Homogenization of parabolic equations with a continuum of space and time scales

TL;DR

This paper develops a homogenization method for linear parabolic equations with multiple scales in space and time, without relying on ergodicity or scale separation, by leveraging harmonic coordinates and a Cordes type condition.

Contribution

It introduces a novel homogenization approach for parabolic operators with continuum scales, using harmonic coordinates and a Cordes condition to reduce computational complexity.

Findings

Homogenization achieved after solving the equation n times.

Harmonic coordinates improve regularity of derivatives.

Method applicable to time-independent media with fewer solutions.

Abstract

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.