Fast Fourier-Galerkin Methods for Nonlinear Boundary Integral Equations

We develop in this paper a fast Fourier-Galerkin method for solving the nonlinear integral equation which is reformulated from a class of nonlinear boundary value problems. By projecting the nonlinear term onto the approximation subspaces, we make the Fourier-Galerkin method more efficient for solving the nonlinear integral equations. A fast algorithm for solving the resulting discrete nonlinear system is designed by integrating together the techniques of matrix compressing, numerical quadrature for oscillatory integrals, and the multilevel augmentation method. We prove that the proposed method enjoys an optimal convergence order and a nearly linear computational complexity. Numerical experiments are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.