| 1. |
- Flodén, Liselott, 1967-, et al.
(författare)
-
A myriad shades of green
- 2009
-
Ingår i: Proceedings of Bridges 2009, Banff, Alberta, Canada.
-
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.
|
|
| 2. |
- Flodén, Liselott, 1967-, et al.
(författare)
-
Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence
- 2011
-
Ingår i: Annals of Functional Analysis. - Mashdad, Iran : TSMG. - 2008-8752. ; 2:1, s. 84-99
-
Tidskriftsartikel (refereegranskat)abstract
- We apply a new version of multiscale convergence named very weak multiscale convergence to find possible frequencies of oscillation in an unknown coefficient of a diffeential equation from its solution. We also use thís notion to study homogenization of a certain linear parabolic problem with multiple spatial and temporal scales
|
|
| 3. |
- Flodén, Liselott, et al.
(författare)
-
Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales
- 2017
-
Ingår i: Progress in Industrial Mathematics at ECMI 2016. - Cham : Springer. - 9783319630816 ; , s. 617-623
-
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.
|
|
| 4. |
- Flodén, Liselott, 1967-, et al.
(författare)
-
Homogenization of parabolic equations with an arbitrary number of scales in both space and time
- 2014
-
Ingår i: Journal of Applied Mathematics. - Boston : Hindawi Publishing Corporation. - 1110-757X .- 1687-0042. ; , s. Art. no. 101685-
-
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
|
|
| 5. |
- Flodén, Liselott, 1967-, et al.
(författare)
-
On the determination of effective properties of certain structures with non-periodic temporal oscillations
- 2009
-
Ingår i: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling. - Wien : Vienna University Press (WUV). - 9783901608353 ; , s. 2627-2630
-
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.
|
|
| 6. |
- Flodén, Liselott, 1967-, et al.
(författare)
-
Two-scale convergence: Some remarks and extensions
- 2013
-
Ingår i: Pure and Applied Mathematics Quarterly. - : International press of Boston. - 1558-8599 .- 1558-8602. ; 9:3, s. 461-486
-
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.
|
|
| 7. |
- Johnsen, Pernilla
(författare)
-
Homogenization of Partial Differential Equations using Multiscale Convergence Methods
- 2021
-
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.
|
|
| 8. |
- Persson, Jens, 1978-
(författare)
-
Homogenization of Some Selected Elliptic and Parabolic Problems Employing Suitable Generalized Modes of Two-Scale Convergence
- 2010
-
Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.
|
|
| 9. |
- Persson, Jens, 1978-
(författare)
-
Selected Topics in Homogenization
- 2012
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main focus of the present thesis is on the homogenization of some selected elliptic and parabolic problems. More precisely, we homogenize: non-periodic linear elliptic problems in two dimensions exhibiting a homothetic scaling property; two types of evolution-multiscale linear parabolic problems, one having two spatial and two temporal microscopic scales where the latter ones are given in terms of a two-parameter family, and one having two spatial and three temporal microscopic scales that are fixed power functions; and, finally, evolution-multiscale monotone parabolic problems with one spatial and an arbitrary number of temporal microscopic scales that are not restricted to be given in terms of power functions. In order to achieve homogenization results for these problems we study and enrich the theory of two-scale convergence and its kins. In particular the concept of very weak two-scale convergence and generalizations is developed, and we study an application of this convergence mode where it is employed to detect scales of heterogeneity.
|
|
