  We develop, analyze, and test an approximate, global data
assimilation/synchronization algorithm based on purely local observations for
the two-dimensional Navier-Stokes equations on the torus. We prove that, for
any error threshold, if the reference flow is analytic with sufficiently large
analyticity radius, then it can be recovered within that threshold. Numerical
computations are included to demonstrate the effectiveness of this approach, as
well as variants with data on moving subdomains. In particular, we demonstrate
numerically that machine precision synchronization is achieved for mobile data
collected from a small fraction of the domain.
