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  • 1.
    Flodén, Liselott
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    G-convergence and homogenization for some monotone operators with multiple scales2005Licentiate thesis, monograph (Other academic)
    Abstract [en]

    This thesis deals with questions concerning the convergence of sequences of functions and operators. G-convergence is studied for elliptic and parabolic equations and the necessary investigations of the properties of certain monotone operators are made. In particular, we consider periodic cases with oscillations in one or several scales including the possibility of rapid oscillations in time. Homogenization procedures for these problems are developed and local problems are identified.

  • 2.
    Flodén, Liselott
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    G-Convergence and Homogenization of some Sequences of Monotone Differential Operators2009Doctoral thesis, monograph (Other academic)
    Abstract [en]

    This thesis mainly deals with questions concerning the convergence of some sequences of elliptic and parabolic linear and non-linear operators by means of G-convergence and homogenization. In particular, we study operators with oscillations in several spatial and temporal scales. Our main tools are multiscale techniques, developed from the method of two-scale convergence and adapted to the problems studied. For certain classes of parabolic equations we distinguish different cases of homogenization for different relations between the frequencies of oscillations in space and time by means of different sets of local problems. The features and fundamental character of two-scale convergence are discussed and some of its key properties are investigated. Moreover, results are presented concerning cases when the G-limit can be identified for some linear elliptic and parabolic problems where no periodicity assumptions are made.

  • 3.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Jonasson, Pernilla
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales2017In: Progress in Industrial Mathematics at ECMI 2016 / [ed] Quintela, P., Barral, P., Gómez, D., Pena, F.J., Rodríguez, J., Salgado, P., Vázquez-Mendéz, M.E., Springer, 2017, p. 617-623Conference paper (Refereed)
    Abstract [en]

    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
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    A strange term in the homogenization of parabolic equations with two spatial and two temporal scales2012In: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965, p. Art. no. 643458-Article in journal (Refereed)
    Abstract [en]

    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.

  • 5.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization of parabolic equations with an arbitrary number of scales in both space and time2014In: Journal of Applied Mathematics, ISSN 1110-757X, E-ISSN 1687-0042, p. Art. no. 101685-Article in journal (Refereed)
    Abstract [en]

    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

  • 6.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Two-scale convergence: Some remarks and extensions2013In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 9, no 3, p. 461-486Article in journal (Refereed)
    Abstract [en]

    We first study the fundamental ideas behind two-scale conver-

    gence to enhance an intuitive understanding of this notion. The classical

    definitions and ideas are motivated with geometrical arguments illustrated

    by illuminating figures. Then a version of this concept, very weak two-scale

    convergence, is discussed both independently and brie°y in the context of

    homogenization. The main features of this variant are that it works also

    for certain sequences of functions which are not bounded in

    L2  and at

    the same time is suited to detect rapid oscillations in some sequences which

    are strongly convergent in

    L2 . In particular, we show how very weak

    two-scale convergence explains in a more transparent way how the oscilla-

    tions of the governing coe±cient of the PDE to be homogenized causes the

    deviation of the

    G-limit from the weak L2 NxN-limit for the sequence of

    coe±cients. Finally, we investigate very weak multiscale convergence and

    prove a compactness result for separated scales which extends a previous

    result which required well-separated scales.

  • 7.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    A myriad shades of green2009In: Proceedings of Bridges 2009, Banff, Alberta, Canada, 2009Conference paper (Refereed)
    Abstract [en]

    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.

  • 8.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence2011In: Annals of Functional Analysis, ISSN 2008-8752, E-ISSN 2008-8752, Vol. 2, no 1, p. 84-99Article in journal (Refereed)
    Abstract [en]

    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

  • 9.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    On the determination of effective properties of certain structures with non-periodic temporal oscillations2009In: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling, Wien: Vienna University Press (WUV), 2009, p. 2627-2630Conference paper (Refereed)
    Abstract [en]

    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.

