  We introduce a localized version of the nudging data assimilation algorithm
for the periodic 2D Navier-Stokes equations in which observations are confined
(i.e., localized) to a window that moves across the entire domain along a
predetermined path at a given speed. We prove that, if the movement is fast
enough, then the algorithm perfectly synchronizes with a reference solution.
The analysis suggests an informed scheme in which the subdomain moves according
to a region where the error is dominant is optimal. Numerical simulations are
presented that compare the efficacy of movement that follows a regular pattern,
one guided by the dominant error, and one that is random.

Analysis of PDEs