Numerische Mathematik

, Volume 121, Issue 3, pp 397–431 | Cite as

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

  • Assyr Abdulle
  • Gilles Vilmart


The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L 2 and the H 1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.

Mathematics Subject Classification (2000)

65N30 65M60 65D30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdulle A.: The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. GAKUTO Int. Ser. Math. Sci. Appl. 31, 135–184 (2009)Google Scholar
  2. 2.
    Abdulle, A.: A priori and a posteriori analysis for numerical homogenization: a unified framework. Ser. Contemp. Appl. Math. CAM, 16, World Scientific Publishing, Singapore, pp. 280–305 (2011)Google Scholar
  3. 3.
    Abdulle A., Vilmart G.: The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods. C. R. Acad. Sci. Paris, Ser. I 349, 1041–1046 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Abdulle A., Vilmart, G.: Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems. preprint, (submitted for publication)
  5. 5.
    Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), (Teubner-Texte Math., vol. 133) Teubner, Stuttgart, pp. 9126 (1993)Google Scholar
  6. 6.
    André N., Chipot M.: Uniqueness and nonuniqueness for the approximation of quasilinear elliptic equations. SIAM J. Numer. Anal. 33(5), 1981–1994 (1996)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Baker G.A., Dougalis V.A.: The effect of quadrature errors on finite element approximations for second order hyperbolic equations. SIAM J. Numer. Anal. 13, 577–598 (1976)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bear J., Bachmat Y.: Introduction to modelling of transport phenomena in porous media. Kluwer Academic, Dordrecht (1991)Google Scholar
  9. 9.
    Brenner, S., Scott, R.: The mathematical theory of finite element methods, 3rd edn. Texts in Applied Mathematics, 15. Springer, New York (2008)Google Scholar
  10. 10.
    Chipot, M.: Elliptic equations: an introductory course. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Basel (2009)Google Scholar
  11. 11.
    Ciarlet P.G.: Basic error estimates for elliptic problems. Handb. Numer. Anal. 2, 17–351 (1991)CrossRefGoogle Scholar
  12. 12.
    Ciarlet P.G., Raviart P.A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element method. In: Aziz, A.K. (ed) Math. Foundation of the FEM with Applications to PDE, pp. 409–474. Academic Press, New York (1972)Google Scholar
  13. 13.
    Douglas J. Jr, Dupont T.: A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29(131), 689–696 (1975)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Douglas J. Jr, Dupont T., Serrin J.: Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form. Arch. Ration. Mech. Anal. 42, 157–168 (1971)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    E, W., Engquist B., Li X., Ren W., Vanden-Eijnden E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)MathSciNetMATHGoogle Scholar
  16. 16.
    Engquist B., Souganidis P.E.: Asymptotic and numerical homogenization. Acta Numer. 17, 147–190 (2008)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Feistauer M., Křížek M., Sobotíková V.: An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type. East-West J. Numer. Math. 1(4), 267–285 (1993)MathSciNetMATHGoogle Scholar
  18. 18.
    Feistauer M., Ženíšek A.: Finite element solution of nonlinear elliptic problems. Numer. Math. 50(4), 451–475 (1987)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Geers M.G.D., Kouznetsova A.G., Brekelmans W.A.M.: Multi-scale computational homogenization: Trends and challenges. J. Comput. Appl. Math. 234(7), 2175–2182 (2010)MATHCrossRefGoogle Scholar
  20. 20.
    Gilbarg D., Trudinger N.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin (2001)Google Scholar
  21. 21.
    Hildebrandt S., Wienholtz E.: Constructive proofs of representation theorems in separable Hilbert space. Comm. Pure Appl. Math. 17, 369–373 (1964)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Hlaváček I., Křížek M., Malý J.: On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type. J. Math. Anal. Appl. 184(1), 168–189 (1994)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Korotov S., Křížek M.: Finite element analysis of variational crimes for a quasilinear elliptic problem in 3D. Numer. Math. 84(4), 549–576 (2000)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13(3), 115–162 (1959)MathSciNetGoogle Scholar
  25. 25.
    Nitsche, J.A.: On L -convergence of finite element approximations to the solution of a nonlinear boundary value problem. Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976), Academic Press, London, pp. 317–325 (1977)Google Scholar
  26. 26.
    Poussin J., Rappaz J.: Consistency, stability, a priori and a posteriori errors for Petrov–Galerkin methods applied to nonlinear problems. Numer. Math. 69(2), 213–231 (1994)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Raviart, P.A.: The use of numerical integration in finite element methods for solving parabolic equations. In: Miller, J.J.H. (ed.) Topics in Numerical Analysis, pp. 233–264. Academic Press, London-New York (1973)Google Scholar
  28. 28.
    Schatz A.H.: An observation concerning Ritz–Galerkin methods with indefinite bilinear forms. Math. Comp. 28, 959–962 (1974)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Schatz A.H., Wang J.P.: Some new error estimates for Ritz–Galerkin methods with minimal regularity assumptions. Math. Comp. 65(213), 19–27 (1996)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Strang G.: Variational crimes in the finite element method. In: Aziz, A.K. (ed) Math. Foundation of the FEM with Applications to PDE, pp. 689–710. Academic Press, New York (1972)Google Scholar
  31. 31.
    Warrick A.W.: Time-dependent linearized infiltration: III. Strip and disc sources. Soil. Sci. Soc. Am. J. 40, 639–643 (1976)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Section de MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.École Normale Supérieure de CachanBruzFrance

Personalised recommendations