Sharp Bounds on Helmholtz Impedance-to-impedance Maps and Application to Overlapping Domain Decomposition

We prove sharp bounds on certain impedance-to-impedance maps (and theircompositions) for the Helmholtz equation with large wavenumber (i.e., athigh-frequency) using semiclassical defect measures. The paper [GGGLS](Gong-Gander-Graham-Lafontaine-Spence, 2022) recently showed that the behaviourof these impedance-to-impedance maps (and their compositions) dictates theconvergence of the parallel overlapping Schwarz domain-decomposition methodwith impedance boundary conditions on the subdomain boundaries. For a modeldecomposition with two subdomains and sufficiently-large overlap, the resultsof this paper combined with those in [GGGLS] show that the parallel Schwarzmethod is power contractive, independent of the wavenumber. For strip-typedecompositions with many subdomains, the results of this paper show that thecomposite impedance-to-impedance maps, in general, behave "badly" with respectto the wavenumber; nevertheless, by proving results about the composite mapsapplied to a restricted class of data, we give insight into thewavenumber-robustness of the parallel Schwarz method observed in the numericalexperiments in [GGGLS].