Convergence of a mobile data assimilation scheme for the 2D Navier-Stokes equations

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