no. 1
A finite element discretization of the contact between two membranes
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 33-52.

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.

DOI : 10.1051/m2an/2008041
Classification : 65N30, 73K10, 73T05
Mots clés : unilateral contact, variational inequalities, finite elements, a priori and a posteriori analysis 
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     title = {A finite element discretization of the contact between two membranes},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {33--52},
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Belgacem, Faker Ben; Bernardi, Christine; Blouza, Adel; Vohralík, Martin. A finite element discretization of the contact between two membranes. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 33-52. doi : 10.1051/m2an/2008041. http://archive.numdam.org/articles/10.1051/m2an/2008041/

[1] M. Ainsworth, J.T. Oden and C.Y. Lee, Local a posteriori error estimators for variational inequalities. Numer. Methods Partial Differential Equations 9 (1993) 23-33. | MR | Zbl

[2] F. Ali Mehmeti and S. Nicaise, Nonlinear interaction problems. Nonlinear Anal. Theory Methods Appl. 20 (1993) 27-61. | MR | Zbl

[3] C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Collection Mathématiques & Applications 45. Springer-Verlag (2004). | MR | Zbl

[4] H. Brezis and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques. Bull. Soc. Math. France 96 (1968) 153-180. | Numdam | MR | Zbl

[5] F. Brezzi, W.W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, II. Mixed methods. Numer. Math. 31 (1978-1979) 1-16. | MR | Zbl

[6] Z. Chen and R.H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000) 527-548. | MR | Zbl

[7] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, New York, Oxford (1978). | MR | Zbl

[8] P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.-L. Lions Eds., North-Holland, Amsterdam (1991) 17-351. | MR | Zbl

[9] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 R2 (1975) 77-84. | Numdam | MR | Zbl

[10] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod & Gauthier-Villars (1974). | MR | Zbl

[11] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | MR | Zbl

[12] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985). | MR | Zbl

[13] J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Vol. IV, P.G. Ciarlet and J.-L. Lions Eds., North-Holland, Amsterdam (1996) 313-485. | MR | Zbl

[14] P. Hild and S. Nicaise, Residual a posteriori error estimators for contact problems in elasticity. ESAIM: M2AN 41 (2007) 897-923. | Numdam | MR | Zbl

[15] J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure Appl. Math. 20 (1967) 493-519. | MR | Zbl

[16] R.H. Nochetto, K.G. Siebert and A. Veeser, Pointwise a posteriori error control for elliptic obstacle problems. Numer. Math. 95 (2003) 163-195. | MR | Zbl

[17] G. Raugel, Résolution numérique par une méthode d'éléments finis du problème de Dirichlet pour le laplacien dans un polygone. C. R. Acad. Sci. Paris Sér. A-B 286 (1978) A791-A794. | MR | Zbl

[18] L. Slimane, A. Bendali and P. Laborde, Mixed formulations for a class of variational inequalities. ESAIM: M2AN 38 (2004) 177-201. | Numdam | MR | Zbl

[19] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996). | Zbl

[20] B.I. Wohlmuth, An a posteriori error estimator for two body contact problems on non-matching meshes. J. Sci. Computing 33 (2007) 25-45. | MR | Zbl

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