Galkowski, Jeffrey Eric; Spence, Euan; (2026) Convergence theory for two-level hybrid Schwarz preconditioners for high-frequency Helmholtz problems. SIAM Journal on Numerical Analysis (In press).

Text simple-arxiv-26-Sept-2025.pdf - Accepted Version Access restricted to UCL open access staff until 2 April 2026. Download (447kB)

Abstract

We give a novel convergence theory for two-level hybrid Schwarz domaindecomposition (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 problemadapted basis functions that have been recently analysed as coarse spaces for the Helmholtz equation, as well as all spaces in which the Galerkin solutions 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 problemadapted basis functions.

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