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Quadratic finite element approximation of the Signorini problem. (English) Zbl 1112.74446
Summary: Applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved with the modeling of the nonpenetration Signorini condition on the discrete solution along the contact region. In this work we describe two nonconforming quadratic finite element approximations of the Poisson-Signorini problem, responding to the crucial practical concern of easy implementation, and we present the numerical analysis of their efficiency. By means of Falk’s Lemma we prove optimal and quasi-optimal convergence rates according to the regularity of the exact solution.

MSC:
74M15 Contact in solid mechanics
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
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[1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[2] M. Benassi and R. E. White, Parallel numerical solution of variational inequalities, SIAM J. Numer. Anal. 31 (1994), no. 3, 813 – 830. · Zbl 0810.65063
[3] F. Ben Belgacem. Numerical Simulation of some Variational Inequalities Arisen from Unilateral Contact Problems by the Finite Element Method, SIAM J. Numer. Anal., 37: 1198-1216, 2000. CMP 2000:12 · Zbl 0974.74055
[4] F. Ben Belgacem. Mixed Finite Element Methods for Signorini’s Problem, submitted. · Zbl 0959.65126
[5] F. Ben Belgacem ans S. C. Brenner. Some Nonstandard Finite Element Estimates with Applications to 3D Poisson and Signorini Problems, Electronic Transactions in Numerical Analysis, 12:134-148, 2001. · Zbl 0981.65131
[6] Faker Ben Belgacem, Patrick Hild, and Patrick Laborde, Extension of the mortar finite element method to a variational inequality modeling unilateral contact, Math. Models Methods Appl. Sci. 9 (1999), no. 2, 287 – 303. · Zbl 0940.74056
[7] C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XI (Paris, 1989 – 1991) Pitman Res. Notes Math. Ser., vol. 299, Longman Sci. Tech., Harlow, 1994, pp. 13 – 51. · Zbl 0797.65094
[8] Franco Brezzi, William W. Hager, and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math. 28 (1977), no. 4, 431 – 443. · Zbl 0369.65030
[9] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[10] P. Coorevits, P. Hild, K. Lhalouani and T. Sassi. Mixed Finite Element Method for Unilateral Problems: Convergence Analysis and Numerical Studies, Math. of Comp., posted on May 21, 2001, PII: S0025-5718(01)01318-7 (to appear in print). · Zbl 1013.74062
[11] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 (French). Travaux et Recherches Mathématiques, No. 21. · Zbl 0298.73001
[12] Richard S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput. 28 (1974), 963 – 971. · Zbl 0297.65061
[13] R. Glowinski, J.-L. Lions and R. Trémolières. Analyse numériques des inéquations variationnelles, Tome 1, Dunod, 1976.
[14] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0695.35060
[15] J. Haslinger and I. Hlavácek. Contact between Elastic Bodies -2.Finite Element Analysis, Aplikace Matematiky, 26: 263-290, 1981.
[16] P. G. Ciarlet and J. L. Lions , Handbook of numerical analysis. Vol. IV, Handbook of Numerical Analysis, IV, North-Holland, Amsterdam, 1996. Finite element methods. Part 2. Numerical methods for solids. Part 2. · Zbl 0864.65001
[17] P. Hild. Problèmes de contact unilatéral et maillages incompatibles, Thèse de l’Université Paul Sabatier, Toulouse 3, 1998.
[18] N. Kikuchi and J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. · Zbl 0685.73002
[19] David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. · Zbl 0457.35001
[20] K. Lhalouani and T. Sassi, Nonconforming mixed variational formulation and domain decomposition for unilateral problems, East-West J. Numer. Math. 7 (1999), no. 1, 23 – 30. · Zbl 0923.73061
[21] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). · Zbl 0212.43801
[22] Mohand Moussaoui and Khadidja Khodja, Régularité des solutions d’un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan, Comm. Partial Differential Equations 17 (1992), no. 5-6, 805 – 826 (French, with English and French summaries). · Zbl 0806.35049
[23] Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. · Zbl 0356.65096
[24] Z.-H. Zhong. Finite Element Procedures for Contact-Impact Problems, Oxford University Press, 1993.
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