The Multiscale Finite Element Method (MsFEM) is developed in the vein of Crouzeix-
Raviart element for solving viscous incompressible
ows in genuine heterogeneous media. Such
ows
are relevant in many branches of engineering, often at multiple scales and at regions where analytical
representations of the microscopic features of the
ows are often unavailable. Full accounts to these
problems heavily depend on the geometry of the system under consideration and are computationally
expensive. Therefore, a method capable of solving multiscale features of the
ow without con ning
itself to ne scale calculations is sought after.
The approximation of boundary condition on coarse element edges when computing the multiscale
basis functions critically in
uences the eventual accuracy of any MsFEM approaches. The weakly
enforced continuity of Crouzeix - Raviart function space across element edges leads to a natural
boundary condition for the multiscale basis functions which relaxes the sensitivity of our method to
complex patterns of obstacles exempt from the needs of implementing any oversampling techniques.
Additionally, the application of penalization method makes it possible to avoid complex unstructured
domain and allows extensive use of simpler Cartesian meshes.
Key words. Crouzeix-Raviart Element, Multiscale Finite Element Method, Stokes Equations,
Penalization Method