Three-scale convergence for processes in heterogeneous media

TY - JOUR

T1 - Three-scale convergence for processes in heterogeneous media

AU - Trucu, D.

AU - Chaplain, M. A. J.

AU - Marciniak-Czochra, A.

N1 - Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - In this article, we propose a new notion of multiscale convergence, called ‘three-scale’, which aims to give a topological framework in which to assess complex processes occurring at three different scales or levels within a heterogeneous medium. This generalizes and extends the notion of two-scale convergence, a well-established concept that is now commonly used for obtaining an averaged, asymptotic value (homogenization) of processes that exist on two different spatial scales. The well-posedness of this new concept is justified via a compactness theorem which ensures that all bounded sequences in L 2(O) are relative compact with respect to the three-scale convergence. This is taken further by giving a boundedness characterization of three-scale convergent sequences and is then continued with the introduction of the notion of ‘strong three-scale convergence’ whose well-posedness is also discussed. Finally, the three-scale convergence of the gradients is established.

AB - In this article, we propose a new notion of multiscale convergence, called ‘three-scale’, which aims to give a topological framework in which to assess complex processes occurring at three different scales or levels within a heterogeneous medium. This generalizes and extends the notion of two-scale convergence, a well-established concept that is now commonly used for obtaining an averaged, asymptotic value (homogenization) of processes that exist on two different spatial scales. The well-posedness of this new concept is justified via a compactness theorem which ensures that all bounded sequences in L 2(O) are relative compact with respect to the three-scale convergence. This is taken further by giving a boundedness characterization of three-scale convergent sequences and is then continued with the introduction of the notion of ‘strong three-scale convergence’ whose well-posedness is also discussed. Finally, the three-scale convergence of the gradients is established.

UR - https://www.scopus.com/pages/publications/84862333063

U2 - 10.1080/00036811.2011.569498

DO - 10.1080/00036811.2011.569498

M3 - Article

AN - SCOPUS:84862333063

SN - 0003-6811

VL - 91

SP - 1351

EP - 1373

JO - Applicable Analysis

JF - Applicable Analysis

IS - 7

ER -