Sufficient resolution of fine-scale variations in material properties is often needed to achieve the high levels of accuracy demanded of computational simulation in biological and geophysical applications. In many cases, this variation is resolved on a scale that is finer than is practical to use for computation, requiring mathematical tools for coarsening (or upscaling) the medium or the model. The mathematical tools of homogenization address the question of determining effective properties of the medium on a coarser scale, but are, in general, not justified for media that arise in nature.

In this talk, I discuss a new approach for upscaling PDE models with heterogeneous media based on variational principles. The variational framework used is based on that of Galerkin finite element discretizations and is closely related to that used in robust multilevel solvers, such as multigrid. As in robust geometric and algebraic multigrid methods, the coarsening procedure is induced by the fine-scale model itself. In this way, we construct a hierarchy of models that resolve the effects of the fine-scale structure at multiple scales. This research is in collaboration with J. David Moulton from Los Alamos National Laboratory.