Supercloseness of Linear DG-FEM and Its Superconvergence Based on the Polynomial Preserving Recovery for Helmholtz Equation

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.