  • 10.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Very weak multiscale convergence2010In: Applied Mathematics Letters, ISSN 0893-9659, E-ISSN 1873-5452, Vol. 23, no 10, p. 1170-1173Article in journal (Refereed)
    Abstract [en]

    We briefly recall the concept of multiscale convergence, which is a generalization of two-scale convergence. Then we investigate a related concept, called very weak multiscale convergence, and prove a compactness result with respect to this type of convergence. Finally we illustrate how this result can be used to study homogenization problems with several scales of oscillations.

  • 11.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    A model for the heat transfer between steel edge and running base in alpine racing skis2006In: WSEAS transactions on heat and mass transfer.,ISSN 1790-5044, 2006, Vol. 1, no 3, p. 256-261p. 256-261Conference paper (Refereed)
  • 12.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the convergence and determination of limits for some sequences of differential operators: Presented at The Ninth International Conference on Integral Methods in Science and Engineering, Niagara Falls, Ontario, Canada, July 23-27 20062006Conference paper (Other academic)
  • 13.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Some remarks on homogenization in perforated domains2008In: Integral Methods in Science and Engineering: Techniques and Applications, Boston: Birkhäuser Verlag, 2008, p. 89-97Chapter in book (Other academic)
  • 14.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Catrakis, R. J.
    On a heat transfer problem related to alpine ski racing2006In: Proceedings of the WSEAS HEAT '06, FLUID '06 : Elounda, Agios Nikolaos, Crete Island, Greece, August 21 - 23, 2006 ; 4th IASME/WSEAS International Conference on Heat Transfer, Thermal Engineering and Environment, 4th IASME/WSEAS International Conference on Fluid Mechanics and Aerodynamics: CD-rom, Elounda, Greece: WSEAS press , 2006, , p. 301-304Conference paper (Refereed)
  • 15.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Troch, Inge
    On some convergemce results and their relation to the impact of impurities on effective heat conduction properties2006In: Proceedings / 5th MATHMOD Vienna : February 8 - 10, 2006, Vienna University of Technology, Austria, Wien, 2006, , p. 16-1-16-7Conference paper (Refereed)
  • 16.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Svanstedt, Nils
    Reiterated homogeneization of monotone parabolic problems2007In: Annali dell'Universita di Ferrara, ISSN 0430-3202, Vol. 53, no 2, p. 217-232Article in journal (Refereed)
    Abstract [en]

    Reiterated homogenization is studied for divergence structure parabolic problems of the form ∂uε ∂t − div a x ε , x ε2 , t, Duε = f . It is shown that under standard assumptions on the function a (y1, y2, t, ξ) the sequence {uε} of solutions converges weakly in L p(0, T ;W1,p 0 (Ω)) to the solution u of the homogenized problem ∂u ∂t − div (b (t, Du)) = f

  • 17.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Jonasson, Pernilla
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scalesIn: Article in journal (Other academic)
    Abstract [en]

    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.

  • 18.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Homogenization of some parabolic operators with several time scales2007In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 52, no 5, p. 431-446Article in journal (Refereed)
    Abstract [en]

    The main focus in this paper is on homogenization of a parabolic problem with two local time scales. Under certain assumptions on the coefficient a, there exists a G-limit b, which we characterize by means of multiscale techniques for r>0, r≠1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.

  • 19.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the effective heat conduction properties of heterogeneous media with multiple scales2005In: IASME Transactions., ISSN 1790-031X, Vol. 2, p. 180-183Article in journal (Refereed)
    Abstract [en]

    We study the asymptotic behaviour of generalized heat equations describing a periodic heterogeneous material with multiple scales when the fineness of the structure goes to zero. For different ratio between the characteristic sizes of the two spatial scales and the single time scale we find different equations, defined on a representative unit, providing us with the connection between the microstructure and the effective properties.

