LAUSR

In this paper we study the supercloseness property of the linear discontinuous Galerkin (DG) finite element method and its superconvergence behavior after post-processing by the polynomial preserving recovery (PPR). The error estimate with explicit dependence on the wave number k, the penalty parameter $$\mu $$μ and the mesh condition parameter $$\alpha $$α is derived. We prove the supercloseness between the DG finite element solution and the linear interpolation and the superconvergence for the recovered gradient by the PPR under the assumption $$k(kh)^2\le C_0$$k(kh)2≤C0 (h is the mesh size) and certain mesh conditions. Furthermore, we estimate the error between the DG numerical gradient and recovered gradient, which motivates us to define the a posteriori error estimator and design a Richardson extrapolation to post-process the recovered gradient by PPR. Finally, some numerical examples are provided to confirm the theoretical results of superconvergence analysis.