A discontinuous Galerkin method for higher-order ordinary differential equations

摘要

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(△t~(p+1) convergence rate in the y~2 norm. We further show that the p-degree discontinuous solution of differential equation of order m and its first m - 1 derivatives are O(△t~(2p+2-m) superconvergent at the end of each step. We also establish that the p-degree discontinuous solution is O(△t~(p+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.