Title: Variational multi-scale methods for computational modeling of
heterogeneous porous media

Abstract: Complex physical phenomena almost always occur on widely
varying scales.  A continuing challenge in mathematical and
computational modeling is to handle all the relevant scales properly.
Fine scale effects can and often do have a profound influence on
coarser scales, so it is imperative to express each modeled phenomenon
appropriately on the scale of interest, and to properly account for
their interactions.  This is especially true of natural porous media.
   We present a relatively recently exploited expansion technique and its
computational implementation for second order elliptic or parabolic
problems in mixed form (i.e., written as a system of first order
equations).  The expansion technique is based analytically on very
general Hilbert space direct sum decompositions, and on the discrete
or computational level it is implemented as a variational multi-scale
method or a type of subgrid upscaling method.
   We take the point of view that the properties of the porous medium can
be expressed at some scale that can be resolved on a very fine grid.
Our subgrid approach is developed to scale up this fine grid
information to coarse scales to allow accurate and efficient
implementation.