A fast Fourier-Galerkin method for solving singular boundary integral equations

We propose in this paper a convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the Fourier basis. The compression leads to a sparse matrix with only O(n log n) nonzero entries, where 2n or 2n + 1 denotes the order of the matrix. Based on this compression strategy, we develop a fast Fourier-Galerkin method for solving a class of singular boundary integral equations. We prove that the fast Fourier-Galerkin method gives the optimal convergence order O( n(-t)), where t denotes the degree of regularity of the exact solution. Moreover, we design a fast scheme for solving the corresponding truncated linear system. We show that solving this system requires only an O( n log(2) n) number of multiplications. We present numerical examples to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed algorithm.