Superconvergence of discontinuous Galerkin methods for nonlinear delay differential equations with vanishing delay

In this paper, we investigate the local superconvergence of the discontinuous Galerkin (DG) solutions on quasi-graded meshes for nonlinear delay differential equations with vanishing delay. It is shown that the optimal order of the DG solution at the mesh points is O(h(2m+1)). By analyzing the supercloseness between the DG solution and the interpolation Pi(h)u of the exact solution, we get the optimal order O(h(m+2)) of the DG solution at characteristic points. We then extend the convergence results of DG solutions to state dependent delay differential equations. Numerical examples are provided to illustrate the theoretical results. (C) 2018 Elsevier B.V. All rights reserved.