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 m th-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 ( h p + 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. [ABSTRACT FROM AUTHOR]