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Mixed formulations for a class of variational inequalities

Published online by Cambridge University Press:  15 February 2004

Leila Slimane ,
Abderrahmane Bendali  and
Patrick Laborde
Leila Slimane
Affiliation:
Université de Moncton, Campus de Shippagen, 218, boulevard J.-D. Gauthier, Shippagen, Nouveau Brunswick, E831P6, Canada.
Abderrahmane Bendali
Affiliation:
Laboratoire MIP, UMR-CNRS 5640, INSA de Toulouse, 135 Avenue de Rangueil, 31077, Toulouse Cedex 4, France. bendali@gmm.insa-tlse.fr.
Patrick Laborde
Affiliation:
Laboratoire MIP, UMR-CNRS 5640, Université Toulouse 3, 118 Route de Narbonne, 31062 Toulouse Cedex 4, France. laborde@mip.ups-tlse.fr.
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Abstract

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.

Keywords

Variational inequalitiesunilateral problemsSignorini problemcontact problemsmixed finite element methodselliptic PDE.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 1 , January 2004 , pp. 177 - 201
Copyright
© EDP Sciences, SMAI, 2004

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References

D.A. Adams, Sobolev spaces. Academic Press, New York (1975).
Baillet, L. and Sassi, T., 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. CrossRef
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).
H. Brezis, Analyse fonctionnelle : Théorie et applications. Masson, Paris (1983).
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, Berlin (1991).
Brezzi, F., Hager, W. and Raviart, P.A., Error estimates for the finite element solution of variational inequalities, Part II. Numer. Math 31 (1978) 1–16. CrossRef
D. Capatina-Papaghiuc, Contribution à la prévention de phénomènes de verrouillage numérique. Ph.D. thesis, Université de Pau, France (1997).
Capatina-Papaghiuc, D. and Raynaud, N., Numerical approximation of stiff transmission problems by mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 32 (1998) 611–629. CrossRef
P.G. Ciarlet, The finite element methods for elliptic problems. North-Holland, Amsterdam (1978).
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.
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).
I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Paris (1974).
Falk, R.C., Error estimates for the approximation of a class of variational inequalities. Math. Comp. 28 (1974) 863–971. CrossRef
Haslinger, J., Mixed formulation of elliptic variational inequalities and its approximation. Appl. Math. 6 (1981) 462–475.
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).
Jarušek, J., Contact problems with bounded friction, coercive case. Czech. Math. J. 33 (1983) 237–261.
N. Kikuchi and J.T. Oden, Contact problems in elasticity: A Study of variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988).
Lhalouani, K. and Sassi, T., Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Num. Math. 7 (1999) 23–30.
J.-L. Lions, Quelques méthodes de résolution de problème aux limites non linéaires. Dunod, Paris (1969).
Mosco, U., Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 (1969) 510–585. CrossRef
Moussaoui, M. and Khodja, K., Régularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Comm. Partial Differential Equations 17 (1992) 805–826. CrossRef
N. Raynaud, Approximation par méthode d'éléments finis de problèmes de transmission raides. Ph.D. thesis, Université de Pau, France (1994).
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).
L. Slimane, Méthodes mixtes et traitement du verrouillage numérique pour la résolution des inéquations variationnelles. Ph.D. thesis, INSA de Toulouse, France (2001).
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.
L. Wang and G. Wang, Dual mixed finite element method for contact problem in elasticity. Math. Num. Sin. 21 (1999).