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Crouzeix-Raviart multiscale finite element method for Navier-Stokes flows in heterogeneous media.

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Qingqing FENG (CEA/DEN/STMF)

We address a nonconforming multiscale finite element method (MsFEM) to solve incompressible Navier-Stokes equations in porous
media. In the work of [B. P. Muljadi, J. Narski, A. Lozinski, and P. Degond, Multiscale Modeling
& Simulation 2015 13:4, 1146-1172], a Crouzeix-Raviart MsFEM has been developed to solve the
Stokes equations in 2D with randomly placed obstacles. We extend the nonconforming MsFEM to
solve Navier-Stokes problems in 2D and 3D. We develop two approaches of MsFEM that:(i) use
basis functions based on Stokes equations, adjunction of stabilizations or not; (ii) use basis
functions based on Oseen equations, adjunction of stabilizations or not. The convection field in the
Oseen term is computed by taking the average of coarse solutions solved in (i). We prove the well-
posedness of cell problems defined in (ii). We compare the accuracy of the two approaches and try
to understand if the convection field should be taken into account in the construction of basis
functions. Besides, we analyse errors of MsFEM in periodic cases and its sensitivity to
discretizations of the domain.