Variational Multiscale Methods for Computational Modeling of
Heterogeneous Porous Media
Todd Arbogast
If a coefficient in a differential equation is heterogeneous, i.e., it
varies
greatly on a small scale, then the solution will vary on that same
small
scale. An example of such a situation is flow in a natural porous
medium,
since normally the rock permeability varies on a small scale, and thus
the
fluid velocity changes greatly from point to point. This presents a
computational challenge, since the numerical grid needs to resolve the
finest scale, but this is much too detailed for even today's
supercomputers
to handle.
Standard finite element approximation of the solution on a coarse grid
fails.
An alternative is to solve the overall partial differential equation by
incorporating the heterogeneity directly into the finite element basis
using multiscale finite elements. These are defined by solving
local, or
subgrid, problems that resolve the fine-scale variation of the
coefficient.
In this way, one can improve the overall resolution of the finite
element
approximation. We present variational aspects of the method,
called the
Variational Multiscale Method, for second order elliptic partial
differential
equations in mixed form (i.e., written as a system of two first order
equations).
The coarse and fine scales are handled by introducing a novel expansion
of
the solution based on a Hilbert space direct sum decomposition. We
present
theoretical convergence results and computational examples which
demonstrate the effectiveness of the method.