  • 20.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the effective heat conduction properties of heterogeneous media with multiple scales2005In: Proceedings. 2005 IASME / WSEAS International Conference on Heat and mass transfer, 2005, p. 180-183Conference paper (Refereed)
    Abstract [en]

    We study the asymptotic behaviour of generalized heat equations describing a periodic heterogeneous material with multiple scales when the fineness of the structure goes to zero. For different ratio between the characteristic sizes of the two spatial scales and the single time scale we find different equations, defined on a representative unit, providing us with the connection between the microstructure and the effective properties.

  • 21.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Reiterated homogenization of some linear and nonlinear monotone parabolic operators2006In: Canadian Applied Mathematics Quarterly, ISSN 1073-1849, Vol. 14, no 2, p. 149-184Article in journal (Refereed)
    Abstract [en]

    This paper concerns the reiterated homogenization of monotone parabolic problems of the form ∂_{t}u^{}-∇⋅a((x/()),(x/²),(t/(^{r})),∇u^{})=f. We prove that under certain assumptions on a, there exists a G-limit b, which is also characterized by means of homogenization for 0<r

  • 22.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Reiterated homogenization of some linear and nonlinear monotone parabolic operators2005Book (Other academic)
    Abstract [en]

    This paper concerns the reiterated homogenization of monotone parabolic problems with multiple scales. We prove that under certain assumptions on the sequences of operators, there exists a G-limit, which is also characterized by means of homogenization for different speeds of oscillations in time.

  • 23.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Some convergence results for evolution problems with multiple scales2006Report (Other academic)
  • 24.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Some convergence results for sequences of oscillating parabolic operators with multiple scales: Presented at the 26-th Midwest-Pacific Differential Equations Conference2005Conference paper (Other academic)
  • 25.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization Of Nonlinear Dissipative Hyperbolic Problems Exhibiting Arbitrarily Many Spatial And Temporal Scales2016In: Networks and Heterogeneous Media, ISSN 1556-1801, E-ISSN 1556-181X, Vol. 11, no 4, p. 627-653Article in journal (Refereed)
    Abstract [en]

    This paper concerns the homogenization of nonlinear dissipative hyperbolic problems partial derivative ttu(epsilon) (x, t) - del . (a(x/epsilon(q1),..., x/epsilon(qn), t/epsilon(r1),..., t/epsilon(rm)) del u(epsilon) (x, t)) +g (x/epsilon(r1),..., x/epsilon(rn), t/epsilon(r1), u(epsilon) (x, t), del u(epsilon) (x, t)) = f (x, t)

    where both the elliptic coefficient a and the dissipative term a are periodic in the n + m first arguments where n and m may attain any non-negative integer value. The homogenization procedure is performed within the framework of evolution multiscale convergence which is a generalization of two-scale convergence to include several spatial and temporal scales. In order to derive the local problems, one for each spatial scale, the crucial concept of very weak evolution multiscale convergence is utilized since it allows less benign sequences to attain a limit. It turns out that the local problems do not involve the dissipative term g even though the homogenized problem does and, due to the nonlinearity property, an important part of the work is to determine the effective dissipative term. A brief illustration of how to use the main homogenization result is provided by applying it to an example problem exhibiting six spatial and eight temporal scales in such a way that a and g have disparate oscillation patterns.

  • 26.
    Flodén, Lotta
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Jonasson, Pernilla
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Zhang, Ye
    Örebro universitet.
    A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem2017In: AIP Conference Proceedings / [ed] Sivasundaram, S, American Institute of Physics (AIP), 2017, Vol. 1798, article id UNSP 020177Conference paper (Refereed)
    Abstract [en]

    We study the homogenization of a hyperbolic-parabolic PDE with oscillations in one fast spatial scale. Moreover, the first order time derivative has a degenerate coefficient passing to infinity when ϵ→0. We obtain a local problem which is of elliptic type, while the homogenized problem is also in some sense an elliptic problem but with the limit for ϵ-1∂tuϵ as an undetermined extra source term in the right-hand side. The results are somewhat surprising and work remains to obtain a fully rigorous treatment. Hence the last section is devoted to a discussion of the reasonability of our conjecture including numerical experiments.

