### Abstract

In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L^{2}-norm under mesh refinement. The order of convergence is proved to be k+3/2, when k-degree piecewise polynomials with k ≥ 1 are used. We further prove that the global effectivity indices in the L^{2}-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.

Original language | English (US) |
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Pages (from-to) | 172-193 |

Number of pages | 22 |

Journal | International Journal of Numerical Analysis and Modeling |

Volume | 11 |

Issue number | 1 |

State | Published - Jan 1 2014 |

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### Keywords

- Discontinuous Galerkin method
- Hyperbolic problems
- Residual-based a posteriori error estimates
- Superconvergence

### ASJC Scopus subject areas

- Numerical Analysis

### Cite this

**Global convergence of a posteriori error estimates for the discontinuous Galerkin method for one-dimensional linear hyperbolic problems.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Global convergence of a posteriori error estimates for the discontinuous Galerkin method for one-dimensional linear hyperbolic problems

AU - Baccouch, Mahboub

PY - 2014/1/1

Y1 - 2014/1/1

N2 - In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be k+3/2, when k-degree piecewise polynomials with k ≥ 1 are used. We further prove that the global effectivity indices in the L2-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.

AB - In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be k+3/2, when k-degree piecewise polynomials with k ≥ 1 are used. We further prove that the global effectivity indices in the L2-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.

KW - Discontinuous Galerkin method

KW - Hyperbolic problems

KW - Residual-based a posteriori error estimates

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=84887030827&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887030827&partnerID=8YFLogxK

M3 - Article

VL - 11

SP - 172

EP - 193

JO - International Journal of Numerical Analysis and Modeling

JF - International Journal of Numerical Analysis and Modeling

SN - 1705-5105

IS - 1

ER -