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Flodén, L., Holmbom, A., Jonasson, P., Lobkova, T., Olsson Lindberg, M. & Zhang, Y. (2017). A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem. In: Sivasundaram, S (Ed.), AIP Conference Proceedings: . Paper presented at 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016, 4 July 2016 through 8 July 2016, University of La Rochelle La Rochelle; France. American Institute of Physics (AIP), 1798, Article ID UNSP 020177.
Open this publication in new window or tab >>A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem
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2017 (English)In: AIP Conference Proceedings / [ed] Sivasundaram, S, American Institute of Physics (AIP), 2017, Vol. 1798, article id UNSP 020177Conference paper, Published 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.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2017
Series
AIP Conference Proceedings, ISSN 0094-243X
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-30455 (URN)10.1063/1.4972769 (DOI)000399203000176 ()2-s2.0-85013657168 (Scopus ID)9780735414648 (ISBN)
Conference
11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016, 4 July 2016 through 8 July 2016, University of La Rochelle La Rochelle; France
Available from: 2017-03-13 Created: 2017-03-13 Last updated: 2018-12-05Bibliographically approved
Flodén, L., Holmbom, A., Jonasson, P., Olsson Lindberg, M., Lobkova, T. & Persson, J. (2017). Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales. In: Quintela, P., Barral, P., Gómez, D., Pena, F.J., Rodríguez, J., Salgado, P., Vázquez-Mendéz, M.E. (Ed.), Progress in Industrial Mathematics at ECMI 2016: . Paper presented at ECMI-16, the 19th European Conference on Mathematics for Industry at Santiago de Compostela, 13-17th June 2016. (pp. 617-623). Springer
Open this publication in new window or tab >>Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales
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2017 (English)In: 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, Published 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.

Place, publisher, year, edition, pages
Springer, 2017
Series
Mathematics in Industry, ISSN 1612-3956 ; 26
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-30694 (URN)10.1007/978-3-319-63082-3_94 (DOI)978-3-319-63081-6 (ISBN)
Conference
ECMI-16, the 19th European Conference on Mathematics for Industry at Santiago de Compostela, 13-17th June 2016.
Available from: 2017-05-02 Created: 2017-05-02 Last updated: 2019-06-20Bibliographically approved
Gulliksson, M., Holmbom, A., Persson, J. & Zhang, Y. (2016). A separating oscillation method of recovering the G-limit in standard and non-standard homogenization problems. Inverse Problems, 32(2), Article ID 025005.
Open this publication in new window or tab >>A separating oscillation method of recovering the G-limit in standard and non-standard homogenization problems
2016 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 32, no 2, article id 025005Article in journal (Refereed) Published
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.

Keywords
homogenization, inverse problems, regularization, G-limit
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-27817 (URN)10.1088/0266-5611/32/2/025005 (DOI)000372370900005 ()2-s2.0-84962255423 (Scopus ID)
Available from: 2016-06-08 Created: 2016-06-07 Last updated: 2017-11-30Bibliographically approved
Holmbom, A. & Lobkova, T. (2016). Homogenization of a hyperbolic-parabolic problem in a perforated domain. In: : . Paper presented at 19th European Conference on Mathematics for Industry, Santiago de Compostela, Spain, June 13-17, 2006..
Open this publication in new window or tab >>Homogenization of a hyperbolic-parabolic problem in a perforated domain
2016 (English)Conference paper, Oral presentation with published abstract (Other academic)
Keywords
Homogenization, two-scale convergence, multiscale convergence, perforated domains
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:miun:diva-28656 (URN)
Conference
19th European Conference on Mathematics for Industry, Santiago de Compostela, Spain, June 13-17, 2006.
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2016-12-29Bibliographically approved
Flodén, L., Holmbom, A., Olsson Lindberg, M. & Persson, J. (2014). Homogenization of parabolic equations with an arbitrary number of scales in both space and time. Journal of Applied Mathematics, Art. no. 101685
Open this publication in new window or tab >>Homogenization of parabolic equations with an arbitrary number of scales in both space and time
2014 (English)In: Journal of Applied Mathematics, ISSN 1110-757X, E-ISSN 1687-0042, p. Art. no. 101685-Article in journal (Refereed) Published
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

