A posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional nonlinear scalar conservation laws

Abstract

In this paper, new a posteriori error estimates for a discontinuous Galerkin (DG) formulation applied to nonlinear scalar conservation laws in one space dimension are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local problem with no boundary condition on each element of the mesh. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree Radau polynomial, when p-degree piecewise polynomials with p≥1 are used. This result allows us to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+5/4. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h1/2) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.