Three-scale convergence for processes in heterogeneous media

D. Trucu, M. A. J. Chaplain, A. Marciniak-Czochra

Research output: Contribution to journalArticlepeer-review

31 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)1351-1373
Number of pages23
JournalApplicable Analysis
Issue number7
Publication statusPublished - 2012


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