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

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.