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.

这项工作为\（\ mathbb {R} ^ {2} \）和\（\ mathbb {R} ^ {3} \）中任何立方网格上的二阶椭圆方程提供了非一致性Rannacher-Turek元素的超收敛估计。特别是，校正后的数值通量显示出与精确通量的Raviart-Thomas插值超级接近。然后，我们根据局部加权平均数设计一个超收敛恢复算子。结合超闭合性和恢复算子，我们证明了恢复的通量超收敛到精确的通量。作为副产品，我们获得了用于一般椭圆方程的Crouzeix-Raviart元素方法的超收敛恢复估计。