## 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^{2} 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^{2}-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^{2}-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 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics