A Superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels

Abstract

We study the numerical solution for Volerra integro-differential equations with smooth and non-smooth kernels. We use a $h$-version discontinuous Galerkin (DG) method and derive nodal error bounds that are explicit in the parameters of interest. In the case of non-smooth kernel, it is justified that the start-up singularities can be resolved at superconvergence rates by using non-uniformly graded meshes. Our theoretical results are numerically validated in a sample of test problems.