Superconvergence of Discontinuous Galerkin Method for Linear Hyperbolic Equations in One Space Dimension∗

In this paper, we study the superconvergence of the error between the discontinuous Galerkin (DG) finite element solution and the exact solution for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise k-th degree polynomials, the error between the DG solution and the exact solution is (k+2)-th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is (k+2)-th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The proof is valid for arbitrary regular meshes and for P polynomials with arbitrary k ≥ 1, and for both periodic boundary conditions and for initial-boundary value problems. We provide numerical experiments of polynomials of degree m = 1 and 2 to demonstrate that the convergent rate is optimal.