Data Assimilation for Navier-Stokes using the Least-Squares Finite-Element Method

TL;DR

This paper presents a least-squares finite-element method for data assimilation in steady-state Navier-Stokes equations, offering a computationally efficient approach that integrates Bayesian analysis and demonstrates accurate results even on coarse meshes.

Contribution

The work introduces a novel least-squares finite-element approach for data assimilation in Navier-Stokes, which is computationally efficient and compatible with Bayesian inference, including Kalman filter equivalence.

Findings

Method achieves good approximation quality on coarse meshes.

Incorporating limited data corrects discretization errors.

Approach is computationally as efficient as a single forward simulation.

Abstract

We investigate theoretically and numerically the use of the Least-Squares Finite-element method (LSFEM) to approach data-assimilation problems for the steady-state, incompressible Navier-Stokes equations. Our LSFEM discretization is based on a stress-velocity-pressure (S-V-P) first-order formulation, using discrete counterparts of the Sobolev spaces $H({\rm div}) \times H^1 \times L^2$ respectively. Resolution of the system is via minimization of a least-squares functional representing the magnitude of the residual of the equations. A simple and immediate approach to extend this solver to data-assimilation is to add a data-discrepancy term to the functional. Whereas most data-assimilation techniques require a large number of evaluations of the forward-simulations and are therefore very expensive, the approach proposed in this work uniquely has the same cost as a single forward run. However, the question arises: what is the statistical model implied by this choice? We answer this within the Bayesian framework, establishing the latent background covariance model and the likelihood. Further we demonstrate that - in the linear case - the method is equivalent to application of the Kalman filter, and derive the posterior covariance. We practically demonstrate the capabilities of our method on a backward-facing step case. Our LSFEM formulation (without data) is shown to have good approximation quality, even on relatively coarse meshes - in particular with respect to mass-conservation and reattachment location. Adding limited velocity measurements from experiment, we show that the method is able to correct for discretization error on very coarse meshes, as well as correct for the influence of unknown and uncertain boundary-conditions.