Superconvergence results for linear second-kind Volterra integral equations

In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order ({mathcal {O}}(h^{r}),) whereas the iterated Galerkin solutions converge with the order ({mathcal {O}}(h^{2r})) in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order ({mathcal {O}}(h^{3r})) in infinity norm. Numerical examples are given to illustrate the theoretical results.