A Discontinuous Galerkin Method for Higher-order Differential Equations

In this paper we propose a new discontinuous finite element method for higher-order initial value problems where the finite element solution exhibits an optimal O(∆tp+1) convergence rate in the L2 norm. We further show that the p-degree discontinuous solution of differential equation of order m and its first m−1 derivatives are O(∆t2p+2−m) superconvergent at the end of each step. We also establish that the p-degree discontinuous solution is O(∆tp+2) superconvergent at the roots of (p + 1 −m)-degree Jacobi polynomial on each step. Finally, we use these results to construct asymptotically correct a posteriori error estimates and present several computational results to validate the theory.