Legendre spectral Galerkin and multi-Galerkin methods for nonlinear Volterra integral equations of Hammerstein type

In this paper, we discuss the superconvergence of the Galerkin solutions for second kind nonlinear integral equations of Volterra-Hammerstein type with a smooth kernel. Using Legendre polynomial bases, we obtain order of convergence O(n-r) for the Legendre Galerkin method in both L2-norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the kernel. The iterated Legendre Galerkin solutions converge with the order O(n-2r), whose convergence order is the same as that of the multi-Galerkin solutions. We also prove that iterated Legendre multi-Galerkin method has order of convergence O(n-3r) in both L2-norm and infinity norm. Numerical examples are given to demonstrate the efficacy of Galerkin and multi-Galerkin methods.