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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.

A postprocess of the standard Galerkin method for the discretization of dissipative equations is presented. The postprocessed Galerkin method uses the same approximate inertial manifold Φ app to approximate the high wave number modes of the solution as in the nonlinear Galerkin (NLG) method. However, in this postprocessed Galerkin method the value of app is calculated only once, after the time integration of the standard Galerkin method is completed, contrary to the NLG in which Φapp evolves with time and affects the time evolution of the lower wave number modes. The postprocessed Galerkin method, which is much cheaper to implement computationally than the NLG, is shown, in the case of Fourier modes, to possess the same rate of convergence (accuracy) as the simplest version of the NLG, which is based on either the Foias­Manley­Temam approximate inertial manifold or the Euler­Galerkin approximate inertial manifold. This is proved for some problems in one and two spatial dimensions, including the Navier­Stokes equations under periodic boundary conditions. The advantages of postprocessing that we present here apply not only to the standard Galerkin method, but also to the computationally more efficient pseudospectral method. [ABSTRACT FROM AUTHOR]