Recovery-Based Error Estimator for the Discontinuous Galerkin Method for Nonlinear Scalar Conservation Laws in One Space Dimension

Abstract

In this paper, we propose and analyze a robust recovery-based error estimator for the original discontinuous Galerkin method for nonlinear scalar conservation laws in one space dimension. The proposed a posteriori error estimator of the recovery-type is easy to implement, computationally simple, asymptotically exact, and is useful in adaptive computations. We use recent results (Meng et al. in SIAM J Numer Anal 50:2336–2356, 2012) to prove that, for smooth solutions, our a posteriori error estimates at a fixed time converge to the true spatial errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p + 1$, when $p$-degree piecewise polynomials with $p\ge 1$ are used. We further prove that the global effectivity index converges to unity at $\mathcal {O}(h)$ rate. Our proofs are valid for arbitrary regular meshes using $P^p$ polynomials with $p\ge 1$, under the condition that $|f'(u)|$ possesses a uniform positive lower bound, where $f(u)$ is the nonlinear flux function. We provide several numerical examples to support our theoretical results, to show the effectiveness of our recovery-based a posteriori error estimates, and to demonstrate that our results hold true for nonlinear conservation laws with general flux functions. These experiments indicate that the restriction on $f(u)$ is artificial.