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  • 1.
    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.

  • 2.
    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

  • 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.
    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.

  • 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, 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.

  • 5.
    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

  • 6.
    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.

  • 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.
    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.

  • 8.
    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.

  • 9.
    Gulliksson, Marten
    et al.
    Univ Orebro, Sch Sci & Technol, Dept Math, SE-70182 Örebro, Sweden.
    Holmbom, Anders
    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.
    Zhang, Ye
    Univ Orebro, Sch Sci & Technol, Dept Math, SE-70182 Orebro, Sweden.
    A separating oscillation method of recovering the G-limit in standard and non-standard homogenization problems2016In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 32, no 2, article id 025005Article in journal (Refereed)
    Abstract [en]

    Reconstructing the homogenized coefficient, which is also called the G-limit, in elliptic equations involving heterogeneous media is a typical nonlinear ill-posed inverse problem. In this work, we develop a numerical technique to determine G-limit that does not rely on any periodicity assumption. The approach is a technique that separates the computation of the deviation of the G-limit from the weak L-2-limit of the sequence of coefficients from the latter. Moreover, to tackle the ill-posedness, based on the classical Tikhonov regularization scheme we develop several strategies to regularize the introduced method. Various numerical tests for both standard and non-standard homogenization problems are given to show the efficiency and feasibility of the proposed method.

  • 10.
    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.

  • 11.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Homogenization of monotone parabolic problems with several temporal scales2012In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 57, no 3, p. 191-214Article in journal (Refereed)
    Abstract [en]

    In this paper we homogenize monotone parabolic problems with two spatial scales and any number of temporal scales. Under the assumption that the spatial and temporal scales are well-separated in the sense explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the “slow” self-similar case), rapid temporal oscillations, and rapid resonant spatial and temporal oscillations (the “rapid” self-similar case), respectively.

  • 12.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Homogenization of Some Selected Elliptic and Parabolic Problems Employing Suitable Generalized Modes of Two-Scale Convergence2010Licentiate thesis, monograph (Other academic)
    Abstract [en]

    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.

  • 13.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Selected Topics in Homogenization2012Doctoral thesis, monograph (Other academic)
    Abstract [en]

    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.

  • 14.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    λ-scale convergence applied to the stationary heat conduction equation with non-periodic thermal conductivity matrix2009In: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling, Wien: Vienna University Press (WUV), 2009, p. 2720-2723Conference paper (Refereed)
    Abstract [en]

    In this contribution we study the homogenization of non-periodic stationary heat conduction problems with homogeneous Dirichlet boundary data by applying the recently developed λ-scale convergence technique developed by Holmbom and Silfver. λ-scale convergence can be seen as either being a special case of scale convergence (developed by Mascarenhas and Toader) or of “generalized” two-scale convergence (developed by Holmbom, Silfver, Svanstedt and Wellander). From either viewpoint, it is a possibly powerful generalization of Nguetseng’s classical, periodic two-scale convergence method. We give a definition of the concept of λ-scale convergence, which is then used to claim a main theorem on homogenization of certain non-periodic stationary heat conduction problems. The original part of the contribution starts by defining a two-dimensional “toy model”. We show that the “toy model” satisfies the right conditions such that the aforementioned main theorem on the homogenization can be employed. In this way we derive the homogenized problem, i.e. the homogenized thermal conductivity matrix, and the local problem. The contribution is concluded by giving a numerical example where we explicitly compute the homogenized thermal conductivity matrix.

1 - 14 of 14
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