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

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