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  • 1.
    Danielsson, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Further Investigations of Convergence Results for Homogenization Problems with Various Combinations of Scales2020Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    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.

  • 2.
    Danielsson, Tatiana
    et al.
    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.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Johnsen, Pernilla
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    On some concepts of convergence and their connectionsManuscript (preprint) (Other academic)
  • 3.
    Danielsson, Tatiana
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Johnsen, Pernilla
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Homogenization of linear parabolic equations with three spatial and three temporal microscopic scales for certain matching between the microscopic scalesIn: Article in journal (Refereed)
  • 4.
    Danielsson, Tatiana
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Johnsen, Pernilla
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Homogenization of the heat equation with a vanishing volumetric heat capacity2019In: Progress in Industrial Mathematics at ECMI 2018 / [ed] Faragó, István, Izsák, Ferenc, Simon, Péter L. (Eds.), 2019Conference paper (Refereed)
  • 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.
    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.

  • 6.
    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 scalesManuscript (preprint) (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.

  • 7.
    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.

  • 8.
    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)
  • 9.
    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.

  • 10.
    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 (Refereed)
    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.

  • 11.
    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.

1 - 11 of 11
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