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 pp-degree LDG solution and its spatial derivative are proportional to (p+1)(p+1)-degree right and left Radau polynomials, respectively. Thus, the discretization errors for the pp-degree LDG solution and its spatial derivative are O(hp+2)O(hp+2) superconvergent at the roots of (p+1)(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 tt converge to the true errors at O(hp+3)O(hp+3) and O(hp+2)O(hp+2) rates, respectively. We also show that the global effectivity indices for the solution and its derivative in the L2L2-norm converge to unity at O(h2)O(h2) and O(h)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)O(hp+2) superconvergent solutions. Our proofs are valid for arbitrary regular meshes and for PpPp polynomials with p≥1p≥1, and for periodic, Dirichlet, and mixed Dirichlet–Neumann boundary conditions. Several numerical simulations are performed to validate the theory.