In this paper, we introduce a fast Fourier-Galerkin method for solving boundary integral equations on torus-shaped surfaces, which are diffeomorphic to a torus. We analyze the properties of the integral operator's kernel to derive the decay pattern of the entries in the representation matrix. Leveraging this decay pattern, we devise a truncation strategy that efficiently compresses the dense representation matrix of the integral operator into a sparser form containing only $\mathcal{O}(N\ln^2 N)$ nonzero entries, where $N$ denotes the degrees of freedom of the discretization method. We prove that this truncation strategy achieves a quasi-optimal convergence order of $\mathcal{O}(N^{-p/2}\ln N)$, with $p$ representing the degree of regularity of the exact solution to the boundary integral equation. Additionally, we confirm that the truncation strategy preserves stability throughout the solution process. Numerical experiments validate our theoretical findings and demonstrate the effectiveness of the proposed method.

翻译：本文提出了一种用于求解环面（与环面微分同胚）上边界积分方程的快速傅里叶-伽辽金方法。我们分析了积分算子核的性质，以推导出表示矩阵元素的衰减模式。利用这种衰减模式，我们设计了一种截断策略，能够将积分算子的稠密表示矩阵高效地压缩为仅包含 $\mathcal{O}(N\ln^2 N)$ 个非零元素的稀疏形式，其中 $N$ 表示离散化方法的自由度。我们证明了该截断策略能够达到 $\mathcal{O}(N^{-p/2}\ln N)$ 的拟最优收敛阶，其中 $p$ 表示边界积分方程精确解的正则性阶数。此外，我们确认该截断策略在整个求解过程中保持了稳定性。数值实验验证了我们的理论结果，并证明了所提方法的有效性。