Mixed formulations for a class of variational inequalities
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 1, pp. 177-201.

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.

DOI : https://doi.org/10.1051/m2an:2004009
Classification : 35J85,  76M30
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: Mathematical Modelling and Numerical Analysis - 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},
zbl = {1100.65059},
mrnumber = {2073936},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004009/}
}
Slimane, Leila; Bendali, Abderrahmane; Laborde, Patrick. Mixed formulations for a class of variational inequalities. ESAIM: Mathematical Modelling and Numerical Analysis - 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] D.A. Adams, Sobolev spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[2] L. Baillet and T. Sassi, 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 1073.74047

[3] F. Ben Belgacem, Y. Renard and L. Slimane, A mixed formulation for the Signorini problem in incompressible elasticity, theory and finite element approximation. Appl. Numer. Math. (to appear). | MR 2134092 | Zbl 1086.74037

[4] H. Brezis, Analyse fonctionnelle : Théorie et applications. Masson, Paris (1983). | MR 697382 | Zbl 0511.46001

[5] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, Berlin (1991). | MR 1115205 | Zbl 0788.73002

[6] F. Brezzi, W. Hager and P.A. Raviart, Error estimates for the finite element solution of variational inequalities, Part II. Numer. Math 31 (1978) 1-16. | Zbl 0427.65077

[7] D. Capatina-Papaghiuc, Contribution à la prévention de phénomènes de verrouillage numérique. Ph.D. thesis, Université de Pau, France (1997).

[8] D. Capatina-Papaghiuc and N. Raynaud, Numerical approximation of stiff transmission problems by mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 32 (1998) 611-629. | Numdam | Zbl 0907.73054

[9] P.G. Ciarlet, The finite element methods for elliptic problems. North-Holland, Amsterdam (1978). | MR 1115235 | Zbl 0999.65129

[10] P. Coorevits, P. Hild, K. Lhalouani and T. Sassi, Mixed finite elemen methods for unilateral problems: convergence analysis and numerical studies. Math. Comp. 71 (2001) 1-25. | Zbl 1013.74062

[11] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). | MR 464857 | Zbl 0298.73001

[12] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Paris (1974). | MR 463993 | Zbl 0281.49001

[13] R.C. Falk, Error estimates for the approximation of a class of variational inequalities. Math. Comp. 28 (1974) 863-971. | Zbl 0297.65061

[14] J. Haslinger, Mixed formulation of elliptic variational inequalities and its approximation. Appl. Math. 6 (1981) 462-475. | Zbl 0483.49003

[15] J. Haslinger, I. Hlaváček and J. Nečas, 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 0873.73079

[16] J. Jarušek, Contact problems with bounded friction, coercive case. Czech. Math. J. 33 (1983) 237-261. | Zbl 0519.73095

[17] N. Kikuchi and J.T. Oden, Contact problems in elasticity: A Study of variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988). | MR 961258 | Zbl 0685.73002

[18] K. Lhalouani and T. Sassi, Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Num. Math. 7 (1999) 23-30. | Zbl 0923.73061

[19] J.-L. Lions, Quelques méthodes de résolution de problème aux limites non linéaires. Dunod, Paris (1969). | MR 259693 | Zbl 0189.40603

[20] U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 (1969) 510-585. | Zbl 0192.49101

[21] M. Moussaoui and K. Khodja, Régularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Comm. Partial Differential Equations 17 (1992) 805-826. | Zbl 0806.35049

[22] N. Raynaud, Approximation par méthode d'éléments finis de problèmes de transmission raides. Ph.D. thesis, Université de Pau, France (1994).

[23] J.E. Robert and J.-M. Thomas, Mixed and Hybrid Methods. Handb. Numer. Anal., Vol. II: Finite Element Methods, Part 1, North-Holland, Amesterdam (1991). | MR 1115239 | Zbl 0875.65090

[24] L. Slimane, Méthodes mixtes et traitement du verrouillage numérique pour la résolution des inéquations variationnelles. Ph.D. thesis, INSA de Toulouse, France (2001).

[25] L. Slimane, A. Bendali and P. Laborde, Mixed formulations for a class of variational inequalities. C.R. Math. Acad. Sci. Paris 334 (2002) 87-92. | Zbl 0998.65063

[26] L. Wang and G. Wang, Dual mixed finite element method for contact problem in elasticity. Math. Num. Sin. 21 (1999). | Zbl 0967.74070