Previous |  Up |  Next


PDE’s of evolution; method of Rothe; two-scale convergence; homogenization of periodic structures
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.
[1] G. Allaire: Homogenization and two-scale convergence. SIAM J.  Math. Anal. 23 (1992), 1482–1512. DOI 10.1137/0523084 | MR 1185639 | Zbl 0770.35005
[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. DOI 10.1137/0521046 | MR 1052874
[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
[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.
[5] J.  Barták, J. Herrmann, V. Lovicar and O. Vejvoda: Partial Differential Equations of Evolution. Ellis Horwood-SNTL, New York-Prague, 1991. MR 1112789
[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. DOI 10.1137/S0036141000370260 | MR 1856245
[7] D. Cioranescu, P. Donato: An Introduction to Homogenization. Oxford University Press, Oxford, 1999. MR 1765047
[8] J. Dalík, J. Svoboda and S. Šťastník: A model of moisture and temperature propagation. Preprint, Techn. Univ. Brno (Faculty of Civil Engineering), 2000.
[9] J. Franců: Monotone operators. A survey directed to applications to differential equations. Appl. Math. 35 (1990), 257–301. MR 1065003
[10] J.  Fučík, A.  Kufner: Nonlinear Differential Equations. Elsevier, Amsterdam, 1980.
[11] A. Holmbom: Homogenization of parabolic equations. An alternative approach and some corrector-type results. Appl. Math. 42 (1997), 321–343. DOI 10.1023/A:1023049608047 | MR 1467553 | Zbl 0898.35008
[12] W. Jäger, J. Kačur: Solution of porous medium type systems by linear approximation schemes. Numer. Math. 60 (1991), 407–427. DOI 10.1007/BF01385729 | MR 1137200
[13] J.  Kačur: Method of Rothe in Evolution Equations. Teubner, Leipzig, 1985. MR 0834176
[14] A. Kufner, O. John and S. Fučík: Function Spaces. Academia, Prague, 1977. MR 0482102
[15] M. L. Mascarenhas, A.-M. Toader: Scale convergence in homogenization. Preprint, Univ. Lisboa, 2000. MR 1841866
[16] A.-M. Matache, Ch.  Schwab: Two-scale finite element method for homogenization problems. Math. Model. Numer. Anal. 26 (2002), 537–572. MR 1932304
[17] V. G. Maz’ya: Sobolev Spaces. Izdat. Leningradskogo universiteta, Leningrad (St. Petersburg), 1985. (Russian)
[18] G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM J.  Math. Anal. 20 (1989), 608–623. DOI 10.1137/0520043 | MR 0990867 | Zbl 0688.35007
[19] K. Rektorys: The Method of Discretization in Time. Reidel, Dordrecht, 1982. Zbl 0522.65059
[20] E. Rothe: Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben. Math. Ann. 102 (1930), 650–670. DOI 10.1007/BF01782368 | MR 1512599
[21] T. Roubíček: Relaxation in Optimization Theory and Variational Calculus. Walter de Gruyter, Berlin, 1997. MR 1458067
[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.
[23] J. Svoboda, J.  Vala: Micromodelling of creep in composites with perfect matrix / particle interfaces. Metallic Materials 36 (1998), 109–129.
[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)
[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.
[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.
[27] J. Vala: Two-scale limits in some nonlinear problems of engineering mechanics. Math. Comput. Simulation 61 (2003), 177–185. DOI 10.1016/S0378-4754(02)00074-5 | MR 1983667
Partner of
EuDML logo