  The adaptation of Crouzeix - Raviart finite element in the context of
multiscale finite element method (MsFEM) is studied and implemented on
diffusion and advection-diffusion problems in perforated media. It is known
that the approximation of boundary condition on coarse element edges when
computing the multiscale basis functions critically influences 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 perforations. Another ingredient to our
method is the application of bubble functions which is shown to be instrumental
in maintaining high accuracy amid dense perforations. Additionally, the
application of penalization method makes it possible to avoid complex
unstructured domain and allows extensive use of simpler Cartesian meshes.

Numerical Analysis