We study continuous data assimilation (CDA) applied to projection and penalty methods for the Navier-Stokes (NS) equations. Penalty and projection methods are more efficient than consistent NS discretizations, however are less accurate due to modeling error (penalty) and splitting error (projection). We show analytically and numerically that with measurement data and properly chosen parameters, CDA can effectively remove these splitting and modeling errors and provide long time optimally accurate solutions.

翻译：我们研究了将连续数据同化（CDA）应用于纳维-斯托克斯（NS）方程的投影法和惩罚法。与严格一致的NS离散格式相比，惩罚法和投影法效率更高，但由于建模误差（惩罚法）和分裂误差（投影法）的存在，其精度较低。我们通过理论分析和数值实验表明，在引入测量数据并合理选取参数的情况下，CDA能够有效消除这些分裂误差和建模误差，从而长时间提供最优精度的解。