# Error analysis of fully discrete mixed finite element data assimilation schemes for the Navier-Stokes equations

## TL;DR

This paper analyzes the error behavior of fully discrete mixed finite element schemes for the Navier-Stokes equations within a data assimilation framework, providing uniform error estimates for various discretization methods.

## Contribution

It introduces a comprehensive error analysis for fully discrete mixed finite element data assimilation schemes, including new bounds independent of inverse viscosity.

## Key findings

- Uniform error estimates for all schemes analyzed.
- Error bounds that do not depend on inverse powers of viscosity.
- Validation of semi-implicit and fully implicit time discretizations.

## Abstract

In this paper we consider fully discrete approximations with inf-sup stable mixed finite element methods in space to approximate the Navier-Stokes equations. A continuous downscaling data assimilation algorithm is analyzed in which measurements on a coarse scale are given represented by different types of interpolation operators. For the time discretization an implicit Euler scheme, an implicit and a semi-implicit second order backward differentiation formula %BDF2 scheme and a semi-implicit BDF2 scheme are considered. Uniform in time error estimates are obtained for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements. For the spatial discretization we consider both the Galerkin method and the Galerkin method with grad-div stabilization. For the last scheme error bounds in which the constants do not depend on inverse powers of the viscosity are obtained.