The Black Box Multigrid Numerical Homogenization Algorithm

Abstract

In mathematical models of flow through porous media, the coefficients typically exhibit severe variations in two or more significantly different length scales. Consequently, the numerical treatment of these problems relies on ahomogenizationorupscalingprocedure to define an approximate coarse-scale problem that adequately captures the influence of the fine-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 conflicting demands of accuracy and efficiency exist for a few specific classes of fine-scale structure (e.g., fine-scale periodic), this is not the case in general. In this paper we propose a new, efficient, numerical approach for thehomogenizationof the permeability in models of single-phase saturated flow. Our approach is motivated by the observation that multiple length scales are captured automatically by robust multilevel iterative solvers, such as Dendy'sblack box multigrid. In particular, the operator-induced variational coarsening in black box multigrid produces coarse-grid operators that capture the essential coarse-scale influence of the medium's fine-scale structure. We derive an explicit local, cell-based, approximate expression for the symmetric, 2×2 homogenized permeability tensor that is defined implicitly by the black box coarse-grid operator. The effectiveness of this black box multigrid numerical homogenization method is demonstrated through numerical examples.