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. The paper [GGGLS] (Gong-Gander-Graham-Lafontaine-Spence, 2022) 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 in [GGGLS] 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 in [GGGLS].

翻译：我们利用半经典亏损测度，证明了赫姆霍兹方程在大波数（即高频）下某些阻抗-阻抗映射（及其复合）的尖锐界。文献[GGGLS]（Gong-Gander-Graham-Lafontaine-Spence，2022）近期表明，这些阻抗-阻抗映射（及其复合）的行为决定了子域边界带阻抗边界条件的并行重叠Schwarz区域分解方法的收敛性。对于具有两个子域且重叠足够大的模型分解，本文结果与[GGGLS]中结论相结合表明，并行Schwarz方法具有与波数无关的幂压缩性。对于含多个子域的条带型分解，本文结果表明复合阻抗-阻抗映射在一般情况下相对于波数表现出"不良"行为；然而，通过证明作用于受限数据类的复合映射的相关结论，我们揭示了[GGGLS]数值实验中观察到的并行Schwarz方法波数鲁棒性的内在机理。