In this paper we consider fully discrete approximations with inf-sup stable mixed finite element methods in space to approximate the Navier-Stokes equations. A continuous downscaling data assimilation algorithm is analyzed in which measurements on a coarse scale are given represented by different types of interpolation operators. For the time discretization an implicit Euler scheme, an implicit and a semi-implicit second-order backward differentiation formula are considered. Uniform-in-time error estimates are obtained for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements. For the spatial discretization we consider both the Galerkin method and the Galerkin method with grad-div stabilization. For the last scheme error bounds in which the constants do not depend on inverse powers of the viscosity are obtained.

中文翻译：

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Navier-Stokes方程全离散混合有限元数据同化方案的误差分析

在本文中，我们考虑在空间中采用ins-up稳定混合有限元方法的完全离散逼近，以近似Navier-Stokes方程。分析了连续降尺度的数据同化算法，其中给出了由不同类型的插值运算符表示的粗略度量。对于时间离散化，考虑隐式欧拉方案，隐式和半隐式二阶后向微分公式。对于完全离散近似值和对应于测量值的参考解决方案之间的误差，所有方法均获得了及时的误差估计。对于空间离散化，我们同时考虑了Galerkin方法和具有grad-div稳定的Galerkin方法。