A novel coarse space applying to the weighted Schwarz method for Helmholtz equations

TL;DR

This paper introduces a new adaptive coarse space for the weighted Schwarz method applied to Helmholtz equations, improving convergence rates and robustness across mesh sizes and wave numbers.

Contribution

A novel coarse space based on eigenvalue functions of local eigenproblems is designed and analyzed for the weighted Schwarz method on Helmholtz problems.

Findings

Uniform convergence independent of mesh size and wave number

The proposed coarse space enhances the efficiency of the Schwarz method

Numerical experiments validate theoretical convergence results

Abstract

In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming nodal finite elements. We design and analyze a new adaptive coarse space for this kind of restricted additive Schwarz method. This coarse space is spanned by some eigenvalue functions of local generalized eigenvalue problems, which are defined by weighted positive semi-definite bilinear forms on subspaces consisting of local discrete Helmholtz-harmonic functions from impedance boundary data. We proved that a two-level hybrid Schwarz preconditioner with the proposed coarse space possesses uniformly convergence independent of the mesh size, the subdomain size and the wave numbers under suitable assumptions. We also introduce an economic coarse space to avoid solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results.