  The error analysis of a proper orthogonal decomposition (POD) data
assimilation (DA) scheme for the Navier-Stokes equations is carried out. A
grad-div stabilization term is added to the formulation of the POD method.
Error bounds with constants independent on inverse powers of the viscosity
parameter are derived for the POD algorithm. No upper bounds in the nudging
parameter of the data assimilation method are required. Numerical experiments
show that, for large values of the nudging parameter, the proposed method
rapidly converges to the real solution, and greatly improves the overall
accuracy of standard POD schemes up to low viscosities over predictive time
intervals.
