2023/10/13 by Martin Averseng, Xavier Claeys, Averseng, Martin +3
Engineering · #65F08 #65N38 #65N55 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2310.09204
openalex publication_date 2023/10/13 · openalex created_date 2023/10/17 · openalex updated_date 2026/07/28
We present a preconditioning method for the linear systems arising from the boundary element discretization of the Laplace hypersingular equation on a 2-dimensional triangulated surface Γ in ℝ3. We allow Γ to belong to a large class of geometries that we call polygonal multiscreens, which can be non-manifold. After introducing a new, simple conforming Galerkin discretization, we analyze a substructuring domain-decomposition preconditioner based on ideas originally developed for the Finite Element Method. The surface Γ is subdivided into non-overlapping regions, and the application of the preconditioner is obtained via the solution of the hypersingular equation on each patch, plus a coarse subspace correction. We prove that the condition number of the preconditioned linear system grows poly-logarithmically with H/h, the ratio of the coarse mesh and fine mesh size, and our numerical results indicate that this bound is sharp. This domain-decomposition algorithm therefore guarantees significant speedups for iterative solvers, even when a large number of subdomains is used.