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We present non-overlapping Domain Decomposition Methods (DDM) based onquasi-optimal transmission operators for the solution of Helmholtz transmissionproblems with piece-wise constant material properties. The quasi-optimaltransmission boundary conditions incorporate readily available approximationsof Dirichlet-to-Neumann operators. These approximations consist of eithercomplexified hypersingular boundary integral operators for the Helmholtzequation or square root Fourier multipliers with complex wavenumbers. We showthat under certain regularity assumptions on the closed interface of materialdiscontinuity, the DDM with quasi-optimal transmission conditions arewell-posed. We present a DDM framework based on Robin-to-Robin (RtR) operatorsthat can be computed robustly via boundary integral formulations. Moreimportantly, the use of quasi-optimal transmission operators results in DDMthat converge in small numbers of iterations even in the challenginghigh-contrast, high-frequency regime of Helmholtz transmission problems.Furthermore, the DDM presented in this text require only minor modifications tohandle the case of transmission problems in partially coated domains, whilestill maintaining excellent convergence properties. We also investigate thedependence of the DDM iterative performance on the number of subdomains.

Domain Decomposition Methods based on quasi-optimal transmission operators for the solution of Helmholtz transmission problems