Three-scale convergence for processes in heterogeneous media

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

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    30 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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