Superconvergence of discontinuous Galerkin solutions for higher-order ordinary differential equations

In this paper, we study the superconvergence properties of the discontinuous Galerkin (DG) method applied to one-dimensional mth-order ordinary differential equations without introducing auxiliary variables. We show that the leading term of the discretization error on each element is proportional to a combination of Jacobi polynomials. Thus, the p  -degree DG solution is O(hp+2)O(hp+2) superconvergent at the roots of specific combined Jacobi polynomials. Moreover, we use these results to compute simple, efficient and asymptotically exact a posteriori error estimates and to construct higher-order DG approximations.