Superconvergence and postprocessing of the continuous Galerkin method for nonlinear Volterra integro-differential equations

Abstract

We propose a novel postprocessing technique for improving the global accuracy of the continuous Galerkin (CG) method for nonlinear Volterra integro-differential equations. The key idea behind the postprocessing technique is to add a higher order Lobatto polynomial of degree k + 1 to the CG approximation of degree k. We first show that the CG method superconverges at the nodal points of the time partition. We further prove that the postprocessed CG approximation converges one order faster than the unprocessed CG approximation in the L2-, H1- and L∞-norms. As a by-product of the postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact. Numerical examples are presented to highlight the superconvergence properties of the postprocessed CG approximations and the robustness of the a posteriori error estimators.