A general setting is proposed for the mixed finite element approximations of elliptic differential problems involving a unilateral boundary condition. The treatment covers the Signorini problem as well as the unilateral contact problem with or without friction. Existence, uniqueness for both the continuous and the discrete problem as well as error estimates are established in a general framework. As an application, the approximation of the Signorini problem by the lowest order mixed finite element method of Raviart-Thomas is proved to converge with a quasi-optimal error bound.
Mots clés : variational inequalities, unilateral problems, Signorini problem, contact problems, mixed finite element methods, elliptic PDE
@article{M2AN_2004__38_1_177_0,
author = {Slimane, Leila and Bendali, Abderrahmane and Laborde, Patrick},
title = {Mixed formulations for a class of variational inequalities},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {177--201},
publisher = {EDP-Sciences},
volume = {38},
number = {1},
year = {2004},
doi = {10.1051/m2an:2004009},
mrnumber = {2073936},
zbl = {1100.65059},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004009/}
}
TY - JOUR AU - Slimane, Leila AU - Bendali, Abderrahmane AU - Laborde, Patrick TI - Mixed formulations for a class of variational inequalities JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 177 EP - 201 VL - 38 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004009/ DO - 10.1051/m2an:2004009 LA - en ID - M2AN_2004__38_1_177_0 ER -
%0 Journal Article %A Slimane, Leila %A Bendali, Abderrahmane %A Laborde, Patrick %T Mixed formulations for a class of variational inequalities %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 177-201 %V 38 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2004009/ %R 10.1051/m2an:2004009 %G en %F M2AN_2004__38_1_177_0
Slimane, Leila; Bendali, Abderrahmane; Laborde, Patrick. Mixed formulations for a class of variational inequalities. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 1, pp. 177-201. doi : 10.1051/m2an:2004009. http://www.numdam.org/articles/10.1051/m2an:2004009/
[1] , Sobolev spaces. Academic Press, New York (1975). | MR | Zbl
[2] and, Méthode d'éléments finis avec hybridisation frontière pour les problèmes de contact avec frottement. C.R. Acad. Sciences Paris Série I 334 (2002) 917-922. | Zbl
[3] , and, A mixed formulation for the Signorini problem in incompressible elasticity, theory and finite element approximation. Appl. Numer. Math. (to appear). | MR | Zbl
[4] , Analyse fonctionnelle : Théorie et applications. Masson, Paris (1983). | MR | Zbl
[5] and, Mixed and hybrid finite element methods. Springer-Verlag, Berlin (1991). | MR | Zbl
[6] , and, Error estimates for the finite element solution of variational inequalities, Part II. Numer. Math 31 (1978) 1-16. | Zbl
[7] , Contribution à la prévention de phénomènes de verrouillage numérique. Ph.D. thesis, Université de Pau, France (1997).
[8] and, Numerical approximation of stiff transmission problems by mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 32 (1998) 611-629. | Numdam | Zbl
[9] , The finite element methods for elliptic problems. North-Holland, Amsterdam (1978). | MR | Zbl
[10] ,, and, Mixed finite elemen methods for unilateral problems: convergence analysis and numerical studies. Math. Comp. 71 (2001) 1-25. | Zbl
[11] and, Les inéquations en mécanique et en physique. Dunod, Paris (1972). | MR | Zbl
[12] and, Analyse convexe et problèmes variationnels. Dunod, Paris (1974). | MR | Zbl
[13] , Error estimates for the approximation of a class of variational inequalities. Math. Comp. 28 (1974) 863-971. | Zbl
[14] , Mixed formulation of elliptic variational inequalities and its approximation. Appl. Math. 6 (1981) 462-475. | Zbl
[15] , and, Numerical methods for unilateral problems in solid mechanics. Handb. Numer. Anal., Vol. IV: Finite Element Methods, Part 2 - Numerical Methods for solids, Part 2, P.G. Ciarlet and J.-L. Lions Eds., North-Holland, Amsterdam (1996). | Zbl
[16] , Contact problems with bounded friction, coercive case. Czech. Math. J. 33 (1983) 237-261. | Zbl
[17] and, Contact problems in elasticity: A Study of variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988). | MR | Zbl
[18] and, Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Num. Math. 7 (1999) 23-30. | Zbl
[19] , Quelques méthodes de résolution de problème aux limites non linéaires. Dunod, Paris (1969). | MR | Zbl
[20] , Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 (1969) 510-585. | Zbl
[21] and, Régularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Comm. Partial Differential Equations 17 (1992) 805-826. | Zbl
[22] , Approximation par méthode d'éléments finis de problèmes de transmission raides. Ph.D. thesis, Université de Pau, France (1994).
[23] and, Mixed and Hybrid Methods. Handb. Numer. Anal., Vol. II: Finite Element Methods, Part 1, North-Holland, Amesterdam (1991). | MR | Zbl
[24] , Méthodes mixtes et traitement du verrouillage numérique pour la résolution des inéquations variationnelles. Ph.D. thesis, INSA de Toulouse, France (2001).
[25] , and, Mixed formulations for a class of variational inequalities. C.R. Math. Acad. Sci. Paris 334 (2002) 87-92. | Zbl
[26] and, Dual mixed finite element method for contact problem in elasticity. Math. Num. Sin. 21 (1999). | Zbl
Cité par Sources :
