TY - JOUR

T1 - Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations

AU - Baccouch, Mahboub

N1 - Funding Information:
The authors would like to thank the two anonymous reviewers for the valuable comments and suggestions which improve the quality of the paper. This research was supported by the University Committee on Research and Creative Activity (UCRCA Proposal 2015-01-F ) at the University of Nebraska at Omaha.
Publisher Copyright:
© 2016 The Author

PY - 2016/8/1

Y1 - 2016/8/1

N2 - 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(hp+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(h2p+1) superconvergent at the downwind points. We use these results to prove that the p-degree DG solution is O(hp+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(hp+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.

AB - 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(hp+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(h2p+1) superconvergent at the downwind points. We use these results to prove that the p-degree DG solution is O(hp+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(hp+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.

KW - A posteriori error estimation

KW - Adaptive mesh refinement

KW - Discontinuous Galerkin method

KW - Nonlinear ordinary differential equations

KW - Superconvergence

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U2 - 10.1016/j.apnum.2016.03.008

DO - 10.1016/j.apnum.2016.03.008

M3 - Article

AN - SCOPUS:84975468498

SN - 0168-9274

VL - 106

SP - 129

EP - 153

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -