Asymptotically exact a posteriori error estimates for a one-dimensional linear hyperbolic problem

In this manuscript we investigate the global convergence of the implicit residual-based a posteriori error estimates of Adjerid et al. (2002) [3]. The authors used the discontinuous Galerkin method to solve one-dimensional transient hyperbolic problems and showed that the local error on each element is proportional to a Radau polynomial. The discontinuous Galerkin error estimates under investigation are computed by solving a local steady problem on each element. Here we prove that, for smooth solutions, these a posteriori error estimates at a fixed time t converge to the true spatial error in the L2 norm under mesh refinement.