  In this paper we analyze a finite element method applied to a continuous
downscaling data assimilation algorithm for the numerical approximation of the
two and three dimensional Navier-Stokes equations corresponding to given
measurements on a coarse spatial scale. For representing the coarse mesh
measurements we consider different types of interpolation operators including a
Lagrange interpolant. We obtain uniform-in-time estimates for the error between
a finite element approximation and the reference solution corresponding to the
coarse mesh measurements. We consider both the case of a plain Galerkin method
and a Galerkin method with grad-div stabilization. For the stabilized method we
prove error bounds in which the constants do not depend on inverse powers of
the viscosity. Some numerical experiments illustrate the theoretical results.
