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 downscal-ing data assimilation algorithm for the numerical approximation of the two- and three-dimensionalNavier–Stokes equations corresponding to given measurements on a coarse spatial scale. For repre-senting the coarse mesh measurements we consider different types of interpolation operators includinga Lagrange interpolant. We obtain uniform-in-time estimates for the error between a finite elementapproximation and the reference solution corresponding to the coarse mesh measurements. We con-sider 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 inversepowers of the viscosity. Some numerical experiments illustrate the theoretical results.