Discrete data assimilation for the three-dimensional Navier-Stokes-α model

We study the data assimilation for the three-dimensional Navier-Stokes-alpha model when the measurements are obtained discretely in time and may be contaminated by systematic errors. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements in an appropriate norm. In the particular case of error-free measurements, we have exponential convergence of the approximating solution toward the reference solution.