Global in time stability and accuracy of IMEX-FEM data assimilation schemes for the Navier-Stokes equations

TL;DR

This paper analyzes the stability and accuracy of IMEX-FEM data assimilation schemes for the Navier-Stokes equations, providing theoretical guarantees and numerical validation for long-term performance.

Contribution

It offers the first comprehensive stability and accuracy analysis of IMEX-FEM data assimilation methods for Navier-Stokes, including theoretical proofs and practical guidelines.

Findings

Proven long-term stability and accuracy under specific conditions

Numerical tests confirm theoretical predictions

Discretization parameters critically affect results

Abstract

We study numerical schemes for incompressible Navier-Stokes equations using IMEX temporal discretizations, finite element spacial discretizations, and equipped with continuous data assimilation (a technique recently developed by Azouani, Olson, and Titi in 2014). We analyze stability and accuracy of the proposed methods, and are able to prove well-posedness, long time stability, and long time accuracy estimates, under restrictions of the time step size and data assimilation parameter. We give results for several numerical tests that illustrate the theory, and show that, for good results, the choice of discretization parameter and element choices can be critical.