### Abstract

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 L^{2}-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 L^{2}-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 L^{2}-norm under mesh refinement. The order of convergence is proved to be p+2. Finally, we prove that the global effectivity index in the L^{2}-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.

Original language | English (US) |
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Pages (from-to) | 129-153 |

Number of pages | 25 |

Journal | Applied Numerical Mathematics |

Volume | 106 |

DOIs | |

State | Published - Aug 1 2016 |

### Keywords

- A posteriori error estimation
- Adaptive mesh refinement
- Discontinuous Galerkin method
- Nonlinear ordinary differential equations
- Superconvergence

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics