A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems

Abstract

A non-overlapping domain decomposition method (DDM) is proposed for the parallel finite-element solution of large-scale time-harmonic wave problems. It is well-known that the convergence rate 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 this 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. We address the question of the cross-point treatment when the HABC operator is used in the transmission condition, or when it is used in the exterior boundary condition, or both. The proposed cross-point treatment relies on corner conditions developed for Padé-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 of the cross-point treatment for settings with regular partitions and homogeneous media. Numerical experiments with distorted partitions and smoothly varying heterogeneous media show the robustness of this treatment.