An optimized Schwarz domain decomposition method with cross-point treatment for time-harmonic acoustic scattering

The parallel finite-element solution of large-scale time-harmonic wave problems is addressed with a non-overlapping optimized Schwarz domain decomposition method (DDM). It is well-known that the efficiency of this kind of method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) have proved to be well-suited, as a good compromise between basic impedance conditions, which lead to suboptimal convergence , and conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain-which are too expensive to compute. However, a direct application of the approach for configurations with interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) is suboptimal and, in some cases, can lead to incorrect results. In this work, we extend a non-overlapping DDM with HABC-based transmission conditions approach to efficiently deal with cross-points for lattice-type partitioning. The proposed cross-point treatment relies on corner conditions developed for Pade-type HABCs. Two-dimensional numerical results with a nodal finite-element discretization are proposed to validate the approach, including convergence studies with respect to the frequency, the mesh size and the number of subdomains. These results demonstrate the efficiency for settings with regular partitions and homogeneous media. Numerical experiments with non-regular partitions and smoothly varying heterogeneous media show the robustness of the approach.