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 \(L^2\)-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 \(L^2\)-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 \(L^2\)-norm at \(\mathcal {O}(h^{p+3/2})\) rate. Finally, we prove that the global effectivity index in the \(L^2\)-norm converge to unity at \(\mathcal {O}(h^{1/2})\) rate. Our proofs are valid for arbitrary regular meshes using \(P^p\) polynomials with \(p\ge 1\). Finally, several numerical examples are given to validate the theoretical results.