In many applications, homogenization (or upscaling) techniques are necessary to develop computationally feasible models on scales coarser than the variation of the coefficients of the continuum model. The accuracy of such techniques depends dramatically on assumptions that underlie the particular upscaling methodology used. For example, decoupling of fine- and coarse-scale effects in the underlying medium may utilize artificial internal boundary conditions on sub-cell problems. Such assumptions, however, may be at odds with the true, fine-scale solution, yielding coarse-scale errors that may be unbounded.

In this work, we present an efficient multilevel upscaling (MLUPS) procedure for single-phase saturated flow through porous media. Coarse-scale models are explicitly created from a given fine-scale model through the application of standard operator-induced variational coarsening techniques. Such coarsenings, which originated with robust multigrid solvers, have been shown to accurately capture the influence of fine-scale heterogeneity over the complete hierarchy of resulting coarse-scale models. Moreover, implicit in this hierarchy is the construction of interpolation operators that provide a natural and complete multiscale basis for the fine-scale problem. Thus, this new multilevel upscaling methodology is similar to the Multiscale Finite Element Method (MSFEM) and, indeed, we show that it attains similar accuracy on a variety of problems. While MSFEM is based on a two-scale approach, MLUPS generates a complete hierarchy of coarse-scale models, resulting in a speed-up factor of approximately 15. In addition, we demonstrate that this new upscaling methodology can use both structured coarsening algorithms, such as Dendy's BoxMG, and fully algebraic algorithms, such as Ruge's AMG.