An h–p Version of the Discontinuous Galerkin Method for Volterra Integro-Differential Equations with Vanishing Delays

We present an h–p version of the discontinuous Galerkin time stepping method for Volterra integro-differential equations with vanishing delays. We derive a priori error bounds in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}- and L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document}-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Moreover, we prove that the h–p version of the discontinuous Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders achieves exponential rates of convergence for solutions with start-up singularities. Numerical experiments are presented to illustrate the theoretical results.