We present a two-scale finite element method (FEM) 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 by Wang 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.

We consider certain computational aspects of upscaling fluid flows through deformable porous media. We start with pore level models and discuss upscaled (homogenized) equations and respective cell problems. Analytical solution of a cell problem in certain geometry, as well as an accurate numerical procedure for the general case, are presented. A microscale (pore level) fluid-structure interaction (FSI) problem is formulated in terms of incompressible Newtonian fluid and a linearized elastic solid. At least three different macroscopic models can be derived from this microscale formulation, depending on the assumptions on the fluid-structure interface. The first two are the well-known linear poroelasticity model [M. Biot, J. Appl. Phys., 12 (1941), pp. 155–164] and its nonlinear extension [C. K. Lee and C. C. Mei, Int. J. Eng. Sci., 35 (1997), pp. 329–352]. Both are derived under the assumption of small (at pore level) displacements of the interface with the difference that the first model excludes pore scale rigid body motions, while the second one accounts for them. A third macroscopic model is explained below. It concerns a particular case, namely, a porous medium formed by long parallel channels with thick elastic walls. An asymptotic solution to the FSI problem is derived for such a geometry, allowing finite (at pore scale) displacements for the interfaces. A nonlinear Darcy-type upscaled equation for the averaged pressure is obtained. The cell problems for each of the above cases, as well as a numerical algorithm for solving these cell problems, are discussed. The microscale cell FSI problems are treated numerically by an iterative procedure which solves sequentially fluid and solid subproblems and couples them via the interface conditions. Numerical and asymptotic solutions are found to converge to each other, thus validating both the numerical solver and the analytical derivation.

A modified finite difference approximation for interface problems in Rn, n=1,2,3, is presented. The essence of the modification falls in the simultaneous discretization of any two normal components of the flux at the opposite faces of the finite volume. In this way, the continuous normal component of the flux through an interface is approximated by finite differences with second-order consistency. The derived scheme has a minimal (2n+1)-point stencil for problems in Rn. Second-order convergence with respect to the discrete H1 -norm is proved for a class of interface problems. Second-order pointwise convergence is observed in a series of numerical experiments with one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) interface problems. The numerical experiments presented demonstrate advantages of the new scheme compared with the known schemes which use arithmetic and harmonic averaging of the discontinuous diffusion coefficient.