Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations

TL;DR

This paper provides uniform-in-time error estimates for finite element methods applied to a downscaling data assimilation algorithm for Navier-Stokes equations, including stabilized variants with viscosity-independent bounds.

Contribution

It introduces uniform error bounds for finite element approximations of Navier-Stokes with data assimilation, including stabilized methods with viscosity-independent constants.

Findings

Error estimates are uniform in time for both plain and stabilized Galerkin methods.

Stabilized method error bounds do not depend on inverse powers of viscosity.

Numerical experiments confirm theoretical error bounds.

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.