Uniform in Time Error Estimates for a Finite Element Method Applied to a Downscaling Data Assimilation Algorithm for the Navier--Stokes Equations

Abstract

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.