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We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance 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 impedance-to-impedance 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 graph-partitioning software.<\/jats:p>","DOI":"10.1007\/s00211-022-01318-8","type":"journal-article","created":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T21:02:42Z","timestamp":1663707762000},"page":"259-306","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation"],"prefix":"10.1007","volume":"152","author":[{"given":"Shihua","family":"Gong","sequence":"first","affiliation":[]},{"given":"Martin J.","family":"Gander","sequence":"additional","affiliation":[]},{"given":"Ivan G.","family":"Graham","sequence":"additional","affiliation":[]},{"given":"David","family":"Lafontaine","sequence":"additional","affiliation":[]},{"given":"Euan A.","family":"Spence","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,20]]},"reference":[{"issue":"5","key":"1318_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00526-021-02047-w","volume":"60","author":"K Ammari","year":"2021","unstructured":"Ammari, K., Amrouche, C.: Resolvent estimates for wave operators in Lipschitz domains. 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