A fast Fourier-Galerkin method solving boundary integral equations for the Helmholtz equation with exponential convergence

A boundary integral equation in general form will be considered, which can be used to solve Dirichlet problems for the Helmholtz equation. The goal of this paper is to develop a fast Fourier-Galerkin method solving these boundary integral equations. To this aim, a scheme for splitting integral operators is presented, which splits the corresponding integral operator into a convolution operator and a compact operator. A truncation strategy is presented, which can compress the dense coefficient matrix to a sparse one having only $\mathcal {O}(n)$ nonzero entries, where n is the order of the Fourier basis functions used in the method. The proposed fast method preserves the stability and optimal convergence order. Moreover, exponential convergence can be obtained under suitable assumptions. Numerical examples are presented to confirm the theoretical results for the approximation accuracy and computational complexity of the proposed method.