Place, publisher, year, edition, pages
Boston: Hindawi Publishing Corporation, 2014
Keywords
Multiscale convergence, very weak multiascale convergence, homogenization theory, parabolic partial differential equations, evolution
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-20903 (URN)10.1155/2014/101685 (DOI)000332561700001 ()2-s2.0-84896941846 (Scopus ID)
Note

Publ online Dec 2013

Available from: 2013-12-28 Created: 2013-12-28 Last updated: 2017-12-06Bibliographically approved
Holmbom, A., Olsson Lindberg, M., Persson, J. & Flodén, L. (2013). A note on parabolic homogenization with a mismatch between the spatial scales. Abstract and Applied Analysis, Art. no. 329704
Open this publication in new window or tab >>A note on parabolic homogenization with a mismatch between the spatial scales
2013 (English)In: Abstract and Applied Analysis, ISSN 1085-3375, E-ISSN 1687-0409, p. Art. no. 329704-Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Hindawi Publishing Corporation, 2013
Keywords
Partial differential equations, parabolic, homogenization, two-scale convergence
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-20017 (URN)10.1155/2013/329704 (DOI)000325558500001 ()2-s2.0-84886475240 (Scopus ID)
Available from: 2013-10-18 Created: 2013-10-18 Last updated: 2017-12-06Bibliographically approved
Flodén, L., Holmbom, A., Olsson Lindberg, M. & Persson, J. (2013). Two-scale convergence: Some remarks and extensions. Pure and Applied Mathematics Quarterly, 9(3), 461-486
Open this publication in new window or tab >>Two-scale convergence: Some remarks and extensions
2013 (English)In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 9, no 3, p. 461-486Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
International press of Boston, 2013
Keywords
Two-scale convergence, multiscale convergence, very weak multiscale convergence, homogenization
National Category
Natural Sciences Mathematics
Identifiers
urn:nbn:se:miun:diva-20401 (URN)10.4310/PAMQ.2013.v9.n3.a4 (DOI)000327544500004 ()2-s2.0-84887587258 (Scopus ID)
Available from: 2013-12-02 Created: 2013-12-02 Last updated: 2017-12-06Bibliographically approved
Flodén, L., Holmbom, A. & Olsson Lindberg, M. (2012). A strange term in the homogenization of parabolic equations with two spatial and two temporal scales. Journal of Function Spaces and Applications, Art. no. 643458
Open this publication in new window or tab >>A strange term in the homogenization of parabolic equations with two spatial and two temporal scales
2012 (English)In: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965, p. Art. no. 643458-Article in journal (Refereed) Published
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.

Keywords
Homogenization, parabolic partial differential equations, two-scale convergence, very weak multiscale convergence
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-14550 (URN)10.1155/2012/643458 (DOI)000301409700001 ()2-s2.0-84864936385 (Scopus ID)
Available from: 2011-09-28 Created: 2011-09-28 Last updated: 2017-12-08Bibliographically approved
Flodén, L., Holmbom, A., Olsson, M. & Persson, J. (2011). Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence. Annals of Functional Analysis, 2(1), 84-99
Open this publication in new window or tab >>Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence
2011 (English)In: Annals of Functional Analysis, ISSN 2008-8752, E-ISSN 2008-8752, Vol. 2, no 1, p. 84-99Article in journal (Refereed) Published
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

Place, publisher, year, edition, pages
Mashdad, Iran: TSMG, 2011
National Category
Natural Sciences Mathematics
Identifiers
urn:nbn:se:miun:diva-13732 (URN)000208879900008 ()2-s2.0-84886564588 (Scopus ID)
Available from: 2011-09-28 Created: 2011-05-02 Last updated: 2017-12-08Bibliographically approved
Flodén, L., Holmbom, A., Olsson, M. & Persson, J. (2010). Very weak multiscale convergence. Applied Mathematics Letters, 23(10), 1170-1173
Open this publication in new window or tab >>Very weak multiscale convergence
2010 (English)In: Applied Mathematics Letters, ISSN 0893-9659, E-ISSN 1873-5452, Vol. 23, no 10, p. 1170-1173Article in journal (Refereed) Published
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.

Keywords
Homogenization; Multiscale convergence; Parabolic; Two-scale convergence; Very weak multiscale convergence
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-11654 (URN)10.1016/j.aml.2010.05.005 (DOI)000280890900007 ()2-s2.0-77955424899 (Scopus ID)
Available from: 2010-06-11 Created: 2010-06-11 Last updated: 2017-12-12Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-6742-5781

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