The finite element method (FEM) is a discretization technique for solving partial differential equations. It widely used in many fields of engineering such as computational fluid dynamics. The superconvergence of finite element solutions is an interesting and useful phenomenon in the scientific computing of real world problems and has become an area of active research in recent years. A general superconvergence of discontinuous Galerkin finite element method for the elliptic problem is established by using L^2-projection method. Regularity assumptions for the elliptic problem with regular partitions are required. Numerical experiments are given to verify the theoretical results.