A fast multiscale Galerkin method for solving a boundary integral equation in a domain with corners

Abstract

A fast multiscale Galerkin method is proposed for solving the boundary integral equation derived from the Dirichlet problem of the Laplace equation in a domain with corners. It is well known that the integral operator in the equation can be split into two operators, one is noncompact, the other is compact. We design two truncation strategies for the representation matrices of these operators, respectively, which compress these two dense matrices to sparse ones having only $\mathcal{𝒪}\left(2^n\right)$ number of nonzero entries, where $2^n$ is the number of the wavelet basis functions used in the method. We prove that the proposed truncation strategies do not ruin the stability and convergence rate of the integral equation. Numerical experiments are presented to verify the theoretical results and demonstrate the effectiveness of the method.