The main aim of this paper is to develop a nonconforming mixed finite element method for the time-dependent Navier–Stokes problem with nonlinear damping term. The superconvergent analysis of a backward Euler fully-discrete scheme is presented, where the constrained nonconforming rotated (CNR) \(Q_1\) element and the \(Q_0\) element are used to approximate the velocity \({\varvec{u}}\) and the pressure p, respectively. By use of the characters of the element pair together with some striking skills, i.e., mean-value skill and a new transforming skill with respect to \(\tau \), the superclose estimates of \(O(h^2+\tau )\) for \({\varvec{u}}\) in broken \(H^1\)-norm and p in \(L^2\)-norm are deduced rigorously. Furthermore, the global superconvergent results are obtained through the interpolated postprocessing technique. Finally, some numerical results are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and \(\tau \), the time step.

本文的主要目的是为带有非线性阻尼项的时变Navier-Stokes问题开发一种非协调混合有限元方法。提出了向后欧拉全离散方案的超收敛分析，其中约束非协调旋转（CNR）\（Q_1 \）元素和\（Q_0 \）元素用于近似速度\（{\ varvec {u} } \和压力p。通过使用元素对的特征以及一些惊人的技巧，即关于（（tau \）的均值技巧和新的变换技巧），\（O（h ^ 2 + \ tau ）的超接近估计）\）为\（{\ varvec【U}} \）中破\（H ^ 1 \）范数和p在\（L ^ 2 \）范数被严格推导。此外，通过插值后处理技术获得了全局超收敛结果。最后，提供一些数值结果以证实理论分析。在这里，h是细分参数，\（\ tau \）是时间步长。