We prove sharp bounds on certain impedance-to-impedance maps (and their
compositions) for the Helmholtz equation with large wavenumber (i.e., at
high frequency) using semiclassical defect measures. Gong et al. (Numer.Math. 152:2 (2022), 259–306) recently showed that the behaviour of these
impedance-to-impedance maps (and their compositions) dictates the convergence of
the parallel overlapping Schwarz domain-decomposition method with impedance
boundary conditions on the subdomain boundaries. For a model decomposition with
two subdomains and sufficiently large overlap, the results of this paper combined
with those of Gong et al. show that the parallel Schwarz method is power
contractive, independent of the wavenumber. For strip-type decompositions
with many subdomains, the results of this paper show that the composite
impedance-to-impedance maps, in general, behave “badly” with respect to the
wavenumber; nevertheless, by proving results about the composite maps applied to a
restricted class of data, we give insight into the wavenumber-robustness of the
parallel Schwarz method observed in the numerical experiments of Gong et
al.
