A discontinuous Galerkin method for higher-order ordinary differential equations

In this paper, we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal O(Δtp+1) convergence rate in the L2L2 norm. We further show that the p-degree discontinuous solution of differential equation of order m and its first m − 1 derivatives are O(Δt2p+2−m) superconvergent at the end of each step. We also establish that the p-degree discontinuous solution is O(Δtp+2) superconvergent at the roots of (p + 1 − m)-degree Jacobi polynomial on each step. Finally, we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates.