We present a two-scale finite element method for solving Brinkman’s and Darcy’s equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes’ equations byWang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy’s equations. In order to reduce the “resonance error” and to ensure convergence to the global fine solution the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems.
Two-level domain decomposition preconditioner for 3D flows in anisotropic highly heterogeneous porous media is presented. Accurate finite volume discretization based on multipoint flux approximation (MPFA) for 3D pressure equation is employed to account for the jump discontinuities of full permeability tensors. DD/MG type preconditioner for above mentioned problem is developed. Coarse scale operator is obtained from a homogenization type procedure. The influence of the overlapping as well as the influence of the smoother and cell problem formulation is studied. Results from numerical experiments are presented and discussed.