The generalized Stokes equations (called also Brinkman equations),
were introduced in 1947 by Brinkman as an enhancement of Darcy law (by
adding dissipative term scaled by the viscosity) to better describe the
flows of incompressible fluids in highly porous media. Examples of such
media are industrial open foams, filters, and insulation materials.

Motivated by applications of such materials we have developed a
numerical method for computing flows in heterogeneous media of high
porosity with complicated internal structure of the permeability.
We will present a two-scale finite element approximation of Brinkman
equations. The method uses two main ingredients: (I) discontinuous
Galerkin finite element method for Stokes equations, proposed and
studied by J.Wang and X.Ye (2007, SINUM) and (II) subgrid approximation
developed by T.Arbogast for Darcy equations (2004, SINUM).

There are two different applications of the proposed method:
(1) numerical upscaling of Brinkman equations on coarse-grid that
incorporates fine-grid features, and (2) an alternating Schwarz
iteration that uses the coarse-grid in an overlapping domain
decomposition setting. A number of numerical examples will be presented
to demonstrate the the performance of both the subgrid method and the
iterative procedure.

The talk is based on a joint ongoing research of the author with O.
Iliev (ITWM, Germany and IMI, Bulgaria) and J. Willems (RICAM, Austria).