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

The multiscale finite element method (MsFEM) is developed in the vein of the Crouzeix--Raviart element for solving viscous incompressible flows in genuine heterogeneous media. Such flows are relevant in many branches of engineering, often at multiple scales and at regions where analytical representations of the microscopic features of the flows are often unavailable. Full accounts of 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 flow without confining itself to fine scale calculations is sought. 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 obstacles exempt from the need to implement any oversampling techniques. Additionally, the application of a penalization method makes it possible to avoid a complex unstructured domain and allows extensive use of simpler Cartesian meshes.