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

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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 discretised 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 is 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, and Spence, Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation], 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.

Original languageEnglish
Pages (from-to)175-215
Number of pages41
JournalMathematics of Computation
Volume92
Issue number339
DOIs
Publication statusPublished - 31 Jan 2023

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