This paper considers continuous data assimilation (CDA) in partial differential equation (PDE) discretizations where nudging parameters can be taken arbitrarily large. We prove that long-time optimally accurate solutions are obtained for such parameters for the heat and Navier-Stokes equations (using implicit time stepping methods), with error bounds that do not grow as the nudging parameter gets large. Existing theoretical results either prove optimal accuracy but with the error scaled by the nudging parameter, or suboptimal accuracy that is independent of it. The key idea to the improved analysis is to decompose the error based on a weighted inner product that incorporates the (symmetric by construction) nudging term, and prove that the projection error from this weighted inner product is optimal and independent of the nudging parameter. We apply the idea to BDF2 - finite element discretizations of the heat equation and Navier-Stokes equations to show that with CDA, they will admit optimal long-time accurate solutions independent of the nudging parameter, for nudging parameters large enough. Several numerical tests are given for the heat equation, fluid transport equation, Navier-Stokes, and Cahn-Hilliard that illustrate the theory.

翻译：本文研究了偏微分方程离散化中的连续数据同化问题，其中松弛参数可任意取大。我们证明了对于热方程和Navier-Stokes方程（采用隐式时间步进方法），使用此类参数可获得长期最优精度的解，其误差界不随松弛参数增大而增长。现有理论结果要么证明了最优精度但误差随松弛参数缩放，要么证明了独立于该参数的次优精度。改进分析的关键思想是基于一个加权内积分解误差，该内积包含了（构造上对称的）松弛项，并证明该加权内积产生的投影误差具有最优性且独立于松弛参数。我们将该思想应用于热方程和Navier-Stokes方程的BDF2-有限元离散化，证明在连续数据同化下，当松弛参数足够大时，这些离散化将产生独立于松弛参数的最优长期精确解。针对热方程、流体输运方程、Navier-Stokes方程和Cahn-Hilliard方程给出了若干数值实验以验证理论。