A Low-Rank Direct Solver for Nonconformal Domain Decomposition Methods

Nonoverlapping and nonconformal domain decomposition methods (DDMs) have established an effective discretization framework for complex electromagnetic (EM) applications, where global iterative solvers are commonly employed to tackle DDM equations. However, DDMs are not free from convergence issues, and excessive iteration numbers can significantly prolong the duration of simulations involving multiple right-hand sides (RHSs). This study explores the possibility in developing a low-rank direct solver for nonconformal DDMs, with the goal of offering a stable and reliable alternative to DDM iterative approaches. Our direct solver consists of two major components: a subdomain solver and a DDM solver. The subdomain solver is developed by extracting an interface skeleton from subdomains, achieved through recursive partitioning and merging of mesh elements. The DDM solver builds upon skeletonization and is constructed through recursive reorderings of interface matrices with the integration of hierarchical matrix techniques and randomized matrix compressions. Both solvers have drawn inspiration from existing mathematical studies but also possess distinct features due to unique matrix structures in DDMs. We validate and demonstrate the effectiveness of our solver through several numerical examples, showcasing its capability to stably handle DDM systems and its efficiency in scenarios involving multiple RHSs.