The Black Box Multigrid Numerical Homogenization Algorithm

In mathematical models of ow through porous media, the coe cients typically exhibit severe variations in two or more signi cantly di erent length scales. Consequently, the numerical treatment of these problems relies on a homogenization or upscaling procedure to de ne an approximate coarse-scale problem that adequately captures the in uence of the ne-scale structure. Inherent in such a procedure is a compromise between its computational cost and the accuracy of the resulting coarse-scale solution. Although techniques that balance the con icting demands of accuracy and e ciency exist for a few speci c classes of ne-scale structure (e.g., ne-scale periodic), this is not the case in general. In this paper we propose a new, e cient, numerical approach for the homogenization of the permeability in models of single-phase saturated ow. Our approach is motivated by the observation that multiple length scales are captured automatically by robust multilevel iterative solvers, such as Dendy's black box multigrid. In particular, the operator-induced variational coarsening in black box multigrid produces coarse-grid operators that capture the essential coarse-scale in uence of the medium's ne-scale structure. We derive an explicit local, cell-based, approximate expression for the symmetric, 2 2 homogenized permeability tensor that is de ned implicitly by the black box coarse-grid operator. The e ectiveness of this black box multigrid numerical homogenization method is demonstrated through numerical examples.