Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations

We develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove that the DG solution exhibits an optimal O(h^p+1) convergence rate in the L²-norm when p-degree piecewise polynomials with p≥1 are used. We further prove that the DG solution is O(h^2p+1) superconvergent at the downwind points. We use these results to prove that the p-degree DG solution is O(h^p+2) super close to a particular projection of the exact solution. This superconvergence result allows us to show that the true error can be divided into a significant part and a less significant part. The significant part of the discretization error for the DG solution is proportional to the (p+1)-degree right Radau polynomial and the less significant part converges at O(h^p+2) rate in the L²-norm. Numerical experiments demonstrate that the theoretical rates are optimal. Based on the global superconvergent approximations, we construct asymptotically exact a posteriori error estimates and prove that they converge to the true errors in the L²-norm under mesh refinement. The order of convergence is proved to be p+2. Finally, we prove that the global effectivity index in the L²-norm converges to unity at O(h) rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement. A local adaptive procedure that makes use of our local a posteriori error estimate is also presented.