Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time
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 prov...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/101685 |
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| author | Liselott Flodén Anders Holmbom Marianne Olsson Lindberg Jens Persson |
| author_facet | Liselott Flodén Anders Holmbom Marianne Olsson Lindberg Jens Persson |
| author_sort | Liselott Flodén |
| collection | DOAJ |
| description | 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<r1<r2. |
| format | Article |
| id | doaj-art-cbd858e3a7484b508af785c5e26d5773 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-cbd858e3a7484b508af785c5e26d57732025-02-03T01:10:34ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/101685101685Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and TimeLiselott Flodén0Anders Holmbom1Marianne Olsson Lindberg2Jens Persson3Department of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 Östersund, SwedenDepartment of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 Östersund, SwedenDepartment of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 Östersund, SwedenDepartment of Quality Technology and Management, Mechanical Engineering and Mathematics, Mid Sweden University, S-83125 Östersund, SwedenThe 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<r1<r2.http://dx.doi.org/10.1155/2014/101685 |
| spellingShingle | Liselott Flodén Anders Holmbom Marianne Olsson Lindberg Jens Persson Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time Journal of Applied Mathematics |
| title | Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time |
| title_full | Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time |
| title_fullStr | Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time |
| title_full_unstemmed | Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time |
| title_short | Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time |
| title_sort | homogenization of parabolic equations with an arbitrary number of scales in both space and time |
| url | http://dx.doi.org/10.1155/2014/101685 |
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