Discrete data assimilation algorithm for the three-dimensional Leray-$\alpha $ model

We study the data assimilation algorithm for the three-dimensional Leray-$\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, which shows exponential convergence up to a term which depends on the size of the errors.