[Lʼeffet de lʼintégration numérique sur la méthode des éléments finis pour des problèmes non-monotones elliptiques, avec application aux méthodes numériques dʼhomogénéisation]
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 et sont démontrées ainsi que lʼunicité de la solution numérique pour un maillage suffisamment fin. Ces résultats permettent lʼanalyse multi-échelles de méthodes dʼhomogénéisation numérique.
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 and the 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.
Accepté le :
Publié le :
@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, Tome 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] On a priori error analysis of fully discrete heterogeneous multiscale FEM, SIAM Multiscale Model. Simul., Volume 4 (2005) no. 2, pp. 447-459
[2] 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] A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems http://infoscience.epfl.ch/record/152102 (preprint)
[5] Fully discrete analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems http://infoscience.epfl.ch/record/163326 (preprint)
[6] Homogénéisation de problèmes quasi-linéaires, Publ. IRMA, Lille, Volume 3 (1981) no. 7, pp. 13-51
[7] Basic Error Estimates for Elliptic Problems, Handb. Numer. Anal., vol. 2, North-Holland, Amsterdam, 1991 (pp. 17–351)
[8] 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] A Galerkin method for a nonlinear Dirichlet problem, Math. Comp., Volume 29 (1975) no. 131, pp. 689-696
[10] Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., Volume 18 (2005) no. 1, pp. 121-156
[11] Finite element solution of nonlinear elliptic problems, Numer. Math., Volume 50 (1987) no. 4, pp. 451-475
[12] 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] 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] An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comp., Volume 28 (1974), pp. 959-962
[15] 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] Two-grid discretization techniques for linear and nonlinear PDE, SIAM J. Numer. Anal., Volume 33 (1996) no. 5, pp. 1759-1777
Cité par Sources :
