| 1. |
- Flodén, Liselott, et al.
(författare)
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Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales
- 2017
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Ingår i: Progress in Industrial Mathematics at ECMI 2016. - Cham : Springer. - 9783319630816 ; , s. 617-623
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Konferensbidrag (refereegranskat)abstract
- We study the homogenization of a certain linear hyperbolic-parabolic problem exhibiting two rapid spatial scales {ε; ε2}. The homogenization is performed by means of evolution multiscale convergence, a generalization of the concept of two-scale convergence to include any number of scales in both space and time. In particular we apply a compactness result for gradients. The outcome of the homogenization procedure is that we obtain a homogenized problem of hyperbolic-parabolic type together with two elliptic local problems, one for each rapid scale, for the correctors.
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| 2. |
- Flodén, Liselott, 1967-, et al.
(författare)
-
Homogenization of parabolic equations with an arbitrary number of scales in both space and time
- 2014
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Ingår i: Journal of Applied Mathematics. - Boston : Hindawi Publishing Corporation. - 1110-757X .- 1687-0042. ; , s. Art. no. 101685-
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Tidskriftsartikel (refereegranskat)abstract
- The main contribution of this paper is the homogenization of the linearparabolic equationtu (x, t) − ·axq1, ...,xqn,tr1, ...,trmu (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 providea characterization of multiscale limits for gradients in an evolution settingadapted to a quite general class of well-separated scales, which we nameby jointly well-separated scales (see Appendix for the proof). We proceedwith a weaker version of this concept called very weak multiscale convergence.We prove a compactness result with respect to this latter typefor jointly well-separated scales. This is a key result for performing thehomogenization of parabolic problems combining rapid spatial and temporaloscillations such as the problem above. Applying this compactnessresult together with a characterization of multiscale limits of sequences ofgradients we carry out the homogenization procedure, where we togetherwith the homogenized problem obtain n local problems, i.e. one for eachspatial microscale. To illustrate the use of the obtained result we apply itto a case with three spatial and three temporal scales with q1 = 1, q2 = 2and 0 < r1 < r2.MSC: 35B27; 35K10
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| 3. |
- Flodén, Liselott, 1967-, et al.
(författare)
-
Two-scale convergence: Some remarks and extensions
- 2013
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Ingår i: Pure and Applied Mathematics Quarterly. - : International press of Boston. - 1558-8599 .- 1558-8602. ; 9:3, s. 461-486
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Tidskriftsartikel (refereegranskat)abstract
- We first study the fundamental ideas behind two-scale conver-gence to enhance an intuitive understanding of this notion. The classicaldefinitions and ideas are motivated with geometrical arguments illustratedby illuminating figures. Then a version of this concept, very weak two-scaleconvergence, is discussed both independently and brie°y in the context ofhomogenization. The main features of this variant are that it works alsofor certain sequences of functions which are not bounded inL2 and atthe same time is suited to detect rapid oscillations in some sequences whichare strongly convergent inL2 . In particular, we show how very weaktwo-scale convergence explains in a more transparent way how the oscilla-tions of the governing coe±cient of the PDE to be homogenized causes thedeviation of theG-limit from the weak L2 NxN-limit for the sequence ofcoe±cients. Finally, we investigate very weak multiscale convergence andprove a compactness result for separated scales which extends a previousresult which required well-separated scales.
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| 4. |
- Johnsen, Pernilla
(författare)
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Homogenization of Partial Differential Equations using Multiscale Convergence Methods
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The focus of this thesis is the theory of periodic homogenization of partial differential equations and some applicable concepts of convergence. More precisely, we study parabolic problems exhibiting both spatial and temporal microscopic oscillations and a vanishing volumetric heat capacity type of coefficient. We also consider a hyperbolic-parabolic problem with two spatial microscopic scales. The tools used are evolution settings of multiscale and very weak multiscale convergence, which are extensions of, or closely related to, the classical method of two-scale convergence. The novelty of the research in the thesis is the homogenization results and, for the studied parabolic problems, adapted compactness results of multiscale convergence type.
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