Previous |  Up |  Next

Article

Title: The method of Rothe and two-scale convergence in nonlinear problems (English)
Author: Vala, Jiří
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 48
Issue: 6
Year: 2003
Pages: 587-606
Summary lang: English
.
Category: math
.
Summary: Modelling of macroscopic behaviour of materials, consisting of several layers or components, cannot avoid their microstructural properties. This article demonstrates how the method of Rothe, described in the book of K. Rektorys The Method of Discretization in Time, together with the two-scale homogenization technique can be applied to the existence and convergence analysis of some strongly nonlinear time-dependent problems of this type. (English)
Keyword: PDE’s of evolution
Keyword: method of Rothe
Keyword: two-scale convergence
Keyword: homogenization of periodic structures
MSC: 35B27
MSC: 35K55
MSC: 74Q10
MSC: 74Q15
idZBL: Zbl 1099.35047
idMR: MR2025966
DOI: 10.1023/B:APOM.0000024496.35738.28
.
Date available: 2009-09-22T18:16:16Z
Last updated: 2020-07-02
Stable URL: http://hdl.handle.net/10338.dmlcz/134553
.
Reference: [1] G. Allaire: Homogenization and two-scale convergence.SIAM J.  Math. Anal. 23 (1992), 1482–1512. Zbl 0770.35005, MR 1185639, 10.1137/0523084
Reference: [2] T.  Arbogast, J. Douglas and U. Hornung: Derivation of the double porosity model of single phase flow via homogenization theory.SIAM J.  Math. Anal. 21 (1990), 823–836. MR 1052874, 10.1137/0521046
Reference: [3] I.  Babuška: Homogenization approach in engineering.In: Lecture Notes in Economics and Mathematical Systems, M. Berkmann, H. P. Kunzi (eds.), Springer, Berlin, 1975, pp. 137–153. MR 0478946
Reference: [4] I. Babuška: Mathematics of the verification and validation in computational engineering.In: Mathematical and Computer Modelling in Science and Engineering. Proc. Int. Conf. in Prague (January  2003), Czech Tech. Univ. Prague, 2003, pp. 5–12.
Reference: [5] J.  Barták, J. Herrmann, V. Lovicar and O. Vejvoda: Partial Differential Equations of Evolution.Ellis Horwood-SNTL, New York-Prague, 1991. MR 1112789
Reference: [6] G.  Bouchitté, I.  Fragalà: Homogenization of thin structures by two-scale method with respect to measures.SIAM J.  Math. Anal. 32 (2001), 1198–1226. MR 1856245, 10.1137/S0036141000370260
Reference: [7] D. Cioranescu, P. Donato: An Introduction to Homogenization.Oxford University Press, Oxford, 1999. MR 1765047
Reference: [8] J. Dalík, J. Svoboda and S. Šťastník: A model of moisture and temperature propagation.Preprint, Techn. Univ. Brno (Faculty of Civil Engineering), 2000.
Reference: [9] J. Franců: Monotone operators. A survey directed to applications to differential equations.Appl. Math. 35 (1990), 257–301. MR 1065003
Reference: [10] J.  Fučík, A.  Kufner: Nonlinear Differential Equations.Elsevier, Amsterdam, 1980.
Reference: [11] A. Holmbom: Homogenization of parabolic equations. An alternative approach and some corrector-type results.Appl. Math. 42 (1997), 321–343. Zbl 0898.35008, MR 1467553, 10.1023/A:1023049608047
Reference: [12] W. Jäger, J. Kačur: Solution of porous medium type systems by linear approximation schemes.Numer. Math. 60 (1991), 407–427. MR 1137200, 10.1007/BF01385729
Reference: [13] J.  Kačur: Method of Rothe in Evolution Equations.Teubner, Leipzig, 1985. MR 0834176
Reference: [14] A. Kufner, O. John and S. Fučík: Function Spaces.Academia, Prague, 1977. MR 0482102
Reference: [15] M. L. Mascarenhas, A.-M. Toader: Scale convergence in homogenization.Preprint, Univ. Lisboa, 2000. MR 1841866
Reference: [16] A.-M. Matache, Ch.  Schwab: Two-scale finite element method for homogenization problems.Math. Model. Numer. Anal. 26 (2002), 537–572. MR 1932304
Reference: [17] V. G. Maz’ya: Sobolev Spaces.Izdat. Leningradskogo universiteta, Leningrad (St. Petersburg), 1985. (Russian)
Reference: [18] G. Nguetseng: A general convergence result for a functional related to the theory of homogenization.SIAM J.  Math. Anal. 20 (1989), 608–623. Zbl 0688.35007, MR 0990867, 10.1137/0520043
Reference: [19] K. Rektorys: The Method of Discretization in Time.Reidel, Dordrecht, 1982. Zbl 0522.65059
Reference: [20] E. Rothe: Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben.Math. Ann. 102 (1930), 650–670. MR 1512599, 10.1007/BF01782368
Reference: [21] T. Roubíček: Relaxation in Optimization Theory and Variational Calculus.Walter de Gruyter, Berlin, 1997. MR 1458067
Reference: [22] K. Segeth: Rothe method and method of lines. A brief discussion.In: Mathematical and Computer Modelling in Science and Engineering. Proc. Int. Conf. in Prague (January 2003), Czech Tech. Univ. Prague, 2003, pp. 316–320.
Reference: [23] J. Svoboda, J.  Vala: Micromodelling of creep in composites with perfect matrix / particle interfaces.Metallic Materials 36 (1998), 109–129.
Reference: [24] J.  Vala: Two-scale convergence in nonlinear evolution problems.In: Programy a algoritmy numerické matematiky. Proc. 11$^{\text{th}}$ Summer School in Dolní Maxov (June 2002), Math. Inst. Acad. Sci. Czech Rep, to appear. (Czech)
Reference: [25] J.  Vala: Method of discretization in time and two-scale convergence for nonlinear problems of engineering mechanics.Mathematical and Computer Modelling in Science and Engineering. Proc. Int. Conf. in Prague (January 2003), Czech Techn. Univ. Prague, 2003, pp. 359–363.
Reference: [26] J.  Vala: Two-scale convergence with respect to measures in continuum mechanics.Equadiff, CD-ROM Proc. 10$^{\text{th}}$ Int. Conf. in Prague (August  2001), Charles University in Prague. To appear.
Reference: [27] J. Vala: Two-scale limits in some nonlinear problems of engineering mechanics.Math. Comput. Simulation 61 (2003), 177–185. MR 1983667, 10.1016/S0378-4754(02)00074-5
.

Files

Files Size Format View
AplMat_48-2003-6_16.pdf 389.3Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo