### Abstract

In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L^{2}-norm. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O(h^{2p+1}) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise P^{p} polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

Original language | English (US) |
---|---|

Pages (from-to) | 160-179 |

Number of pages | 20 |

Journal | Applied Numerical Mathematics |

Volume | 115 |

DOIs | |

State | Published - May 1 2017 |

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### Keywords

- A priori error estimates
- Discontinuous Galerkin method
- Nonlinear second-order ordinary differential equations equation
- Superconvergence

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

**Superconvergence of the discontinuous Galerkin method for nonlinear second-order initial-value problems for ordinary differential equations.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Superconvergence of the discontinuous Galerkin method for nonlinear second-order initial-value problems for ordinary differential equations

AU - Baccouch, Mahboub

PY - 2017/5/1

Y1 - 2017/5/1

N2 - In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O(h2p+1) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

AB - In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O(h2p+1) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

KW - A priori error estimates

KW - Discontinuous Galerkin method

KW - Nonlinear second-order ordinary differential equations equation

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=85010433462&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85010433462&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2017.01.007

DO - 10.1016/j.apnum.2017.01.007

M3 - Article

AN - SCOPUS:85010433462

VL - 115

SP - 160

EP - 179

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -