Crouzeix-raviart multiscale finite element method for Stokes flows in heterogeneous media

This paper addresses a non-conforming Multiscale Finite Element Method (MsFEM) for viscous incompressible flows in genuine heterogeneous media. The multiscale method relies upon the coupling of two grid scales: one coarse grid and one fine grid. The principe is to compute on the fine grid effective properties of the media in the form of basis functions, which are then used to solve problems on the coarse grid [1]. It's known that when computing multiscale basis functions, the approximation of boundary conditions on coarse element edges influences critically the accuracy of the MsFEM. Based on the work of [2], the weakly enforced conformity between coarse elements leads to a natural boundary condition on coarse element edges. This relaxes the sensitivity of the method to complex patterns of obstacles, without any need of oversampling techniques. Two methods have been developped and compared making use of a study of open flows in a heterogeneous domain.