Global convergence of a posteriori error estimates for the discontinuous Galerkin method for one-dimensional linear hyperbolic problems

In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates 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 k+3/2, when k-degree piecewise polynomials with k ≥ 1 are used. We further prove that the global effectivity indices in the L²-norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These 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 k + 5/4 and 1/2, respectively. Several numerical simulations are performed to validate the theory.