Superconvergence of Legendre spectral projection methods for Fredholm-Hammerstein integral equations

In this paper, we consider the multi-Galerkin and multi-collocation methods for solving the Fredholm-Hammerstein integral equation with a smooth kernel, using Legendre polynomial bases. We show that Legendre multi-Galerkin and Legendre multi-collocation methods have order of convergence O(n−3r+34) and O(n−2r+12), respectively, in uniform norm, where n is the highest degree of Legendre polynomial employed in the approximation and r is the smoothness of the kernel. Also, one step of iteration method is used to improve the order of convergence and we prove that iterated Legendre multi-Galerkin and iterated Legendre multi-collocation methods have order of convergence O(n−4r) and O(n−2r), respectively, in uniform norm. Numerical examples are given to illustrate the theoretical results.