The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods

Comptes Rendus Mathematique

Volumn 349, Issue 19-20, 2011, Pages 1041-1046

  Abdulle, Assyr ^a   Vilmart, Gilles ^a  

^a EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 80054010824     PISSN: 1631073X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.crma.2011.09.005     Document Type: Article

References (16)

1. Abdulle A. On a priori error analysis of fully discrete heterogeneous multiscale FEM. SIAM Multiscale Model. Simul. 2005, 4(2):447-459.
2. Abdulle A. The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. GAKUTO Internat. Ser. Math. Sci. Appl. 2009, 31:135-184.
3. A. Abdulle, A priori and a posteriori error analysis for numerical homogenization: a unified framework, Ser. Contemp. Appl. Math. CAM, vol. 16, World Scientific Publishing Co. Pte. Ltd., Singapore, 2011, in press.
4. Abdulle A., Vilmart G. A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems preprint. http://infoscience.epfl.ch/record/152102.
5. Abdulle A., Vilmart G. Fully discrete analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems preprint. http://infoscience.epfl.ch/record/163326.
6. Boccardo L., Murat F. Homogénéisation de problèmes quasi-linéaires. Publ. IRMA, Lille 1981, 3(7):13-51.
7. Ciarlet P.G. Basic Error Estimates for Elliptic Problems. Handb. Numer. Anal. 1991, vol. 2. North-Holland, Amsterdam, pp. 17-351.
8. Ciarlet P.G., Raviart P.A. The combined effect of curved boundaries and numerical integration in isoparametric finite element method. Math. Foundation of the FEM with Applications to PDE 1972, 409-474. Academic Press, New York, NY. A.K. Aziz (Ed.).
9. Douglas J., Dupont T. A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 1975, 29(131):689-696.
10. E W., Ming P., Zhang P. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc. 2005, 18(1):121-156.
11. Feistauer M., Ženíšek A. Finite element solution of nonlinear elliptic problems. Numer. Math. 1987, 50(4):451-475.
12. 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. 1993, 1(4):267-285.
13. Hlaváček I., Křížek M., Malý J. On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type. J. Math. Anal. Appl. 1994, 184(1):168-189.
14. Schatz A.H. An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp. 1974, 28:959-962.
15. Strang G. Variational crimes in the finite element method. Math. Foundation of the FEM with Applications to PDE 1972, 689-710. Academic Press, New York, NY. A.K. Aziz (Ed.).
16. Xu J. Two-grid discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 1996, 33(5):1759-1777.