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The Multiscale Finite Element Method (MsFEM) is developed in the vein ofCrouzeix-Raviart element for solving viscous incompressible flows in genuineheterogeneous media. Such flows are relevant in many branches of engineering,often at multiple scales and at regions where analytical representations of themicroscopic features of the flows are often unavailable. Full accounts to theseproblems heavily depend on the geometry of the system under consideration andare computationally expensive. Therefore, a method capable of solvingmultiscale features of the flow without confining itself to fine scalecalculations is sought after.  The approximation of boundary condition on coarse element edges whencomputing the multiscale basis functions critically influences the eventualaccuracy of any MsFEM approaches. The weakly enforced continuity of Crouzeix -Raviart function space across element edges leads to a natural boundarycondition for the multiscale basis functions which relaxes the sensitivity ofour method to complex patterns of obstacles exempt from the needs ofimplementing any oversampling techniques. Additionally, the application ofpenalization method makes it possible to avoid complex unstructured domain andallows extensive use of simpler Cartesian meshes.

Non-Conforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media. Part I: Methodologies and Numerical Experiments