In this paper, based on the scalar auxiliary variable (SAV) approach, the superconvergence error analysis is investigated for the time-dependent Navier–Stokes equations. In which, an equivalent system of the Navier–Stokes equations with three variables and a fully-discrete scheme is developed with semi-implicit Euler discretization for the temporal direction and low-order bilinear-constant finite element discretization for the spatial direction, respectively. With the help of the high-precision estimations of the bilinear-constant finite element pair on the rectangular meshes, the superclose error estimates for velocity in -norm and pressure in -norm are obtained by treating the trilinear term carefully and skillfully. The global superconvergence results are also derived in terms of a simple and efficient interpolation post-processing technique. Finally, some numerical results are provided to demonstrate the correctness of the theoretical analysis.

中文翻译：

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时变纳维-斯托克斯方程线性化半隐式双线性常数SAV有限元法的超收敛误差分析

本文基于标量辅助变量（SAV）方法，研究了瞬态 Navier-Stokes 方程的超收敛误差分析。其中，建立了三变量纳维-斯托克斯方程的等效系统和全离散格式，分别对时间方向进行半隐式欧拉离散化，对空间方向进行低阶双线性常数有限元离散化。借助矩形网格上双线性常数有限元对的高精度估计，通过对三线性项的仔细处理，得到了范数速度和范数压力的超接近误差估计。全局超收敛结果也是通过简单高效的插值后处理技术得出的。最后，提供了一些数值结果来证明理论分析的正确性。