We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier-Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper [Li et al. {\it CMAME} 2023], and we extend the methodology by improving the analysis to be in the $L^2$ norm instead of a weighted $H^1$ norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration.

翻译：本文考虑在可获得解的部分测量数据条件下，针对不可压缩定常（或非定常问题中固定时间步）Navier-Stokes方程的非线性求解器。通过向Picard迭代中添加一个同化项将测量数据融入解中，该项惩罚真实解与求解器解的粗网格插值差异，其实现方式类似于连续数据同化(CDA)应用于时间相关偏微分方程的方法。该问题在文献[Li et al. {\it CMAME} 2023]中已有研究，本文通过两方面扩展该方法：一是将分析改进至$L^2$范数（而非权重依赖于粗网格尺寸的加权$H^1$范数），二是考虑含噪声测量数据情形。对于含噪声测量数据，我们证明CDA-Picard方法在噪声强度范围内保持稳定性和收敛性。数值实验验证了理论结果，并表明当使用含噪声数据时，一个非常有效的策略是利用CDA-Picard为经典牛顿迭代提供初始猜测。