A fast Fourier-Galerkin method for solving integral equations of second kind with weakly singular kernels

Abstract

We propose in this paper a convenient way to compress the dense matrix representation of a compact integral operator with a weakly singular kernel under the Fourier basis. This compression leads to a sparse matrix with only ${\mathcal{O}}(n\log n)$ number of nonzero entries, where 2n+1 denotes the order of the matrix. Based on this compression strategy, we develop a fast Fourier-Galerkin method for solving second kind integral equations with weakly singular kernels. We prove that the approximate order of the truncated equation remains optimal and that the spectral condition number of the coefficient matrix of the truncated linear system is uniformly bounded. Furthermore, we develop a fast algorithm for solving the corresponding truncated linear system, which preserves the optimal order of the approximate solution with only ${\mathcal{O}}(n\log^{2}n)$ number of multiplications required. Numerical examples complete the paper.