In this talk, we first introduce some backgrounds of delay differential equations and then consider the Galerkin method to solve the vanishing delay differential equations under uniform and quasi-graded mesh. The global convergence and local superconvergence results are obtained. Based on the local superconvergence results, several postprocessing techniques to accelerate the global convergence are proposed. State dependent delay differential equations are also considered. Theoretical expectations are confirmed by numerical experiments.
