This work presents superconvergence estimates of the nonconforming Rannacher--Turek element for second order elliptic equations on any cubical meshes in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. In particular, a corrected numerical flux is shown to be superclose to the Raviart--Thomas interpolant of the exact flux. We then design a superconvergent recovery operator based on local weighted averaging. Combining the supercloseness and the recovery operator, we prove that the recovered flux superconverges to the exact flux. As a by-product, we obtain a superconvergent recovery estimate of the Crouzeix--Raviart element method for general elliptic equations.

翻译：这项工作为以$\mathb{R ⁇ 2}$和$\mathb{R ⁇ 3}$为单位的二次单层椭圆方程中不兼容的 RAnnacher- Turek 元素提供了超趋同估计值。 特别是, 校正的数字通量显示与精确通量的Raviart- Thomas 内插器的超相近。 然后我们根据本地加权平均率设计一个超级趋同的回收操作器。 将超近度和回收操作器结合起来, 我们证明已回收的通量与精确通量的超级通融。 作为副产品, 我们获得了关于普通椭圆方程的Crouzix- Raviart 元素方法的超趋同性回收估计值。