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Finite element solution of nonlinear elliptic problems

  • Convergence of the SSOR Method for Nonlinear Systems of Simultaneous Equations
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Summary

The study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed Dirichlet-Neumann boundary conditions is presented. In the discretization variational crimes are commited (approximation of the given domain by a polygonal one, numerical integration). With the assumption that the corresponding operator is strongly monotone and Lipschitz-continuous and that the exact solutionu∈H 1(Ω), the convergence of the method is proved; under the additional assumptionu∈H 2(Ω), the rate of convergenceO(h) is derived without the use of Green's theorem.

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References

  1. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland 1978

    Google Scholar 

  2. Ciarlet, P.G., Raviart, P.A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element method. In: The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, ed.), pp. 409–474, New York: Academic Press 1972

    Google Scholar 

  3. Ciarlet, P.G., Schultz, M.H., Varga, R.S.: Numerical methods of higher order accuracy for nonlinear boundary value problems. V. Monotone operator theory. Numer. Math.13, 51–77 (1969)

    Google Scholar 

  4. Feistauer, M.: On irrotational flows through cascades of profiles in a layer of variable thickness, Apl. Mat.29, 423–458 (1984)

    Google Scholar 

  5. Feistauer, M.: Mathematical and numerical study of nonlinear problems in fluid mechanics. In: Proc. Conf. Equadiff 6, Brno 1985. (J. Vosmanský, M. Zlámal, eds.), pp. 3–16 Berlin, Heidelberg: Springer 1986

    Google Scholar 

  6. Feistauer, M.: On the finite element approximation of a cascade flow problem. Numer. Math. (To appear)

  7. Feistauer, M., Felcman, J., Vlášek, Z.: Finite element solution of flows through cascades of profiles in a layer of variable thickness. Apl. Mat.31, 309–339 (1986)

    Google Scholar 

  8. Frehse, J., Rannacher, R.: Optimal uniform convergence for the finite element approximation of a quasilinear elliptic boundary value problem. In: Proc. Symp.: Formulations and Computational Algorithms in Finite Element Analysis. M.I.T. 793–812 (1976)

  9. Frehse, J., Rannacher, R.: AsymptoticL ∞-error estimate for linear finite element approximations of quasilinear boundary value problems. SIAM J. Numer. Anal.15, 418–431 (1978)

    Google Scholar 

  10. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. TATA Inst. of Fund. Research Bombay, Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  11. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Series in Comput. Physics, Berlin, Heidelberg, New York, Tokyo: Springer 1984

    Google Scholar 

  12. Glowinski, R., Marrocco, A.: Analyse numérique du champ magnétique d'un alternateur par éléments finis et surrelaxation punctuelle non linéaire. Comput. Methods Appl. Mech. Eng.3, 55–85 (1974)

    Google Scholar 

  13. Glowinski, R., Marrocco, A.: Sur l'approximation par éléments finis d'ordre un, et la résolution par pénalisation-dualité, d'une classe des problémes de Dirichlet non lináires. RAIRO Anal. Numér.9, R-2, 41–76 (1975)

    Google Scholar 

  14. Melkes, F.: The finite element method for nonlinear problems. Apl. Mat.15, 177–189 (1970)

    Google Scholar 

  15. Nečas, J.: Les Méthodes Directes en Théorie des Equations Elliptiques. Academia, Prague, Masson Paris 1967

    Google Scholar 

  16. Nečas, J.: Introduction to the Theory of Nonlinear Elliptic Equations. Teubner-Texte zur Mathematik, Band52, Leipzig 1983

  17. Noor, M.A., Whiteman, J.R.: Error bounds for finite element solution of mildly nonlinear elliptic boundary value problems. Numer. Math.26, 107–116 (1976)

    Google Scholar 

  18. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press 1970

    Google Scholar 

  19. Strang, G.: Variational crimes in the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. (A.K. Aziz, ed.), pp. 689–710. New York: Academic Press 1972

    Google Scholar 

  20. Strang, G., Fix, G.: An Analysis of the Finite Element Method. Englewood Cliffs: Prentice-Hall 1973

    Google Scholar 

  21. Ženíŝek, A.: Curved triangular finiteC m-elements. Apl. Mat.,23, 346–377 (1978)

    Google Scholar 

  22. Ženíŝek, A.: Nonhomogeneous boundary conditions and curved triangular finite elements. Apl. Mat.26, 121–141 (1981)

    Google Scholar 

  23. Ženíŝek, A.: Discrete forms of Friedrichs' inequalities in the finite element method. RAIRO Numer. Anal.15, 265–286 (1981)

    Google Scholar 

  24. Ženíŝek, A.: How to avoid the use of Green's theorem in the Ciarlet's and Raviart's theory of variational crimes. M2AN (to appear)

  25. Zlámal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal.10, 229–240 (1973)

    Google Scholar 

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Authors and Affiliations

  1. Charles University Prague, Sokolovská 83, CS-18600, Praha 8, Czechoslovakia

    Miloslav Feistauer

  2. Technical University Brno, Obránc⫲ míru 21, CS-60200, Brno, Czechoslovakia

    Alexander Ženíšek

Authors
  1. Miloslav Feistauer
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  2. Alexander Ženíšek
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Feistauer, M., Ženíšek, A. Finite element solution of nonlinear elliptic problems. Numer. Math. 50, 451–475 (1986). https://doi.org/10.1007/BF01396664

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