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 L²-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 p+ 1 order of convergence for the solution and its spatial derivative in the L²-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 p+ 1 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 p+ 3 / 2 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 (p+ 1) -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 L²-norm at O(h^{p + 3 / 2}) rate. Finally, we prove that the global effectivity index in the L²-norm converge to unity at O(h^{1 / 2}) rate. Our proofs are valid for arbitrary regular meshes using P^p polynomials with p≥ 1. Finally, several numerical examples are given to validate the theoretical results.