  This paper addresses the issue of homogenization of linear divergence form
parabolic operators in situations where no ergodicity and no scale separation
in time or space are available. Namely, we consider divergence form linear
parabolic operators in $\Omega \subset \R^n$ with $L^\infty(\Omega \times
(0,T))$-coefficients. It appears that the inverse operator maps the unit ball
of $L^2(\Omega\times (0,T))$ into a space of functions which at small (time and
space) scales are close in $H^1$-norm to a functional space of dimension $n$.
It follows that once one has solved these equations at least $n$-times it is
possible to homogenize them both in space and in time, reducing the number of
operations counts necessary to obtain further solutions. In practice we show
that under a Cordes type condition that the first order time derivatives and
second order space derivatives of the solution of these operators with respect
to harmonic coordinates are in $L^2$ (instead of $H^{-1}$ with Euclidean
coordinates). If the medium is time independent then it is sufficient to solve
$n$ times the associated elliptic equation in order to homogenize the parabolic
equation.
