Hybrid finite element methods for the Signorini problem

Mathematics of Computation

Volumn 72, Issue 243, 2003, Pages 1117-1145

  Ben Belgacem, F ^a   Renard, Y ^a  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Bubble stabilization; Finite element approximation; Mixed formulation; Variational inequalities

Indexed keywords

EID: 0037972493     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01490-X     Document Type: Article

References (33)

1. R. A. Adams. Sobolev Spaces, Academic Press, 1975, MR 56:9247
2. F. Ben Belgacem. Discrétisations 3D non conformes par la méthode de décomposition de domaine des éléments avec joints : Analyse mathématique et mise en æuvre 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 Method, SIAM J. Numer. Anal., 37: 1198-1216, 2000. MR 2002c:74066
4. F. Ben Belgacem. The Mortar Finite Element Method with Lagrange Multipliers, Numer. Math. 84, 173-197, 1999. MR 2000k:65204
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, 200. MR 2002c:65187
6. F. Ben Belgacem, P. Hild and P. Laborde. Extension of the Mortar Finite Element to a Variational Inequality Modeling Unilateral Contact, Math. Models and Methods in Appl. Sci., 9: 287-303, 1999. MR 99m:73066
7. C. Bernardi, N. Débit and Y. Maday. Coupling Finite Element and Spectral Methods: First Results, Math. Comp., 54, 21-39, 1990. MR 90f:65201
8. C. Bernardi and V. Girault. A Local Regularisation Operator for Triangular and Quadrangular Finite Elements, SIAM J. Numer. Anal., 35, 1893-1916, 1998. MR 99g:65107
9. C. Bernardi, Y. Maday et A. T. Patera. A New Non Conforming Approach to Domain Decomposition: The Mortar Element Method. Collège de France Seminar, Pitman, H. Brezis, J.-L. Lions, 1990. MR 95a:65201
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, (P. Bjrstad, M. Espedal, D. Keyes, Eds.), Domain Decomposition Press, Bergen, 1-11, 1998, (IX International Conference, Bergen, Norway, June 1996).
11. F. Brezzi, W. W. Hager and P. A. Raviart. Error estimates for the finite element solution of variational inequalities, Numer. Math., 28, 431-443, 1977. MR 56:7254
12. F. Brezzi and D. Marini. Error Estimates for the Three-field Formulation with Bubble Stabilization, Math. Comp., 70, 911-934, 2001. MR 2002b:65159
13. P.-G. Ciarlet. The Finite Element Method for Elliptic Problems, North Holland, 1978. MR 58:25001
14. P. Clément. Approximation by Finite Elements Functions Using Local Regularization, RAIRO, Anal. Numer., 9, 77-84, 1975. MR 53:4569
15. P. Coorevits, P. Hild, K. Lhalouani and T. Sassi. Mixed Finite Element Method for Unilateral Problem: Convergence Analysis and Numerical Studies, Math. Comp., 71, 1-25, 2002. MR 2002g:74043
16. 2-Projection on Finite Element Function Spaces, Math. Comp., 48 521-532, 1987. MR 88f:41016
17. G. Duvaut and J.-L. Lions. Les inéquations en mécanique et en physique, Dunod, 1972. MR 57:4778
18. R. S. Falk. Error Estimates for the Approximation of a Class of Variational Inequalities, Math. Comp., 28: 963-971, 1974. MR 52:12323
19. R. Glowinski, J.-L. Lions and R. Trémolières. Analyse numériques des inéquations variation-nelles, Tome 1, Dunod, 1976. MR 58:31697a
20. P. Grisvard. Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman, 1985. MR 86m:35044
21. J. Haslinger and I. Hlaváček, Contact between Elastic Bodies -2, Finite Element Analysis, Aplikace Matematiky, 26, 263-290, 1981. MR 83c:73052a
22. J. Haslinger, I, Hlaváček and J. Nečas. Numerical Methods for Unilateral Problems in Solid Mechanics, in Handbook of Numerical Analysis, Vol. IV, Part 2, Eds. P.G. Ciarlet and J.-L. Lions, North Holland, 1996. MR 98b: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, 1988. MR 89j:73097
25. D. Kinderlehrer and G. Stamppachia. An Introduction to Variational Inequalities and their Applications, Academic Press, 1980. MR 81g:49013
26. K. Lhalouani and T. Sassi. Nonconforming Mixed Variational Formulation and Domain Decomposition for Unilateral Problems, East West J. Numer. Math., 7: 23-30, 1999. MR 2000b:65217
27. J.-L. Lions and E. Magenes. Problèmes aux limites non homogènes, Dunod, 1968. MR 40:512; MR 40:513
28. M. Moussaoui and K. Khodja. Régularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan, Commun. Part. Diff. Eq., 17: 805-826, 1992. MR 93b:35055
29. P.-A. Raviart and J.-M. Thomas. Primal Hybrid Finite Element Methods for 2nd Order Elliptic Equations, Math. Comp., 31, 391-396, 1977. MR 55:4747
30. P. Seshaiyer and M. Suri. Uniform h-p Convergence Results for the Mortar Finite Element Method, Math. Comp., 69, 521-546, 2000. MR 2000i:65192
31. G. Strang and G. J. Fix. An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. MR 56:1747
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.