Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension

In this paper we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to transient convection-diffusion problems in one space dimension. We show that the leading terms of the local discretization errors for the p-degree LDG solution and its spatial derivative are proportional to (p+1)-degree right and left Radau polynomials, respectively. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are O(hp+²) superconvergent at the roots of (p+1)-degree right and left Radau polynomials, respectively. The superconvergence results are used to construct asymptotically correct a posteriori error estimates. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at O(hp+³) and O(hp+²) rates, respectively. We also show that the global effectivity indices for the solution and its derivative in the ^L2-norm converge to unity at O(^h2) and O(h) rates, respectively. Finally, we show that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hp+²) superconvergent solutions. Our proofs are valid for arbitrary regular meshes and for ^Pp polynomials with p≥1, and for periodic, Dirichlet, and mixed Dirichlet-Neumann boundary conditions. Several numerical simulations are performed to validate the theory.