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Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time
Joint Authors
Holmbom, Anders
Persson, Jens
Flodén, Liselott
Olsson Lindberg, Marianne
Source
Journal of Applied Mathematics
Issue
Vol. 2014, Issue 2014 (31 Dec. 2014), pp.1-16, 16 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2014-02-24
Country of Publication
Egypt
No. of Pages
16
Main Subjects
Abstract EN
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
American Psychological Association (APA)
Flodén, Liselott& Holmbom, Anders& Olsson Lindberg, Marianne& Persson, Jens. 2014. Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time. Journal of Applied Mathematics،Vol. 2014, no. 2014, pp.1-16.
https://search.emarefa.net/detail/BIM-446394
Modern Language Association (MLA)
Flodén, Liselott…[et al.]. Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time. Journal of Applied Mathematics No. 2014 (2014), pp.1-16.
https://search.emarefa.net/detail/BIM-446394
American Medical Association (AMA)
Flodén, Liselott& Holmbom, Anders& Olsson Lindberg, Marianne& Persson, Jens. Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time. Journal of Applied Mathematics. 2014. Vol. 2014, no. 2014, pp.1-16.
https://search.emarefa.net/detail/BIM-446394
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-446394