 
Summary: Workshop: Gemischte und nichtstandard
FiniteElementeMethoden mit Anwendungen 5
Abstracts
Mixed variational multiscale methods and multiscale finite elements
Todd Arbogast
(joint work with Kirsten J. Boyd)
A longstanding open problem in applied mathematics is to accurately approxi
mate a function that possesses scales smaller than the level of practical discretiza
tion. In this work, we consider the approximation on a coarse grid of spacing H
of the solution (u, p) to a second order elliptic problem in mixed form:
u = K p in ,
· u = f in ,
u · = 0 on ,
where Rd
, d = 2 or 3, is a bounded domain, K is uniformly elliptic and
bounded, and is the outward unit normal vector. We assume that K and possibly
f exhibit microstructure (i.e., variability or heterogeneity) on a small scale < H,
which induces similar scale variation into the solution.
In 1983, Babuska and Osborn [7] gave a practical strategy for problems in one
dimension by defining what they called the generalized finite element method,
