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Abstract
We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using firstorder absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1, 2, 3) that it is welldefined in a tensor product of appropriate local function spaces, each with L^{2} impedance boundary data. We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedancetoimpedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2d, for rectangular domains and stripwise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedancetoimpedance maps that ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. At least for a model case with two subdomains, these results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedancetoimpedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graphpartitioning software.
Original language  English 

Pages (fromto)  259306 
Number of pages  48 
Journal  Numerische Mathematik 
Volume  152 
Issue number  2 
Early online date  20 Sept 2022 
DOIs  
Publication status  Published  31 Oct 2022 
Bibliographical note
Funding Information:We gratefully acknowledge support from the UK Engineering and Physical Sciences Research Council Grants EP/R005591/1 (DL and EAS) and EP/S003975/1 (SG, IGG, and EAS). This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath.
ASJC Scopus subject areas
 Computational Mathematics
 Applied Mathematics
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 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council