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Meccanica

, Volume 28, Issue 2, pp 121–128 | Cite as

Time discretization schemes for the Stefan problem in a concentrated capacity

  • Enrico Magenes
  • Claudio Verdi
Article

Abstract

We consider two different time discretization algorithms for a nonlinear parabolic PDE arising in heat conduction phenomena with phase changes in two adjoining bodies Ω and Γ, where Γ can be considered as the boundary of Ω. Stability, convergence and error estimate results are given for both algorithms.

Key words

Free boundary problems Time discretization Error estimates 

Sommario

Si studiano due algoritmi di discretizzazione nel tempo di un sistema di equazioni a derivate parziali non lineari paraboliche che governa la conduzione del calore, in presenza di cambiamento di fase, in due corpi congiunti Ω e Γ, di cui Γ possa essere considerato come la frontiera di Ω, Vengono dati risultati di stabilità, convergenza e maggiorazione dell'errore per entrambi gli algoritmi.

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References

  1. 1.
    Amiez, G. and Grémaud, P. A., ‘On a numerical approach of Stefanlike problems’,Numer. Math.,59 (1991) 71–89.Google Scholar
  2. 2.
    Andreucci, D., ‘Existence and uniqueness of solution to a concentrated capacity problem with change of phase’,European J. Appl. Math.,1, (1990) 339–351.Google Scholar
  3. 3.
    Berger, A. E., Brézis, H. and Rogers, J. C. W., ‘A numerical method for solving the problemu t - Δf(u) 0’RAIRO Modél. Math. Anal. Numér.,13 (1979) 297–312.Google Scholar
  4. 4.
    De Rham, G.,Variétés Differentiables, Hermann, Paris, 1955.Google Scholar
  5. 5.
    Fasano, A., Primicerio, M. and Rubinstein, L., ‘A model problem for heat conduction with a free boundary in a concentrated capacity’,J. Inst. Maths. Appl.,26 (1980) 327–347.Google Scholar
  6. 6.
    Jäger, W. and Kačur, J., ‘Solution of porous medium type systems by linear approximation schemes’,Numer. Math.,60 (1991) 407–427.Google Scholar
  7. 7.
    Lions, J. L. and Magenes, E.,Non-Homogeneous Boundary Value Problems and Applications, Vols 1 and 2, Springer-Verlag, Berlin, 1972.Google Scholar
  8. 8.
    Magenes, E., ‘Remarques sur l'approximation des problèmes non linéaires paraboliques’, inAnalyse Mathématique et Applications, Gauthier-Villars, Paris, 1988, pp. 298–318.Google Scholar
  9. 9.
    Magenes, E., ‘Numerical approximation of nonlinear evolution problems’, inFrontiers in Pure and Applied Mathematics (ed. R. Dautray), North-Holland, Amsterdam, 1991, pp. 193–207.Google Scholar
  10. 10.
    Magenes, E., ‘Some new results on a Stefan problem in a concentrated capacity’,Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9,3 (1992) 23–34.Google Scholar
  11. 11.
    Magenes, E., ‘The Stefan problem in a concentrated capacity’, inProc. Internat. Symp. on Problemi attuali dell'Analisi e della Fisica Matematica dedicated to G. Fichera, Taormina, 1992 (to appear).Google Scholar
  12. 12.
    Magenes, E., Nochetto, R.H. and Verdi, C., ‘Energy error estimates for a linear scheme to approximate nonlinear parabolic equations’,RAIRO Modél. Math. Anal. Numér.,21 (1987) 655–678.Google Scholar
  13. 13.
    Nochetto, R. H., ‘A stable extrapolation method for multidimensional degenerate problems’,Math. Comput.,53 (1989) 455–470.Google Scholar
  14. 14.
    Nochetto, R. H. and Verdi, C., ‘An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation’,Math. Comput.,51 (1988) 27–53.Google Scholar
  15. 15.
    Shillor, M., ‘Existence and continuity of a weak solution to the problem of a free boundary in a concentrated capacity’,Proc. Roy. Soc. Edinburgh Sect. A,100 (1985) 271–280.Google Scholar
  16. 16.
    Slodička, M., ‘Solution of nonlinear parabolic problems by linearization’, Preprint n. M3-92Comenius Univ. Fuc. Math. Phys. (1992) 1–9.Google Scholar
  17. 17.
    Verdi, C. and Visintin, A., ‘Error estimates for a semi-explicit numerical scheme for Stefan-type problems’,Numer. Math.,52 (1988) 165–185.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Enrico Magenes
    • 1
  • Claudio Verdi
    • 2
    • 3
  1. 1.Dipartimento di Matematico, and Istituto di Analisi Numerica del CNRUniversitú di PaviaPaviaItaly
  2. 2.Dipartimento di MatematicaUniversità di MilanoMilano
  3. 3.Istituto di Analisi Numerica del CNRPaviaItaly

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