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International Journal for Uncertainty Quantification

Impact factor: 1.000

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2018021021
Forthcoming Article

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

Alexander Schwarz
University of Duisburg-Essen
Richard P. Dwight
TU Delft


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) x H1 x 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. A simple and immediate approach to extend this LSFEM to data-assimilation is o add a data-discrepancy term to the Least-Squares 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. In numerical examples 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, as well as correct for the influence of unknown and uncertain boundary conditions.