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

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 stressvelocity-pressure (S-V-P) first-order formulation, using discrete counterparts of the Sobolev spaces H(div) × H 1 × L 2 for the variables respectively. In general S-V-P formulations are promising when the stresses are of special interest, e.g. for non-Newtonian, multiphase or turbulent flows. Resolution of the system is via minimization of a leastsquares 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 datadiscrepancy term to the functional. Whereas most data-assimilation techniques require a large number of evaluations of the forward-simulation 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.

Vectorial variational problems in L ∞ constrained by the Navier–Stokes equations*

“…In this paper, inspired by the methodology of data assimilation, especially variational data assimilation in continuous time (for relevant works we refer e.g. to [13,19,26,28,29,43,45,50]), we seek to minimise the misfit functional (u, p, y) → (1 − λ) Q(•, •, u, ∇u, p) − q + λ y over all admissible triplets (u, p, y) which satisfy (1.1), for a fixed weight λ ∈ (0, 1). The role of this weight is to obtain essentially a Pareto family of extremals, one for each value λ, even though in this paper we do not pursue further this viewpoint of vector-valued minimisation (the interested reader may e.g.…”

Vectorial variational problems in $L^\infty$ constrained by the Navier-Stokes equations

“…In this paper, inspired by the methodology of data assimilation, especially variational data assimilation in continuous time (for relevant works we refer e.g. to [13,19,26,29,30,44,46,52]), we seek to minimise the misfit functional (u, p, y) → (1 − λ) Q(•, •, u, ∇u, p) − q + λ y over all admissible triplets (u, p, y) which satisfy (1.1), for a fixed weight λ ∈ (0, 1). The role of this weight is to obtain essentially a Pareto family of extremals, one for each value λ, even though in this paper we do not pursue further this viewpoint of vector-valued minimisation (the interested reader may e.g.…”

Least‐squares formulation to solve non‐Newtonian fluid flow and application of data assimilation in 2D

“…For more details on this approach see e.g. [6]. Considering additionally the dependence of the dynamic viscosity η on the shear rate γ = 2(∇ s v : ∇ s v), the resulting functional reads…”