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This paper discusses a numerical subgrid resolution approach for solving the Stokes-Brinkman system of equations, which is describing coupled ow in plain and in highly porous media. Various scientic 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 ltration. In many complicated lters, the lter medium or the lter element geometry are too ne to be resolved by a feasible computational grid. The subgrid approach presented in the paper is aimed at describing how these ne 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 coecients 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 eciency, both in terms of computational time and memory usage, are presented. Comparison with solutions on full ne grid (when possible) are presented in order to evaluate the accuracy. Advantages and limitations of the considered subgrid approach are discussed.

The performance of oil filters used in the automotive industry can be significantly improved, especially when computer simulation is an essential component of the design process. In this paper, we consider parallel numerical algorithms for solving mathematical models describing the process of filtration, filtering out solid particles from liquid oil. The Navier-Stokes-Brinkmann system of equations is used to describe the laminar flow of incompressible isothermal oil. The space discretization in the complicated filter geometry is based on the finite-volume method. Special care is taken for an accurate approximation of velocity and pressure on the interface between the fluid and the porous media. The time discretization used here is a proper modification of the fractional time step discretization (cf. Chorin scheme) of the Navier-Stokes equations, where the Brinkmann term is considered at both, prediction and correction substeps. A data decomposition method is used to develop a parallel algorithm, where the domain is distributed among processors by using a structured reference grid. The MPI library is used to implement the data communication part of the algorithm. A theoretical model is proposed for the estimation of the complexity of the given parallel algorithm and a scalability analysis is done on the basis of this model. Results of computational experiments are presented, and the accuracy and efficiency of the parallel algorithm is tested on real industrial geometries.