Post-processing discontinuous Galerkin solutions to Volterra integro-differential equations: Analysis and simulations

摘要

This paper presents a superconvergence extraction technique for Volterra integro-differential equations with smooth and non-smooth kernels. Specifically, extracting superconvergence is done via a post-processed discontinuous Galerkin (DG) method obtained from interpolating the DG solution using Lagrange polynomials at the nodal points. A global superconvergence error bound (in the L∞-norm) is established. For a non-smooth kernel, a family of non-uniform time meshes is used to compensate for the singular behaviour of the exact solution near t=0. The derived theoretical results are numerically validated in a sample of test problems, demonstrating higher-than-expected convergence rates.