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 $$L^2$$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 $$p+1$$p+1 order of convergence for the solution and its spatial derivative in the $$L^2$$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 $$p+1$$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 $$p+3/2$$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 $$(p+1)$$(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 $$L^2$$L2-norm at $$\mathcal {O}(h^{p+3/2})$$O(hp+3/2) rate. Finally, we prove that the global effectivity index in the $$L^2$$L2-norm converge to unity at $$\mathcal {O}(h^{1/2})$$O(h1/2) rate. Our proofs are valid for arbitrary regular meshes using $$P^p$$Pp polynomials with $$p\ge 1$$p≥1. Finally, several numerical examples are given to validate the theoretical results.