Superconvergence of Discontinuous Galerkin Solutions for Delay Differential Equations of Pantograph Type

This paper is concerned with the superconvergence of the discontinuous Galerkin solutions for delay differential equations with proportional delays vanishing at $t = 0$. Two types of superconvergence are analyzed here. The first is based on interpolation postprocessing to improve the global convergence order by finding the superconvergence points of discontinuous Galerkin solutions. The second type follows from the integral iteration which just requires a local integration procedure applied to the discontinuous Galerkin solution, thus increasing the order of convergence. The theoretical results are illustrated by a broad range of numerical examples.