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Hybrid finite element methods for the Signorini problem. (English) Zbl 1023.74043
Summary: We study three mixed linear finite element methods for numerical simulation of two-dimensional Signorini problem. Applying Falk’s lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them. Two of these finite elements provide optimal results under reasonable regularity assumptions on Signorini solution, and the numerical investigation shows that the third method also provides optimal accuracy.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74M15 Contact in solid mechanics
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##### References:
 [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] F. Ben Belgacem. Discrétisations 3D non conformes par la méthode de décomposition de domaine des éléments avec joints : Analyse mathématique et mise en \oeuvre pour le problème de Poisson. Thèse de l’Université Pierre et Marie Curie, Paris VI. Note technique EDF, ref. HI72/93017 (1993). [3] F. Ben Belgacem, Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods, SIAM J. Numer. Anal. 37 (2000), no. 4, 1198 – 1216. · Zbl 0974.74055 [4] Faker Ben Belgacem, The mortar finite element method with Lagrange multipliers, Numer. Math. 84 (1999), no. 2, 173 – 197. · Zbl 0944.65114 [5] Faker Ben Belgacem and Susanne C. Brenner, Some nonstandard finite element estimates with applications to 3D Poisson and Signorini problems, Electron. Trans. Numer. Anal. 12 (2001), 134 – 148. · 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] Christine Bernardi, Naïma Debit, and Yvon Maday, Coupling finite element and spectral methods: first results, Math. Comp. 54 (1990), no. 189, 21 – 39. · Zbl 0685.65098 [8] C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35 (1998), no. 5, 1893 – 1916. · Zbl 0913.65007 [9] 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 [10] F. Brezzi, L. P. Franca, D. Marini and A. Russo. Stabilization Techniques for Domain Decomposition with Nonmatching Grids, Domain Decomposition Methods in Sciences and Engineering, , Domain Decomposition Press, Bergen, $$1-11$$, $$1998$$, (IX International Conference, Bergen, Norway, June 1996). [11] 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 [12] Franco Brezzi and Donatella Marini, Error estimates for the three-field formulation with bubble stabilization, Math. Comp. 70 (2001), no. 235, 911 – 934. · Zbl 0970.65118 [13] 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 [14] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77 – 84 (English, with Loose French summary). · Zbl 0368.65008 [15] Patrice Coorevits, Patrick Hild, Khalid Lhalouani, and Taoufik Sassi, Mixed finite element methods for unilateral problems: convergence analysis and numerical studies, Math. Comp. 71 (2002), no. 237, 1 – 25. · Zbl 1013.74062 [16] M. Crouzeix and V. Thomée, The stability in \?_{\?} and \?\textonesuperior _{\?} of the \?$$_{2}$$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521 – 532. · Zbl 0637.41034 [17] 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 [18] Richard S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput. 28 (1974), 963 – 971. · Zbl 0297.65061 [19] R. Glowinski, J.-L. Lions, and R. Trémolières, Analyse numérique des inéquations variationnelles. Tome 1, Dunod, Paris, 1976 (French). Théorie générale premiéres applications; Méthodes Mathématiques de l’Informatique, 5. R. Glowinski, J.-L. Lions, and R. Trémolières, Analyse numérique des inéquations variationnelles. Tome 2, Dunod, Paris, 1976 (French). Applications aux phénomènes stationnaires et d’évolution; Méthodes Mathématiques de l’Informatique, 5. [20] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0695.35060 [21] Jaroslav Haslinger and Ivan Hlaváček, Contact between elastic bodies. II. Finite element analysis, Apl. Mat. 26 (1981), no. 4, 263 – 290 (English, with Czech summary). With a loose Russian summary. Jaroslav Haslinger and Ivan Hlaváček, Contact between elastic bodies. III. Dual finite element analysis, Apl. Mat. 26 (1981), no. 5, 321 – 344 (English, with Czech summary). With a loose Russian summary. [22] 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 [23] P. Hild. Problèmes de contact unilatéral et maillages incompatibles, Thèse de l’Université Paul Sabatier, Toulouse $$3$$, $$1998$$. [24] 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 [25] 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 [26] 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 [27] 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). J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968 (French). · Zbl 0212.43801 [28] Hiroyuki Usami, Nonexistence results of entire solutions for superlinear elliptic inequalities, J. Math. Anal. Appl. 164 (1992), no. 1, 59 – 82. · Zbl 0761.35025 [29] P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp. 31 (1977), no. 138, 391 – 413. · Zbl 0364.65082 [30] Padmanabhan Seshaiyer and Manil Suri, Uniform \?\? convergence results for the mortar finite element method, Math. Comp. 69 (2000), no. 230, 521 – 546. · Zbl 0944.65113 [31] 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 [32] J.-M. Thomas. Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes. Thèse (1977), Université Pierre et Marie Curie, PARIS VI. [33] Z.-H. Zhong. Finite Element Procedures for Contact-Impact Problems, Oxford University Press, $$1993$$.
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