Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation
نویسندگان
چکیده
Abstract We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition unity. provide novel analysis this method at PDE level (without discretization). First, we formulate as fixed point iteration, and show (in dimensions 1, 2, 3) that it well-defined in tensor product appropriate local function spaces, each $$L^2$$ L2 impedance boundary data. then obtain bound on norm operator terms norms certain impedance-to-impedance maps arising from interactions subdomains. These bounds conditions under which (some power of) contraction. In 2-d, rectangular domains strip-wise decompositions (with subdomain only its immediate neighbours), present two techniques verifying assumptions ensure contractivity operator. The first through semiclassical analysis, gives rigorous estimates valid frequency tends to infinity. At least model case subdomains, these results verify required sufficiently large overlap. For more realistic decompositions, directly compute by solving canonical (local) eigenvalue problems. give numerical experiments illustrate theory. also iterative remains convergent and/or provides good preconditioner cases not covered theory, including general such those obtained via automatic graph-partitioning software.
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ژورنال
عنوان ژورنال: Numerische Mathematik
سال: 2022
ISSN: ['0945-3245', '0029-599X']
DOI: https://doi.org/10.1007/s00211-022-01318-8