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.

将考虑一般形式的边界积分方程，该方程可用于求解亥姆霍兹方程的Dirichlet问题。本文的目的是开发一种快速傅立叶-加勒金方法来求解这些边界积分方程。为此，提出了一种用于拆分积分算子的方案，该方案将相应的积分算子拆分为卷积算子和紧凑算子。提出了一种截断策略，该策略可以将密集系数矩阵压缩为只有\（\ mathcal {O}（n）\）个非零条目的稀疏矩阵，其中n是该方法中使用的傅立叶基础函数的阶数。所提出的快速方法保留了稳定性和最佳收敛顺序。此外，可以在适当的假设下获得指数收敛。数值算例验证了所提方法的逼近精度和计算复杂度的理论结果。