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

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.