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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
Flodén, L. & Persson, J. (2016). Homogenization Of Nonlinear Dissipative Hyperbolic Problems Exhibiting Arbitrarily Many Spatial And Temporal Scales. Networks and Heterogeneous Media, 11(4), 627-653
Open this publication in new window or tab >>Homogenization Of Nonlinear Dissipative Hyperbolic Problems Exhibiting Arbitrarily Many Spatial And Temporal Scales
2016 (English)In: Networks and Heterogeneous Media, ISSN 1556-1801, E-ISSN 1556-181X, Vol. 11, no 4, p. 627-653Article in journal (Refereed) Published
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.

Keywords
Homogenization theory, nonlinear dissipative hyperbolic problems, multiscale convergence
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-29510 (URN)10.3934/nhm.2016012 (DOI)000387912500004 ()2-s2.0-85002870806 (Scopus ID)
Available from: 2016-12-12 Created: 2016-12-12 Last updated: 2017-11-29Bibliographically 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
Persson, J. (2012). Homogenization of monotone parabolic problems with several temporal scales. Applications of Mathematics, 57(3), 191-214
Open this publication in new window or tab >>Homogenization of monotone parabolic problems with several temporal scales
2012 (English)In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 57, no 3, p. 191-214Article in journal (Refereed) Published
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.

Keywords
homogenization, H-convergence, multiscale convergence, parabolic, monotone
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:miun:diva-17126 (URN)10.1007/s10492-012-0013-z (DOI)000310077500002 ()2-s2.0-84868321017 (Scopus ID)
Available from: 2012-10-02 Created: 2012-10-02 Last updated: 2017-12-07Bibliographically approved
Persson, J. (2012). Selected Topics in Homogenization. (Doctoral dissertation). Östersund: Mittuniversitetet
Open this publication in new window or tab >>Selected Topics in Homogenization
2012 (English)Doctoral 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.

Abstract [sv]

Huvudsakligt fokus i avhandlingen ligger på homogeniseringen av vissa elliptiska och paraboliska problem. Mer precist så homogeniserar vi: ickeperiodiska linjära elliptiska problem i två dimensioner med homotetisk skalning; två typer av evolutionsmultiskaliga linjära paraboliska problem, en med två mikroskopiska skalor i både rum och tid där de senare ges i form av en tvåparameterfamilj, och en med två mikroskopiska skalor i rum och tre i tid som ges i form av fixa potensfunktioner; samt, slutligen, evolutionsmultiskaliga monotona paraboliska problem med en mikroskopisk skala i rum och ett godtyckligt antal i tid som inte är begränsade till att vara givna i form av potensfunktioner. För att kunna uppnå homogeniseringsresultat för dessa problem så studerar och utvecklar vi teorin för tvåskalekonvergens och besläktade begrepp. Speciellt så utvecklar vi begreppet mycket svag tvåskalekonvergens med generaliseringar, och vi studerar en tillämpningav denna konvergenstyp där den används för att detektera förekomsten av heterogenitetsskalor.

Place, publisher, year, edition, pages
Östersund: Mittuniversitetet, 2012. p. x + 168
Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 127
Keywords
homogenization theory, H-convergence, two-scale convergence, very weak two-scale convergence, multiscale convergence, very weak multiscale convergence, evolution-multiscale convergence, very weak evolution-multiscale convergence, λ-scale convergence, non-periodic linear elliptic problems, evolution-multiscale linear parabolic problems, evolution-multiscale monotone parabolic problems, detection of scales of heterogeneity
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:miun:diva-16230 (URN)978-91-87103-19-3 (ISBN)
Public defence
2012-06-11, Q221, Akademigatan 1, Östersund, 13:00 (Swedish)
Opponent
Supervisors
Available from: 2012-05-23 Created: 2012-05-18 Last updated: 2013-11-01Bibliographically 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
Persson, J. (2010). Homogenization of Some Selected Elliptic and Parabolic Problems Employing Suitable Generalized Modes of Two-Scale Convergence. (Licentiate dissertation). Östersund: Mittuniversitetet
Open this publication in new window or tab >>Homogenization of Some Selected Elliptic and Parabolic Problems Employing Suitable Generalized Modes of Two-Scale Convergence
2010 (English)Licentiate 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.

Place, publisher, year, edition, pages
Östersund: Mittuniversitetet, 2010. p. viii + 124
Series
Mid Sweden University licentiate thesis, ISSN 1652-8948 ; 45
Keywords
H-convergence, G-convergence, homogenization, multiscale analysis, two-scale convergence, multiscale convergence, elliptic partial differential equations, parabolic partial differential equations, monotone operators, heterogeneous media, non-periodic media
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:miun:diva-11991 (URN)978-91-86073-90-9 (ISBN)
Presentation
2010-10-06, Q221, Hus Q, Akademigatan 1, Östersund, 13:00 (Swedish)
Opponent
Supervisors
Available from: 2010-09-20 Created: 2010-09-20 Last updated: 2013-11-01Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-9984-2424

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