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

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.

Superconvergence of the local discontinuous galerkin method applied to the one‐dimensional second‐order wave equation

“…We expect that a similar special projection of the initial conditions of Yang and Shu and a new technique will be needed to obtain the optimal rate of superconvergence. In Part II of this work, , we will use the L 2 optimal error estimates and the superconvergence results proved in this article to show that the a posteriori error estimates of Baccouch , at a fixed time t , converge to the true spatial error in the L 2 ‐norm under mesh refinement.…”

“…These results will be needed to prove that the LDG discretization error estimates converge to the true spatial errors under mesh refinement. This will be reported in Part II of this work .…”

“…In Part II of this work, , we will use these superconvergence results to prove that the a posteriori LDG error estimates for the solution and its spatial derivative, introduced by the author in , converge to the true spatial errors under mesh refinement.…”

“…The superconvergence results of this article will be used to prove that the LDG discretization error estimates for the solution and its derivative introduced by the author in converge to the true spatial errors under mesh refinement. This will be discussed in the second part of this work . Even though the analysis in this article is restricted to the classical wave equation in one space dimension, the same superconvergence results can be directly generalized to second‐order hyperbolic partial differential equations (with constant and variable coefficients) in one space variable.…”

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

“…We provided a mathematical justification in our earlier studies for one-dimensional convection [21,63,64], convection-diffusion [65], and wave equation [66]. Indeed, for one-dimensional problems, we proved that the error estimates without the time change of e u converge to the true errors under mesh refinement.…”