| 1. |
- Danielsson, Tatiana
(författare)
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Further Investigations of Convergence Results for Homogenization Problems with Various Combinations of Scales
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is based on six papers. We study the homogenization of selected parabolic problems with one or more microscopic scales in space and time, respectively. The approaches are prepared by means of certain methods, like two-scale convergence, multiscale convergence and also the evolution setting of multiscale convergence and very weak multiscale convergence. Paper I treats a linear parabolic homogenization problem with rapid spatial and temporal oscillations in perforated domains. Suitable results of two-scale convergence type are established. Paper II deals with further development of compactness results which can be used in the homogenization procedure engaging a certain limit condition. The homogenization procedure deals with a parabolic problem with a certain matching between a fast spatial and a fast temporal scale and a coefficient passing to zero that the time derivative is multiplied with. Papers III and IV are further generalizations of Paper II and investigate homogenization problems with different types of matching between the microscopic scales. Papers III and IV deal with one and two rapid scales in both space and time respectively. Paper V treats the nonlinearity of monotone parabolic problems with an arbitrary number of spatial and temporal scales by applying the perturbed test functions method together with multiscale convergence and very weak multiscale convergence adapted to the evolution setting. In Paper VI we discuss the relation between two-scale convergence and the unfolding method and potential extensions of existing results. The papers above are summarized in Chapter 4. Chapter 1 gives a brief introduction to the topic and Chapters 2 and 3 are surveys over some important previous results.
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| 2. |
- Danielsson, Tatiana, et al.
(författare)
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Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales
- 2024
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 69:1, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of epsilon. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems. At the end of the paper an example is given to concretize the use of the main result.
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| 3. |
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| 4. |
- Flodén, Liselott, 1967-, et al.
(författare)
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A myriad shades of green
- 2009
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Ingår i: Proceedings of Bridges 2009, Banff, Alberta, Canada.
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Konferensbidrag (refereegranskat)abstract
- We discuss the possible application of techniques inspired by the theories of G-convergence and homogenization to understand mixtures of colors and how they appear as observed by the human eye. The ideas are illustrated by pictures describing the equivalent of a convergence process for different kinds of mixtures of colors.
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| 5. |
- Flodén, Liselott, 1967-, et al.
(författare)
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A strange term in the homogenization of parabolic equations with two spatial and two temporal scales
- 2012
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Ingår i: Journal of Function Spaces and Applications. - : Hindawi Limited. - 0972-6802 .- 1758-4965. ; , s. Art. no. 643458-
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Tidskriftsartikel (refereegranskat)abstract
- We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficient a((x/()),(t/²)) in the elliptic part and spatial oscillations in the coefficient ((x/())) that is multiplied with the time derivative ∂_{t}u^{}. We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation in ((x/())) and the temporal oscillation in a((x/()),(t/²)) and disappears if either of these oscillations is removed.
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| 6. |
- 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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| 7. |
- Flodén, Liselott, 1967-, et al.
(författare)
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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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| 8. |
- Flodén, Liselott, 1967-, et al.
(författare)
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On the determination of effective properties of certain structures with non-periodic temporal oscillations
- 2009
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Ingår i: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling. - Wien : Vienna University Press (WUV). - 9783901608353 ; , s. 2627-2630
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Konferensbidrag (refereegranskat)abstract
- We investigate the homogenization of an evolution problem modelled by a parabolic equation, where the coefficient describing the structure is periodic in space but may vary in time in a non-periodic way. This is performed applying a generalization of two-scale convergence called λ-scale convergence. We give a result on the characterization of the λ-scale limit of gradients under certain boundedness assumptions. This is then applied to perform the homogenization procedure. It turns out that, under a certain condition on the rate of change of the temporal variations, the effective property of the given structure can be determined the same way as in periodic cases.
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| 9. |
- Flodén, Liselott, 1967-, et al.
(författare)
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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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| 10. |
- 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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