Application of Hierarchical Low-Rank Approximation Based on Skeletonization to Iterative Solver for Boundary Element Method

Abstract

This work presents a fast iterative solver for 2D Helmholtz' transmission problems, based on skeletonization, a technique commonly used in fast direct solvers. The skeletonization-based iterative technique utilizes the explicit form of hierarchical off-diagonal low-rank matrices to efficiently perform adjoint matrix-vector multiplication. This approach enables the use of a variant of IDR(s)Stab(l), which was difficult to implement with the fast multipole method. Numerical examples demonstrate the superiority of the proposed method, which combines IDR(s)Stab(l) and skeletonization, over the combination of ordinary Krylov subspace methods such as BiCGStab or GMRES with skeletonization.