A Fast Fourier-Galerkin Method for Solving Boundary Integral Equations on Torus-Shaped Surfaces

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.