### 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) |
---|---|

Pages (from-to) | 129-153 |

Number of pages | 25 |

Journal | Applied Numerical Mathematics |

Volume | 106 |

DOIs | |

State | Published - Aug 1 2016 |

### Fingerprint

### 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

### Cite this

**Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

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

AU - Baccouch, Mahboub

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

UR - http://www.scopus.com/inward/record.url?scp=84975468498&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975468498&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2016.03.008

DO - 10.1016/j.apnum.2016.03.008

M3 - Article

AN - SCOPUS:84975468498

VL - 106

SP - 129

EP - 153

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -