Title: Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number,
Abstract: We study superconvergence property of the linear finite element method with
the polynomial preserving recovery (PPR) and Richardson extrapolation for the
two dimensional Helmholtz equation. The $H^1$-error estimate with explicit
dependence on the wave number $k$ {is} derived.
First, we prove that under the assumption $k(kh)^2\leq C_0$ ($h$ 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 $H^1$-seminorm, although the pollution error still exists. Second, we
prove a similar result for the recovered gradient by PPR and found that the PPR
can only improve the interpolation error and has no effect on the pollution
error. Furthermore, we estimate the error between the finite element gradient
and recovered gradient and discovered that the pollution error is canceled
between these two quantities. Finally, we apply the Richardson extrapolation to
recovered gradient and demonstrate numerically that PPR combined with the
Richardson extrapolation can reduce the interpolation and pollution errors
simultaneously, and therefore, leads to an asymptotically exact {\it a
posteriori} error estimator. All theoretical findings are verified by numerical
tests.