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 - http://www.scopus.com/inward/record.url?scp=84862333063&partnerID=8YFLogxK
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 -