Superconvergence analysis of DG-FEM based on the polynomial preserving recovery for Helmholtz equation with high wave number

We study superconvergence property of the linear discontinuous Galerkin finite element method with the polynomial preserving recovery (PPR) and Richardson extrapolation for the two dimensional Helmholtz equation. The error estimate with explicit dependence on the wave number $k$, the penalty parameter $\mu$ and the mesh condition parameter $\alpha$ is derived. First, we prove that under the assumption $k(kh{)}^2\le C_0$ ($h$ is the mesh size) and certain mesh condition, the estimate between the finite element solution and the linear interpolation of the exact solution is superconvergent under the $\text{norme}\cdot$-seminorm. Second, we prove a superconvergence result for the recovered gradient by PPR. Furthermore, we estimate the error between the finite element gradient and recovered gradient, which motivate us to define the a posteriori error estimator. Finally, Some numerical examples are provided to confirm the theoretical results of superconvergence analysis. All theoretical findings are verified by numerical tests.