Superconvergence and a posteriori error estimates of the DG method for scalar hyperbolic problems on Cartesian grids

In this paper, we analyze the discontinuous Galerkin (DG) finite element method for the steady two-dimensional transport-reaction equation on Cartesian grids. We prove the L2 stability and optimal L2 error estimates for the DG scheme. We identify a special numerical flux for which the L2-norm of the solution is of order p + 1, when tensor product polynomials of degree at most p are used. We further prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be p + 1/2. We also provide a very simple derivative recovery formula which is O(hp+1) superconvergent approximation to the directional derivative. Moreover, we establish an O(h2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element. Finally, we prove that the proposed a posteriori DG error estimates converge to the true errors in the L2-norm at O(hp+1) rate and that the global effectivity index converges to unity at O(h) rate. Our results are valid without the flow condition restrictions. We perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal.