The method of Rothe and two-scale convergence in nonlinear problems

Jiří Vala

Applications of Mathematics (2003)

  • Volume: 48, Issue: 6, page 587-606
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Vala, Jiří. "The method of Rothe and two-scale convergence in nonlinear problems." Applications of Mathematics 48.6 (2003): 587-606. <http://eudml.org/doc/33170>.

@article{Vala2003,
abstract = {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.},
author = {Vala, Jiří},
journal = {Applications of Mathematics},
keywords = {PDE’s of evolution; method of Rothe; two-scale convergence; homogenization of periodic structures; evolution PDE; Rothe method; two-scale convergence; homogenization; periodic structures},
language = {eng},
number = {6},
pages = {587-606},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The method of Rothe and two-scale convergence in nonlinear problems},
url = {http://eudml.org/doc/33170},
volume = {48},
year = {2003},
}

TY - JOUR
AU - Vala, Jiří
TI - The method of Rothe and two-scale convergence in nonlinear problems
JO - Applications of Mathematics
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 6
SP - 587
EP - 606
AB - 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.
LA - eng
KW - PDE’s of evolution; method of Rothe; two-scale convergence; homogenization of periodic structures; evolution PDE; Rothe method; two-scale convergence; homogenization; periodic structures
UR - http://eudml.org/doc/33170
ER -

References

top
  1. 10.1137/0523084, SIAM J.  Math. Anal. 23 (1992), 1482–1512. (1992) Zbl0770.35005MR1185639DOI10.1137/0523084
  2. 10.1137/0521046, SIAM J.  Math. Anal. 21 (1990), 823–836. (1990) MR1052874DOI10.1137/0521046
  3. Homogenization approach in engineering, In: Lecture Notes in Economics and Mathematical Systems, M. Berkmann, H. P. Kunzi (eds.), Springer, Berlin, 1975, pp. 137–153. (1975) MR0478946
  4. 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. (2003) 
  5. Partial Differential Equations of Evolution, Ellis Horwood-SNTL, New York-Prague, 1991. (1991) MR1112789
  6. 10.1137/S0036141000370260, SIAM J.  Math. Anal. 32 (2001), 1198–1226. (2001) MR1856245DOI10.1137/S0036141000370260
  7. An Introduction to Homogenization, Oxford University Press, Oxford, 1999. (1999) MR1765047
  8. A model of moisture and temperature propagation, Preprint, Techn. Univ. Brno (Faculty of Civil Engineering), 2000. (2000) 
  9. Monotone operators. A survey directed to applications to differential equations, Appl. Math. 35 (1990), 257–301. (1990) MR1065003
  10. Nonlinear Differential Equations, Elsevier, Amsterdam, 1980. (1980) 
  11. 10.1023/A:1023049608047, Appl. Math. 42 (1997), 321–343. (1997) Zbl0898.35008MR1467553DOI10.1023/A:1023049608047
  12. 10.1007/BF01385729, Numer. Math. 60 (1991), 407–427. (1991) MR1137200DOI10.1007/BF01385729
  13. Method of Rothe in Evolution Equations, Teubner, Leipzig, 1985. (1985) MR0834176
  14. Function Spaces, Academia, Prague, 1977. (1977) MR0482102
  15. Scale convergence in homogenization, Preprint, Univ. Lisboa, 2000. (2000) MR1841866
  16. Two-scale finite element method for homogenization problems, Math. Model. Numer. Anal. 26 (2002), 537–572. (2002) MR1932304
  17. Sobolev Spaces, Izdat. Leningradskogo universiteta, Leningrad (St. Petersburg), 1985. (Russian) (1985) 
  18. 10.1137/0520043, SIAM J.  Math. Anal. 20 (1989), 608–623. (1989) Zbl0688.35007MR0990867DOI10.1137/0520043
  19. The Method of Discretization in Time, Reidel, Dordrecht, 1982. (1982) Zbl0522.65059
  20. 10.1007/BF01782368, Math. Ann. 102 (1930), 650–670. (1930) MR1512599DOI10.1007/BF01782368
  21. Relaxation in Optimization Theory and Variational Calculus, Walter de Gruyter, Berlin, 1997. (1997) MR1458067
  22. 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. (2003) 
  23. Micromodelling of creep in composites with perfect matrix / particle interfaces, Metallic Materials 36 (1998), 109–129. (1998) 
  24. Two-scale convergence in nonlinear evolution problems, In: Programy a algoritmy numerické matematiky. Proc. 11 Summer School in Dolní Maxov (June 2002), Math. Inst. Acad. Sci. Czech Rep, to appear. (Czech) (to appear) 
  25. 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. (2003) 
  26. Two-scale convergence with respect to measures in continuum mechanics, Equadiff, CD-ROM Proc. 10 Int. Conf. in Prague (August  2001), Charles University in Prague. To appear. 
  27. 10.1016/S0378-4754(02)00074-5, Math. Comput. Simulation 61 (2003), 177–185. (2003) MR1983667DOI10.1016/S0378-4754(02)00074-5

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.