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Numerische Mathematik

, Volume 58, Issue 1, pp 51–77 | Cite as

The finite element method for nonlinear elliptic equations with discontinuous coefficients

  • Alexander Ženíšek
Article

Summary

The study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed Dirichlet-Neumann boundary conditions. The change in domain and numerical integration are taken into account. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz-continuous the following convergence results are proved: 1. the rate of convergenceO(hε) if the exact solutionuH1 (Ω) is piecewise of classH1+ε (0<ε≦1);2. the convergence without any rate of convergence ifuH1 (Ω) only.

Subject classifications

AMS(MOS) 65N30 CR G1.8 

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References

  1. 1.
    Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing5, 207–213 (1970)Google Scholar
  2. 2.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland (1978)Google Scholar
  3. 3.
    Doktor, P.: On the density of smooth functions in certain subspaces of Sobolev space. Commentationes Mathematicae Universitatis Carolinae14, 609–622 (1973)Google Scholar
  4. 4.
    Feistauer, M.: On the finite element approximation of functions with noninteger derivatives. Numer. Funct. Anal. Optimization10, 91–110 (1989)Google Scholar
  5. 5.
    Feistauer, M., Sobotíková, V.: Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. Preprint, Charles Univ., Prague 1988 (to appear in M2AN)Google Scholar
  6. 6.
    Feistauer, M., Ženíšek, A.: Finite element solution of nonlinear elliptic problems. Numer. Math.50, 451–475 (1987)Google Scholar
  7. 7.
    Feistauer, M., Ženíšek, A.: Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math.52, 147–163 (1988)Google Scholar
  8. 8.
    Glowinski, R., Marrocco, A.: Analyse numérique du champ magnétique d'un alternateur par elements finis et surrelaxation punctuelle non lineaire. Comput. Methods Appl. Mech. Eng.3, 55–85 (1974).Google Scholar
  9. 9.
    Glowinski, R., Marrocco, A.: Numerical solution of two-dimensional magnetostatic problems by augmented Lagrangian methods. Comput. Methods Appl. Mech. Eng.12, 33–46 (1977)Google Scholar
  10. 10.
    Jamet, P.: Estimations d'erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér.10, 43–61 (1976)Google Scholar
  11. 11.
    Kreisinger, V., Adam, J.: Magnetic fields in nonlinear anisotropic ferromagnetics. Acta Technica ČSAV, 209–241 (1984)Google Scholar
  12. 12.
    Kufner, A., John, O., Fučik, S.: Function Spaces. Prague: Academia 1977Google Scholar
  13. 13.
    Marrocco, A.: Analyse numérique de problémes d'électrotechnique. Ann. Sci. Math. Québ.1, 271–296 (1977)Google Scholar
  14. 14.
    Melkes, F., Chrobáček, K., Rak L.: A stationary magnetic field computation in electrical machines. Technika elektrických strojú, VÚES Brno1–2, 22–29 (1983) (in Czech.)Google Scholar
  15. 15.
    Nečas, J.: Les méthodes directes en théorie des equations elliptiques. Academia Prague, Masson Paris 1967Google Scholar
  16. 16.
    Oganesian, L.A., Ruhovec, L.A.: Variational-difference methods for the solution of elliptic problems. Jerevan: Izd. Akad. Nauk ArSSR 1979 (in Russian)Google Scholar
  17. 17.
    Rektorys, K.: Variational methods in mathematics, science and engineering, 2nd ed. Dordrecht-Boston: Reidel 1979Google Scholar
  18. 18.
    Ženíšek, A.: Discretes forms of Friedrichs' inequalities in the finite element method. RAIRO Numer. Anal.15, 265–286 (1981)Google Scholar
  19. 19.
    Ženíšek, A.: How to avoid the use of Green's theorem in the Ciarlet-Raviart theory of variational crimes. M2AN21, 171–191 (1987)Google Scholar
  20. 20.
    Ženíšek, A.: Finite element variational crimes in parabolic-elliptic problems. Part I. Nonlinear schemes. Numer. Math.55, 343–376 (1989)Google Scholar
  21. 21.
    Zlámal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal.10, 229–240 (1973)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Alexander Ženíšek
    • 1
  1. 1.Department of MathematicsTechnical University BrnoBrnoCzechoslovakia

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