Multi-scale differential equations are problems in which the variables
(for example space or time) can have different lenght scales.
The standard analytical approach is to replace these equations by so
called
homogenized equations, in which the fine scales are averaged out.
It can be successful for several applications, but is also limited
by restrictive assumptions on the media. The information about the small
scales is also lost in this approach.
The direct numerical solution of differential equations with multiple
scales
is often difficult due to the work for resolving the smallest scale.
We will present in this talk a strategy which allows the use of
finite difference methods for the numerical solution of parabolic
multi-scale
problems, based on a coupling of macroscopic and microscopic models for
the
original equation.