A posteriori error estimates and adaptivity for the discontinuous Galerkin solutions of nonlinear second-order initial-value problems

In this paper, we propose and analyze an efficient and reliable a posteriori error estimator of residual-type for the discontinuous Galerkin (DG) method applied to nonlinear second-order initial-value problems for ordinary differential equations. This estimator is simple, efficient, and asymptotically exact. We use our recent optimal L2 error estimates and superconvergence results of Baccouch [15] to show that the significant parts of the DG discretization errors are proportional to the (p+1)-degree right Radau polynomial, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimator which is obtained by solving a local residual problem with no initial condition on each element. We prove that, for smooth solutions, the proposed a posteriori error estimator converges to the actual error in the L2-norm with order of convergence p+2. Computational results indicate that the theoretical order of convergence is sharp. By adding the a posteriori error estimate to the DG solution, we obtain a post-processed approximation which superconverges with order p+2 in the L2-norm. Moreover, we demonstrate the effectiveness of the this error estimator. Finally, we present a local adaptive mesh refinement (AMR) procedure that makes use of our local a posteriori error estimate. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1. Several numerical results are presented to validate the theoretical results.