Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number
Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number
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发表时间：
2017-03
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通讯作者：
Yu Du;Haijun Wu;Zhimin Zhang
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作者：
Yu Du;Haijun Wu;Zhimin Zhang
研究二维Helmholtz方程保多项式恢复线性有限元方法的超收敛性。本文导出了与波数k有显式依赖关系的H^1误差估计。首先，我们证明了在假设k（kh）^2\leqC_0 $（$h$为网格尺寸）和一定的网格条件下，虽然污染误差仍然存在，但有限元解与精确解的线性插值之间的估计在$H^1$-ε下是超收敛的.其次，我们证明了一个类似的结果，恢复梯度的PPR，发现PPR只能改善插值误差，并没有影响污染误差。此外，我们估计了有限元梯度和恢复梯度之间的误差，发现这两个量之间的污染误差被抵消。最后，我们将Richardson外推应用于恢复梯度，并通过数值计算证明了PPR与Richardson外推相结合可以同时减小插值误差和污染误差，从而得到一个渐近精确的后验误差估计.所有的理论研究结果进行了验证的数值试验。
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.