A Hybrid Two-Level Weighted Schwarz Method for Helmholtz Equations

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 716-743, April 2025.
Abstract. In this paper we are concerned with a weighted additive Schwarz method with local impedance boundary conditions for a family of Helmholtz problems in two or three dimensions. These problems are discretized by the finite element method with conforming nodal finite elements. We design and analyze an adaptive coarse space for this kind of weighted additive Schwarz method. This coarse space is constructed by some eigenfunctions of local generalized eigenvalue problems posed on the subspaces consisting of local discrete Helmholtz-harmonic functions from impedance boundary data on [math], where [math] is a considered subdomain. Such a generalized eigenvalue problem is defined by two different bilinear forms, [math] and [math], where [math] denotes a weight operator related to [math], and [math] with [math] being the wave number. We prove that a hybrid two-level weighted Schwarz preconditioner with the proposed coarse space possesses uniform convergence independent of the mesh size, the subdomain size, and the wave numbers under suitable assumptions. This result seems the first rigorous convergence result on two-level weighted Schwarz method with local impedance boundary conditions for Helmholtz equations. We also introduce an economical coarse space to avoid solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results.