Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time
| Autor: | Liselott Flodén, Anders Holmbom, Marianne Olsson Lindberg, Jens Persson |
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| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | Journal of Applied Mathematics, Vol 2014 (2014) |
| Druh dokumentu: | article |
| ISSN: | 1110-757X 1687-0042 |
| DOI: | 10.1155/2014/101685 |
| Popis: | The main contribution of this paper is the homogenization of the linear parabolic equation ∂tuε(x,t)-∇·(a(x/εq1,...,x/εqn,t/εr1,...,t/εrm)∇uε(x,t))=f(x,t) exhibiting an arbitrary finite number of both spatial and temporal scales. We briefly recall some fundamentals of multiscale convergence and provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtain n local problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales with q1=1, q2=2, and 0 |
| Databáze: | Directory of Open Access Journals |
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