Numerische Mathematik

, Volume 34, Issue 4, pp 411–429 | Cite as

The numerical solution of the inverse Stefan problem

  • Peter Jochum


The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.

Subject Classifications

AMS(MOS): 65N10 CR: 5.15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babuska, I., Prager, M., Vitasek, E.: Numerical processes in differential equations. New York: John Wiley 1966Google Scholar
  2. 2.
    Barrodale, I., Young, A.: Algorithms for bestL 1 andL linear approximations on a discrete set. Numer. Math.8, 295–306 (1966)Google Scholar
  3. 3.
    Bonnerot, R., Jamet, P.: A second order finite element method for the one-dimensional Stefan problem. Internat. J. Numer. Methods Engrg.8, 811–820 (1974)Google Scholar
  4. 4.
    Budak, B.M., Vasil'eva, V.I.: The solution of the inverse Stefan problem. U.S.S.R. Computional Math. and Math. Phys.13, 130–151 (1974)Google Scholar
  5. 5.
    Budak, B.M., Vasil'eva, V.I.: On the solution of Stefan's converse problem II. U.S.S.R. Computional Math. and Math. Phys.13, 97–110 (1974)Google Scholar
  6. 6.
    Cannon, J.R.: A Cauchy problem for the heat equation. Ann. Mat. Pura Appl.65, 377–387 (1964)Google Scholar
  7. 7.
    Cannon, J.R., Douglas, J.: The Cauchy problem for the heat equation. SIAM J. Numer. Anal.4, 317–337 (1967)Google Scholar
  8. 8.
    Cannon, J.R., Ewing, R.E.: A direct numerical procedure for the Cauchy problem for the heat equation. J. Math. Anal. Appl.56, 7–17 (1976)Google Scholar
  9. 9.
    Cannon, J.R., Primicerio, M.: Remarks on the one-phase Stefan problem for the heat equation with flux prescribed on the fixed boundary. J. Math. Anal. Appl.35, 361–373 (1971)Google Scholar
  10. 10.
    Cheney, E.W.: Introduction to approximation theory. New York: McGraw-Hill, 1966Google Scholar
  11. 11.
    Cromme, L.: Eine Klasse von Verfahren zur Ermittlung bester nicht-linearer Tschebyscheff-Approximationen. Numer. Math.25, 447–459 (1976)Google Scholar
  12. 12.
    Hoffmann, K.-H., Klostermair, A.: Approximation mit Lösungen von Differentialgleichungen. In: ‘Approximationstheorie’, Lecture Notes in Mathematics 556. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  13. 13.
    Jochum, P.: Differentiable dependence upon the data in a one-phase Stefan problem. J. Math. Meth. Appl. Sci.2, 73–90 (1980)Google Scholar
  14. 14.
    Jochum, P.: The inverse Stefan problem as a problem of nonlinear approximation theory. J. Approximation Theory (1980, in press)Google Scholar
  15. 15.
    Künzi, H.P., Krelle, W.: Nichtlineare Programmierung. Berlin-Göttingen-Heidelberg: Springer 1962Google Scholar
  16. 16.
    Meinardus, G.: Approximation von Funktionen und ihre numerische Behandlung. Berlin-Heidelberg-New York: Springer 1964Google Scholar
  17. 17.
    Ockendon, J.R., Hodkins, W.R. (ed.): Moving boundary problems in heat flow and diffusion. Proceedings Conf. Oxford 1974, Oxford: Clarendon Press 1974Google Scholar
  18. 18.
    Ortega, J., Rheinboldt, W.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1972Google Scholar
  19. 19.
    Osborne, M.R., Watson, G.A.: An algorithm for minimax approximation in nonlinear case. Comput. J.12, 63–68 (1969)Google Scholar
  20. 20.
    Pucci, C.: On the improperly posed Cauchy problem for parabolic equations. Symposium on the numerical treatment of partial differential equations with real characteristics. Basel: Birkhäuser, 1959Google Scholar
  21. 21.
    Saguez, C.: Contrôle optimale de systèmes gouvernés par des inéquations variationelles. Applications a des problèmes de frontière libre. Rapport de Recherche 191, IRIA (1976). Contrôle optimale d'inéquation variationelles avec observation de domaines. Rapport de Recherche 286, IRIA (1978)Google Scholar
  22. 22.
    Schabak, R.: On alternation numbers in nonlinear Chebychev approximation. J. Approximation Theory23, 379–391 (1978)Google Scholar
  23. 23.
    Wulbert, D.: Uniqueness and differential characterization of approximations from manifolds of functions. Amer. Math. J.93, 350–366 (1971)Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Peter Jochum
    • 1
  1. 1.Mathematisches Institut der Universität MünchenMünchen 2Germany (Fed. Rep.)

Personalised recommendations