Three-scale convergence for processes in heterogeneous media |
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Authors: | D. Trucu M.A.J. Chaplain A. Marciniak-Czochra |
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Affiliation: | 1. Division of Mathematics , University of Dundee , Dundee , DD1 4HN , UK trucu@maths.dundee.ac.uk;3. Division of Mathematics , University of Dundee , Dundee , DD1 4HN , UK;4. Interdisciplinary Center of Scientific Computing and BIOQUANT , University of Heidelberg , Im Neuenheimer Feld 267, 69120, Heidelberg , Germany |
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Abstract: | 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(Ω) 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. |
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Keywords: | multiscale analysis heterogeneous media composite media microscale mesoscale macroscale |
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