  • 27.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Flodén, Liselott
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    A note on parabolic homogenization with a mismatch between the spatial scales2013In: Abstract and Applied Analysis, ISSN 1085-3375, E-ISSN 1687-0409, p. Art. no. 329704-Article in journal (Refereed)
    Abstract [en]

    We consider the homogenization of the linear parabolic problem rho(x/epsilon(2))partial derivative(t)u(epsilon)(x,t) - del . (a(x/epsilon(1), t/epsilon(2)(1))del u(epsilon) (x,t)) = f(x,t) which exhibits a mismatch between the spatial scales in the sense that the coefficient a(x/epsilon(1), t/epsilon(2)(1)) of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient rho(x/epsilon(2)) of the time derivative contains a faster spatial scale. It is shown that the faster spatialmicroscale does not give rise to any corrector termand that there is only one local problemneeded to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

  • 28.
    Johansson, Helena
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Österholm, Magnus
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Flodén, Liselott
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Heidtmann, Pia
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Teachers’ and students’ perception of the gap between secondary and tertiary mathematics2018In: Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education / [ed] Bergqvist, E., Österholm, M., Granberg, C., & Sumpter, L., Umeå, Sweden: PME , 2018, Vol. 5, p. 77-77Conference paper (Refereed)
  • 29.
    Stenmark, Petter
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Management and Mechanical Engineering.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Barthelson, Mats
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Ecotechnology and Sustainable Building Engineering.
    Flodén, Liselott
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Ständiga förbättringar i högre utbildning: En modell för systematisk kursutveckling2018Conference paper (Other (popular science, discussion, etc.))
    Abstract [sv]

    Utmaningarna för lärare i högre utbildning att fullgöra sitt pedagogiska uppdrag har uppmärksammats och debatterats i olika sammanhang den senaste tiden (e.g. SvD, 2018-01-21; DN, 2017-09-27). Enligt Högskoleverket (2009) har skillnaderna i förkunskaper bland studenter ökat och studentgrupperna blivit större. Brommesson et al. (2016) menar att den förändrade studentpopulationen innebär en utmaning för den enskilde läraren i konkreta undervisningssituationer. Därtill förändras samhällets kompetensbehov över tid och teknisk utveckling ger möjligheter att bedriva undervisning under andra former än tidigare. Kurser och utbildningar behöver därför utvecklas över tid, för att fortsätta vara aktuella och anpassade efter de behov och förutsättningar som finns. Vid utveckling av kurser och utbildningar krävs både pedagogiska idéer och en fungerande struktur som gör denna utveckling möjlig, där idéerna tas tillvara för att senare omsättas i praktiken. Det har vid ett pedagogiskt utvecklingsprojekt vid mittuniversitetet uppmärksammats några problem med nuvarande stöd:  Resurser och information finns inte på ett ställe  Kunskapsutbyte är begränsat  Det finns inget systematiskt stöd Syfte med detta bidrag är att presentera, diskutera, och få feedback på, en modell för systematisk kursutveckling över tid, för att bemöta ovanstående problem. En modell för ständiga förbättringar presenteras, där pedagogisk utveckling och kursadministration samverkar för att på ett systematiskt sätt tillvarata idéer och lärdomar, och underlätta kunskapsutbyte. Visuellt består modellen av ett administrativt och ett pedagogiskt kurshjul som tillsammans ska täcka upp allt som behövs för att planera, genomföra och utveckla en kurs - från deadlines för administrativa moment till idéer om undervisningsformer och utvärdering. Idén är att man enkelt ska kunna klicka sig fram på en webbsida, till allt som är aktuellt/intressant i det stadie kursen befinner sig - planering, genomförande eller utvärdering och utveckling. Vi kommer att argumentera för att en fungerande kursutveckling underlättas av visuellt stöd och en tydlig arbetsgång. Målgrupp för den här presentationen är i huvudsak undervisande lärare som vill hitta vägar för att mer systematiskt arbeta med kursutveckling.

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