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We consider fully discrete numerical schemes for a downscaling dataassimilation algorithm aimed at approximating the velocity field of the 2DNavier-Stokes equations corresponding to given coarse mesh observationalmeasurements. The time discretization is done by considering semi- andfully-implicit Euler schemes, and the spatial discretization is based on aspectral Galerkin method. The two fully discrete algorithms are shown to beunconditionally stable, with respect to the size of the time step, number oftime steps and the number of Galerkin modes. Moreover, explicit, uniform intime error estimates between the fully discrete solution and the referencesolution corresponding to the observational coarse mesh measurements areobtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensionalNavier-Stokes equations, subject to the no-slip Dirichlet or periodic boundaryconditions, are used in this work as a paradigm. The complete analysis that ispresented here can be extended to other two- and three-dimensional dissipativesystems under the assumption of global existence and uniqueness.

Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm