预测某些物理系统和生物系统的未来状态在气象学、海洋学、油藏管理、神经科学、医学科学、金融业等多个不同的领域里都是至关重要的。降尺度数据同化包括一套将理论模型与观测数据适当结合的技术，以获得对系统未来状态的更加准确的预测。在这篇文章中，我们首先整理经典的混合有限元理论并最终证明 Taylor-Hood 元满足离散 inf-sup 条件；然后介绍了应用于不可压 Navier-Stokes 方程的连续数据同化算法 (CDA-NSE) 和文献中相关的理论分析结果，并使用 Taylor-Hood 元对文献中的算例做了复现和进一步延伸的计算研究。在结论里展望了未来的工作方向。

Predicting the future state of certain physical and biological systems is crucial in many different fields, such as meteorology, oceanography, reservoir management, neuroscience, medical science, and finance. Downscaling data assimilation includes a set of techniques that appropriately combine theoretical models with observational data to obtain more accurate predictions of the future state of the system. In this paper, we first sort out the classical mixed finite element theory and finally prove that Taylor-Hood elements satisfy the discrete inf-sup condition. Then, the continuous data assimilation algorithm (CDA-NSE) applied to incompressible Navier-Stokes equations is introduced and the relevant theoretical analysis results in the literature are presented. The examples in the literature are reproduced and further extended by Taylor-Hood element. In the conclusion, the future direction of work is prospected.