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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.
    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.

  • 3.
    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.

  • 4.
    Holmbom, Anders
    et al.
    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 a hyperbolic-parabolic problem in a perforated domain2016Conference paper (Other academic)
  • 5.
    Johnsen, Pernilla
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales2018In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 63, no 5, p. 503-521Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to the study of the linear parabolic problem  by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient  in front of the timederivative. First, we have an elliptic homogenized problem although the problem studiedis parabolic. Secondly, we get a parabolic local problem even though the problem has adifferent relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence  different from the standard setting are used, which means that these results are also of independent interest.

  • 6.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Homogenization of linear parabolic equations with a certain resonant matching between rapid spatial and temporal oscillations in periodically perforated domains2019In: Acta Mathematicae Applicatae Sinica (English Series), ISSN 0168-9673, E-ISSN 1618-3932, Vol. 35, no 2, p. 340-358Article in journal (Other academic)
    Abstract [en]

    In this article, we study homogenization of a parabolic linear problem governed by a coefficient matrix with rapid spatial and temporal oscillations in periodically perforated domains with homogeneous Neumann data on the boundary of the holes. We prove results adapted to the problem for a characterization of multiscale limits for gradients and very weak multiscale convergence.

  • 7.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization Results for Parabolic and Hyperbolic-Parabolic Problems and Further Results on Homogenization in Perforated Domains2017Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis is based on four papers. The main focus is on homogenization of selected parabolic problems with time oscillations, and hyperbolic-parabolic problems without time oscillations. The approaches are prepared by means of certain methods, such as two-scale convergence, multiscale convergence and evolution multiscale convergence. We also discuss further results on homogenization of evolution problems in perforated domains.

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