Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection–diffusion problems

Abstract

In this paper, new a posteriori error estimates for the local discontinuous Galerkin (LDG) formulation applied to transient convection–diffusion problems in one space dimension are presented and analyzed. These error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree right Radau polynomial while the leading error term for the solution’s derivative is proportional to a (p+1)-degree left Radau polynomial, when polynomials of degree at most p are used. These results are used 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. More precisely, we prove that our LDG error estimates converge to the true spatial errors at O(hp+5/4) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h1/2) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.