Partial Differential Equations/Numerical Analysis
The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods
Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1041-1046.

A finite element method with numerical quadrature is considered for the solution of a class of second-order quasilinear elliptic problems of nonmonotone type. Optimal a priori error estimates for the H1 and the L2 norms are derived. The uniqueness of the finite element solution is established for a sufficiently fine mesh. Our results permit the analysis of numerical homogenization methods.

On considère des méthodes dʼéléments finis avec intégration numérique par quadrature pour des problèmes elliptiques quasi-linéaires de type non-monotone. Les vitesses de convergence optimales pour les normes H1 et L2 sont démontrées ainsi que lʼunicité de la solution numérique pour un maillage suffisamment fin. Ces résultats permettent lʼanalyse multi-échelles de méthodes dʼhomogénéisation numérique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.09.005
Abdulle, Assyr 1; Vilmart, Gilles 1

1 Section de mathématiques, École polytechnique fédérale de Lausanne, station 8, CH-1015 Lausanne, Switzerland
@article{CRMATH_2011__349_19-20_1041_0,
     author = {Abdulle, Assyr and Vilmart, Gilles},
     title = {The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1041--1046},
     publisher = {Elsevier},
     volume = {349},
     number = {19-20},
     year = {2011},
     doi = {10.1016/j.crma.2011.09.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.09.005/}
}
TY  - JOUR
AU  - Abdulle, Assyr
AU  - Vilmart, Gilles
TI  - The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1041
EP  - 1046
VL  - 349
IS  - 19-20
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.09.005/
DO  - 10.1016/j.crma.2011.09.005
LA  - en
ID  - CRMATH_2011__349_19-20_1041_0
ER  - 
%0 Journal Article
%A Abdulle, Assyr
%A Vilmart, Gilles
%T The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods
%J Comptes Rendus. Mathématique
%D 2011
%P 1041-1046
%V 349
%N 19-20
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.09.005/
%R 10.1016/j.crma.2011.09.005
%G en
%F CRMATH_2011__349_19-20_1041_0
Abdulle, Assyr; Vilmart, Gilles. The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods. Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1041-1046. doi : 10.1016/j.crma.2011.09.005. http://www.numdam.org/articles/10.1016/j.crma.2011.09.005/

[1] Abdulle, A. On a priori error analysis of fully discrete heterogeneous multiscale FEM, SIAM Multiscale Model. Simul., Volume 4 (2005) no. 2, pp. 447-459

[2] Abdulle, A. The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs, GAKUTO Internat. Ser. Math. Sci. Appl., Volume 31 (2009), pp. 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 http://infoscience.epfl.ch/record/152102 (preprint)

[5] Abdulle, A.; Vilmart, G. Fully discrete analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems http://infoscience.epfl.ch/record/163326 (preprint)

[6] Boccardo, L.; Murat, F. Homogénéisation de problèmes quasi-linéaires, Publ. IRMA, Lille, Volume 3 (1981) no. 7, pp. 13-51

[7] Ciarlet, P.G. Basic Error Estimates for Elliptic Problems, Handb. Numer. Anal., vol. 2, North-Holland, Amsterdam, 1991 (pp. 17–351)

[8] Ciarlet, P.G.; Raviart, P.A. The combined effect of curved boundaries and numerical integration in isoparametric finite element method (Aziz, A.K., ed.), Math. Foundation of the FEM with Applications to PDE, Academic Press, New York, NY, 1972, pp. 409-474

[9] Douglas, J. Jr.; Dupont, T. A Galerkin method for a nonlinear Dirichlet problem, Math. Comp., Volume 29 (1975) no. 131, pp. 689-696

[10] E, W.; Ming, P.; Zhang, P. Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., Volume 18 (2005) no. 1, pp. 121-156

[11] Feistauer, M.; Ženíšek, A. Finite element solution of nonlinear elliptic problems, Numer. Math., Volume 50 (1987) no. 4, pp. 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., Volume 1 (1993) no. 4, pp. 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., Volume 184 (1994) no. 1, pp. 168-189

[14] Schatz, A.H. An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comp., Volume 28 (1974), pp. 959-962

[15] Strang, G. Variational crimes in the finite element method (Aziz, A.K., ed.), Math. Foundation of the FEM with Applications to PDE, Academic Press, New York, NY, 1972, pp. 689-710

[16] Xu, J. Two-grid discretization techniques for linear and nonlinear PDE, SIAM J. Numer. Anal., Volume 33 (1996) no. 5, pp. 1759-1777

Cited by Sources: