Numerical Solutions for second-kind Volterra integral equations by Galerkin methods

Abstract

In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent papers [6] and [9] by H. Brunner, Q. Lin and N. Yan. Moreover, using an interpolation post-processing technique, we obtain a global superconvergence of the O(h2r)-convergence rate in the piecewise-polynomial space of degree not exceeding (r−1). As a by-product of our results, all these higher order numerical methods can also provide an a posteriori error estimator, which gives critical and useful information in the code development.