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 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 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.

中文翻译：

---

一种具有高阶传输条件和交叉点处理的亥姆霍兹问题的非重叠域分解方法

摘要 针对大规模时谐波问题的并行有限元求解，提出了一种非重叠域分解方法(DDM)。众所周知，这种方法的收敛速度很大程度上取决于在子域之间的接口上强制执行的传输条件。基于高阶吸收边界条件 (HABC) 的局部条件已被证明是非常合适的，作为导致次优收敛的基本阻抗条件与基于精确狄利克雷-诺依曼 (DtN) 的条件之间的良好折衷与子域的互补相关的映射——计算成本太高。然而，将此方法直接应用于具有内部交叉点（其中两个以上子域相交）和边界交叉点（属于外部边界和至少两个子域的点）的配置是次优的，并且在某些情况下，可以导致错误的结果。在这项工作中，我们使用基于 HABC 的传输条件方法扩展了非重叠 DDM，以有效处理格型分区的交叉点。我们解决了当 HABC 算子用于传输条件时，或当它用于外部边界条件时，或两者兼而有之时的交叉点处理问题。建议的交叉点处理依赖于为 Pade 型 HABC 开发的角​​条件。提出了具有节点有限元离散化的二维数值结果来验证该方法，包括关于频率、网格大小和子域数量的收敛研究。这些结果证明了交叉点处理对于具有规则分区和均质介质的设置的效率。带有扭曲分区和平滑变化的异质介质的数值实验表明了这种处理的鲁棒性。