### Abstract

In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L<sup>2</sup> error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862-901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the L<sup>2</sup>-norm under mesh refinement. The order of convergence is proved to be p + 3 / 2, when p-degree piecewise polynomials with p ≥ 1 are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O (h p + 3 / 2) superconvergent solutions. Our computational results show higher O (h p + 2) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L<sup>2</sup>-norm converge to unity at O (h 1 / 2) rate while numerically they exhibit O (h 2) and O (h) rates, respectively. Numerical experiments are shown to validate the theoretical results.

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

Pages (from-to) | 1461-1491 |

Number of pages | 31 |

Journal | Numerical Methods for Partial Differential Equations |

Volume | 31 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2015 |

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### Keywords

- local discontinuous Galerkin method
- residual-based a posteriori error estimates
- second-order wave equation
- superconvergence

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics
- Analysis

### Cite this

**Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one-dimensional second-order wave equation.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one-dimensional second-order wave equation

AU - Baccouch, Mahboub

PY - 2015/9/1

Y1 - 2015/9/1

N2 - In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L2 error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862-901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, 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 p + 3 / 2, when p-degree piecewise polynomials with p ≥ 1 are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O (h p + 3 / 2) superconvergent solutions. Our computational results show higher O (h p + 2) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L2-norm converge to unity at O (h 1 / 2) rate while numerically they exhibit O (h 2) and O (h) rates, respectively. Numerical experiments are shown to validate the theoretical results.

AB - In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L2 error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862-901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, 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 p + 3 / 2, when p-degree piecewise polynomials with p ≥ 1 are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O (h p + 3 / 2) superconvergent solutions. Our computational results show higher O (h p + 2) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L2-norm converge to unity at O (h 1 / 2) rate while numerically they exhibit O (h 2) and O (h) rates, respectively. Numerical experiments are shown to validate the theoretical results.

KW - local discontinuous Galerkin method

KW - residual-based a posteriori error estimates

KW - second-order wave equation

KW - superconvergence

UR - http://www.scopus.com/inward/record.url?scp=84937878092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937878092&partnerID=8YFLogxK

U2 - 10.1002/num.21955

DO - 10.1002/num.21955

M3 - Article

VL - 31

SP - 1461

EP - 1491

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 5

ER -