TY - JOUR

T1 - Data assimilation for Navier-Stokes using the least-squares finite-element method

AU - Schwarz, Alexander

AU - Dwight, Richard P.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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(div) ×H1 ×L2 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 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 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.

AB - 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(div) ×H1 ×L2 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 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 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.

KW - Bayesian inference

KW - Data assimilation

KW - Fluid dynamics

KW - Inverse problems

KW - Least-squares finite-element method

KW - Navier-Stokes

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=85052380283&partnerID=8YFLogxK

UR - http://resolver.tudelft.nl/uuid:013ba330-8ad1-441c-85cf-ca824d4f2e00

U2 - 10.1615/Int.J.UncertaintyQuantification.2018021021

DO - 10.1615/Int.J.UncertaintyQuantification.2018021021

M3 - Article

AN - SCOPUS:85052380283

VL - 8

SP - 383

EP - 403

JO - International Journal of Uncertainty Quantification

JF - International Journal of Uncertainty Quantification

SN - 2152-5080

IS - 5

ER -