Uniform-in-Time Error Estimates for the Postprocessing Galerkin Method Applied to a Data Assimilation Algorithm

We apply the postprocessing Galerkin method to a recently introduced continuous data assimilation (downscaling) algorithm for obtaining a numerical approximation of the solution of the two-dimensional Navier--Stokes equations corresponding to given measurements from a coarse spatial mesh. Under suitable conditions on the relaxation (nudging) parameter, the resolution of the coarse spatial mesh, and the resolution of the numerical scheme, we obtain uniform-in-time estimates for the error between the numerical approximation given by the postprocessing Galerkin method and the reference solution corresponding to the measurements. Our results are valid for a large class of interpolant operators, including low Fourier modes and local averages over finite volume elements. Notably, we use here the two-dimensional Navier--Stokes equations as a paradigm, but our results apply equally to other evolution equations, such as the Boussinesq system of Benard convection and other oceanic and atmospheric circulation models.