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Homogenization of Parabolic Equations with a Continuum of Space and Time Scales

Owhadi, Houman and Zhang, Lei (2007) Homogenization of Parabolic Equations with a Continuum of Space and Time Scales. SIAM Journal on Numerical Analysis, 46 (1). pp. 1-36. ISSN 0036-1429. https://resolver.caltech.edu/CaltechAUTHORS:OWHsiamjna07

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Abstract

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 $\Omega \subset \mathbb{R}^n$ 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 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 $L^2$ (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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/math/0512504arXivDiscussion Paper
https://doi.org/10.1137/060670420DOIUNSPECIFIED
https://doi.org/10.1137/060670420DOIUNSPECIFIED
ORCID:
AuthorORCID
Owhadi, Houman0000-0002-5677-1600
Zhang, Lei0000-0001-9031-4318
Additional Information:© 2007 Society for Industrial and Applied Mathematics. Reprinted with permission. Received by the editors September 22, 2006; accepted for publication (in revised form) August 10, 2007; published electronically December 12, 2007. We thank two anonymous referees for detail comments and suggestions.
Subject Keywords:multiscale problem; compensation; homogenization; upscaling; compression
Issue or Number:1
Record Number:CaltechAUTHORS:OWHsiamjna07
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:OWHsiamjna07
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9431
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:30 Dec 2007
Last Modified:09 Mar 2020 13:19

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