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.