Convergence of Restricted Additive Schwarz with impedance transmission conditions for discretised Helmholtz problems

TL;DR

This paper provides the first convergence analysis of the ORAS method for discretized Helmholtz problems, demonstrating its effectiveness as an iterative solver and preconditioner, especially in 2D strip domains with small mesh sizes.

Contribution

It introduces a convergence analysis for ORAS applied to finite element Helmholtz systems, connecting discrete and non-discrete Schwarz methods, and characterizes convergence via impedance-to-impedance maps.

Findings

ORAS inherits Schwarz method convergence properties in small mesh regimes

Convergence is independent of polynomial order in 2D strip domains

Weighted finite-element error analysis shows operator norm convergence of impedance maps

Abstract

The Restricted Additive Schwarz method with impedance transmission conditions, also known as the Optimised Restricted Additive Schwarz (ORAS) method, is a simple overlapping one-level parallel domain decomposition method, which has been successfully used as an iterative solver and as a preconditioner for discretized Helmholtz boundary-value problems. In this paper, we give, for the first time, a convergence analysis for ORAS as an iterative solver -- and also as a preconditioner -- for nodal finite element Helmholtz systems of any polynomial order. The analysis starts by showing (for general domain decompositions) that ORAS as an unconventional finite element approximation of a classical parallel iterative Schwarz method, formulated at the PDE (non-discrete) level. This non-discrete Schwarz method was recently analysed in [Gong, Gander, Graham, Lafontaine, Spence, arXiv 2106.05218], and the present paper gives a corresponding discrete version of this analysis. In particular, for domain decompositions in strips in 2-d, we show that, when the mesh size is small enough, ORAS inherits the convergence properties of the Schwarz method, independent of polynomial order. The proof relies on characterising the ORAS iteration in terms of discrete `impedance-to-impedance maps', which we prove (via a novel weighted finite-element error analysis) converge as $h\rightarrow 0$ in the operator norm to their non-discrete counterparts.