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

Numerische Mathematik

Volumn 121, Issue 3, 2012, Pages 397-431

  Abdulle, Assyr ^a   Vilmart, Gilles ^a,b  

^a EPFL   (Switzerland)

^b UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84862249018     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-011-0438-4     Document Type: Article

References (31)

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).
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).
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).
4. Abdulle A., Vilmart, G.: Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems. preprint, http://infoscience. epfl. ch/record/163326 (submitted for publication).
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).
6. André N., Chipot M.: Uniqueness and nonuniqueness for the approximation of quasilinear elliptic equations. SIAM J. Numer. Anal. 33(5), 1981-1994 (1996).
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).
8. Bear J., Bachmat Y.: Introduction to modelling of transport phenomena in porous media. Kluwer Academic, Dordrecht (1991).
9. Brenner, S., Scott, R.: The mathematical theory of finite element methods, 3rd edn. Texts in Applied Mathematics, 15. Springer, New York (2008).
10. Chipot, M.: Elliptic equations: an introductory course. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Basel (2009).
11. Ciarlet P. G.: Basic error estimates for elliptic problems. Handb. Numer. Anal. 2, 17-351 (1991).
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).
13. Douglas J. Jr, Dupont T.: A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29(131), 689-696 (1975).
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).
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).
16. Engquist B., Souganidis P. E.: Asymptotic and numerical homogenization. Acta Numer. 17, 147-190 (2008).
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).
18. Feistauer M., Ženíšek A.: Finite element solution of nonlinear elliptic problems. Numer. Math. 50(4), 451-475 (1987).
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).
20. Gilbarg D., Trudinger N.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin (2001).
21. Hildebrandt S., Wienholtz E.: Constructive proofs of representation theorems in separable Hilbert space. Comm. Pure Appl. Math. 17, 369-373 (1964).
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).
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).
24. Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13(3), 115-162 (1959).
25. ∞-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).
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).
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).
28. Schatz A. H.: An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp. 28, 959-962 (1974).
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).
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).
31. Warrick A. W.: Time-dependent linearized infiltration: III. Strip and disc sources. Soil. Sci. Soc. Am. J. 40, 639-643 (1976).