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


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.