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We study superconvergence property of the linear finite element method withthe polynomial preserving recovery (PPR) and Richardson extrapolation for thetwo dimensional Helmholtz equation. The $H^1$-error estimate with explicitdependence on the wave number $k$ {is} derived.  First, we prove that under the assumption $k(kh)^2\leq C_0$ ($h$ is the meshsize) and certain mesh condition, the estimate between the finite elementsolution and the linear interpolation of the exact solution is superconvergentunder the $H^1$-seminorm, although the pollution error still exists. Second, weprove a similar result for the recovered gradient by PPR and found that the PPRcan only improve the interpolation error and has no effect on the pollutionerror. Furthermore, we estimate the error between the finite element gradientand recovered gradient and discovered that the pollution error is canceledbetween these two quantities. Finally, we apply the Richardson extrapolation torecovered gradient and demonstrate numerically that PPR combined with theRichardson extrapolation can reduce the interpolation and pollution errorssimultaneously, and therefore, leads to an asymptotically exact {\it aposteriori} error estimator. All theoretical findings are verified by numericaltests.

Superconvergence analysis of linear FEM based on the polynomial  preserving recovery and Richardson extrapolation for Helmholtz equation with  high wave number