RESUMEN
This paper
addresses the issue of the
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 Ω ⊂
Rn with L∞(Ω ラ (0, T))-coefficients.
It appears that the inverse
operator maps the unit ball
of L2(Ω ラ (0, T)) into a space of functions
which at small (time and space) scales are close in H1 norm to
a functional space of dimension n. It
follows that once one has solved these equations at least n times it
is posible to homogenize them both in space and
in time, reducing the number of operation
counts necessary to obtain further
solutions. In practice we show 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
caloric coordinates are in L2 (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.
1. Introduction and main results.
2. Proofs.
3. Numerical experiments.