arising from the plane wave discretizations of three dimensional Helmholtz equations with large wave numbers. For simplicity, we consider only the plane wave weighted least squares (PWLS) method for Helmholtz equations. The main goal of this paper is to construct efficient multilevel preconditioners for solving the resulting Helmholtz systems. To this end, we first build a multilevel space decomposition for the plane wave discretization space based on overlapping domain decompositions. Then, corresponding to the space decomposition, we construct two additive multilevel preconditioners with smoothers for the underlying Helmholtz systems. In these preconditioners, each subproblem to be solved has a very small number of degrees of freedom, which just equals the number of the plane wave basis functions on one element. Moreover, the preconditioners possess the optimal computational complexity per iteration. We apply the proposed multilevel preconditio...