The Discontinuous Galerkin Finite Element Method for Ordinary Differential Equations

Abstract

We present an analysis of the discontinuous Galerkin (DG) finite element method for nonlinear ordinary differential equations (ODEs). We prove that the DG solution is $(p + 1)$th order convergent in the $L^2$-norm, when the space of piecewise polynomials of degree $p$ is used. A $(2p+1)$th order superconvergence rate of the DG approximation at the downwind point of each element is obtained under quasi-uniform meshes. Moreover, we prove that the DG solution is superconvergent with order $p+2$ to a particular projection of the exact solution. The superconvergence results are used to show that the leading term of the DG error is proportional to the $(p + 1)$-degree right Radau polynomial. These results allow us to develop a residual-based a posteriori error estimator which is computationally simple, efficient, and asymptotically exact. The proposed a posteriori error estimator is proved to converge to the actual error in the $L^2$-norm with order $p+2$. Computational results indicate that the theoretical orders of convergence are optimal. Finally, a local adaptive mesh refinement procedure that makes use of our local a posteriori error estimate is also presented. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement