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(h2m+1). By analyzing the supercloseness between the DG solution and the interpolation Πhu of the exact solution, we get the optimal order O(hm+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.