Convergence theory for two-level hybrid Schwarz preconditioners for high-frequency Helmholtz problems

TL;DR

This paper develops a new convergence theory for two-level hybrid Schwarz preconditioners applied to high-frequency Helmholtz problems, accommodating arbitrary subdomain sizes and boundary conditions, and providing the first rigorous results for non-problem-adapted coarse spaces.

Contribution

It introduces a novel convergence framework for two-level hybrid Schwarz methods for Helmholtz equations, covering general subdomains, boundary conditions, and coarse spaces.

Findings

Convergence conditions ensure preconditioned matrix is close to identity.

Applicable to arbitrary subdomain sizes and boundary conditions.

First rigorous results for coarse spaces with piecewise polynomial functions.

Abstract

We give a novel convergence theory for two-level hybrid Schwarz domain-decomposition (DD) methods for finite-element discretisations of the high-frequency Helmholtz equation. This theory gives sufficient conditions for the preconditioned matrix to be close to the identity, and covers DD subdomains of arbitrary size, arbitrary absorbing layers/boundary conditions on both the global and local Helmholtz problems, and coarse spaces not necessarily related to the subdomains. The assumptions on the coarse space are satisfied by the approximation spaces using problem-adapted basis functions that have been recently analysed as coarse spaces for the Helmholtz equation, as well as all spaces that are known to be quasi-optimal via a Schatz-type argument. As an example, we apply this theory when the coarse space consists of piecewise polynomials; these are then the first rigorous convergence results about a two-level Schwarz preconditioner applied to the high-frequency Helmholtz equation with a coarse space that does not consist of problem-adapted basis functions.