Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension (Journal of Computational Mathematics, 2018)
In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the $L^2$-norm to $(p + 1)$-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be $p + 2$, when piecewise polynomials of degree at most $p$ are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to $(p+1)$-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the $L^2$-norm at $\mathcal{O}(h^{p+2})$ rate. Finally, we prove that the global effectivity indices in the $L^2$-norm converge to unity at $\mathcal{O}(h)$ rate. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be $p+3/2$ and $1/2$, respectively. Our proofs are valid for arbitrary regular meshes using $P^p$ polynomials with $p ≥ 1$. Several numerical experiments are performed to validate the theoretical results.