On a numerical subgrid upscaling algorithm for Stokes-Brinkman equations

This paper discusses a numerical subgrid resolution approach for solving the Stokes-Brinkman system of equations, which is describing coupled flow in plain and in highly porous media. Various scientific and industrial problems are described by this system, and often the geometry and/or the permeability vary on several scales. A particular target is the process of oil filtration. In many complicated filters, the filter medium or the filter element geometry are too fine to be resolved by a feasible computational grid. The subgrid approach presented in the paper is aimed at describing how these fine details are accounted for by solving auxiliary problems in appropriately chosen grid cells on a relatively coarse computational grid. This is done via a systematic and a careful procedure of modifying and updating the coefficients of the Stokes-Brinkman system in chosen cells. This numerical subgrid approach is motivated from one side from homogenization theory, from which we borrow the formulations for the so called cell problem, and from the other side from the numerical upscaling approaches, such as Multiscale Finite Volume, Multiscale Finite Element, etc. Results on the algorithm’s efficiency, both in terms of computational time and memory usage, are presented. Comparison with solutions on full fine grid (when possible) are presented in order to evaluate the accuracy. Advantages and limitations of the considered subgrid approach are discussed.