A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids

Abstract

In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651–1671, 2015). We first prove that the DG solution converges in the $L^2$-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be $p+2$, when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two $(p+1)$-degree right Radau polynomials in the x and y directions. The less significant part converges to zero at ${\mathcal {O}}\left( h^{p+2}\right)$. These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the $L^2$-norm at ${\mathcal {O}}\left( h^{p+2}\right)$ rate. Finally, we prove that the global effectivity indices in the $L^2$-norm converge to unity at ${\mathcal {O}}(h)$ rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.