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 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. (c) 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1461-1491, 2015