### Abstract

In this paper, we provide the optimal convergence rate of a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L^{2}-norm to (p + 1)-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be p + 2, when piecewise polynomials of degree at most p are used. We use these results to show that the leading error terms on each element for the solution and its derivative are proportional to (p + 1)-degree right and left Radau polynomials. These new results enable us to construct residual-based a posteriori error estimates of the spatial errors. We further prove that, for smooth solutions, these a posteriori LDG error estimates converge, at a fixed time, to the true spatial errors in the L^{2}-norm at O(h^{p+2}) rate. Finally, we show that the global effectivity indices in the L^{2}-norm converge to unity at O(h) rate. The current 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 p+3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.

Original language | English (US) |
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Pages (from-to) | 355-380 |

Number of pages | 26 |

Journal | International Journal of Numerical Analysis and Modeling |

Volume | 14 |

Issue number | 3 |

State | Published - 2017 |

### Keywords

- A posteriori error estimation
- Local discontinuous Galerkin method
- Radau points
- Second-order wave equation
- Superconvergence

### ASJC Scopus subject areas

- Numerical Analysis