 
Summary: Mixed Multiscale Methods for Heterogeneous
Elliptic Problems
Todd Arbogast
Abstract We consider a second order elliptic problem written in mixed form, i.e., as
a system of two first order equations. Such problems arise in many contexts, includ
ing flow in porous media. The coefficient in the elliptic problem (the permeability
of the porous medium) is assumed to be spatially heterogeneous. The emphasis here
is on accurate approximation of the solution with respect to the scale of variation in
this coefficient. Homogenization and upscaling techniques alone are generally inad
equate for this problem. As an alternative, multiscale numerical methods have been
developed. They can be viewed in one of three equivalent frameworks: as a Galerkin
or finite element method with nonpolynomial basis functions, as a variational mul
tiscale method with standard finite elements, or as a domain decomposition method
with restricted degrees of freedom on the interfaces. We treat each case, and dis
cuss the advantages of the approach for devising effective local multiscale methods.
Included is recent work on methods that incorporate information from homogeniza
tion theory and effective domain decomposition methods.
1 Elliptic Systems with a Heterogeneous Coefficient
We consider a second order elliptic problem, which we write in mixed form, i.e., as
the following system of two first order equations:
