Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection-Diffusion Problems

In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection-diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the -norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve order of convergence for the solution and its spatial derivative in the -norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order towards a special Gauss-Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the -norm at rate. Finally, we prove that the global effectivity index in the -norm converge to unity at rate. Our proofs are valid for arbitrary regular meshes using polynomials with . Finally, several numerical examples are given to validate the theoretical results.