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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems
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by Shihua Gong, Ivan G. Graham and Euan A. Spence;
Math. Comp. 92 (2023), 175-215
DOI: https://doi.org/10.1090/mcom/3772
Published electronically: October 6, 2022

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. 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.
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Bibliographic Information
  • Shihua Gong
  • Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
  • Address at time of publication: School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China
  • MR Author ID: 1240731
  • ORCID: 0000-0003-3650-2283
  • Email: gongshihua@cuhk.edu.cn
  • Ivan G. Graham
  • Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
  • MR Author ID: 76020
  • ORCID: 0000-0002-5730-676X
  • Email: i.g.graham@bath.ac.uk
  • Euan A. Spence
  • Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
  • MR Author ID: 858295
  • ORCID: 0000-0003-1236-4592
  • Email: e.a.spence@bath.ac.uk
  • Received by editor(s): October 29, 2021
  • Received by editor(s) in revised form: June 3, 2022
  • Published electronically: October 6, 2022
  • Additional Notes: The authors gratefully acknowledge support from the UK Engineering and Physical Sciences Research Council Grants EP/R005591/1 (EAS) and EP/S003975/1 (SG, IGG, and EAS)
  • © Copyright 2022 American Mathematical Society
  • Journal: Math. Comp. 92 (2023), 175-215
  • MSC (2020): Primary 35J05, 65N55, 65N22; Secondary 65F08
  • DOI: https://doi.org/10.1090/mcom/3772
  • MathSciNet review: 4496963