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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Quadratic finite element approximation of the Signorini problem
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by Z. Belhachmi and F. Ben Belgacem PDF
Math. Comp. 72 (2003), 83-104 Request permission

Abstract:

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.
References
  • Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
  • M. Benassi and R. E. White, Parallel numerical solution of variational inequalities, SIAM J. Numer. Anal. 31 (1994), no. 3, 813–830. MR 1275115, DOI 10.1137/0731044
  • 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.
  • F. Ben Belgacem. Mixed Finite Element Methods for Signorini’s Problem, submitted.
  • 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.
  • 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. MR 1674556, DOI 10.1142/S0218202599000154
  • 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. MR 1268898
  • 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. MR 448949, DOI 10.1007/BF01404345
  • Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
  • 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).
  • G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques, No. 21, Dunod, Paris, 1972 (French). MR 0464857
  • Richard S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput. 28 (1974), 963–971. MR 0391502, DOI 10.1090/S0025-5718-1974-0391502-8
  • Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
  • P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
  • P. G. Ciarlet and J. L. Lions (eds.), 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. MR 1422502
  • P. Hild. Problèmes de contact unilatéral et maillages incompatibles, Thèse de l’Université Paul Sabatier, Toulouse 3, 1998.
  • 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. MR 961258, DOI 10.1137/1.9781611970845
  • 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. MR 567696
  • 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. MR 1683934
  • 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). MR 0247243
  • 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). MR 1177293, DOI 10.1080/03605309208820864
  • Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
  • Z.-H. Zhong. Finite Element Procedures for Contact-Impact Problems, Oxford University Press, 1993.
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Additional Information
  • Z. Belhachmi
  • Affiliation: Méthodes Mathématiques pour l’Analyse des Systèmes, CNRS-UPRES-A-7035, Université de Metz , ISGMP, Batiment A, Ile du Saulcy, 57045 Metz, France
  • Email: belhach@poncelet.sciences.univ-metz.fr
  • F. Ben Belgacem
  • Affiliation: Mathématiques pour l’Industrie et la Physique, Unité Mixte de Recherche CNRS–UPS–INSAT–UT1 (UMR 5640), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France
  • Email: belgacem@mip.ups-tlse.fr
  • Received by editor(s): April 20, 2000
  • Received by editor(s) in revised form: April 10, 2001
  • Published electronically: December 5, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Math. Comp. 72 (2003), 83-104
  • MSC (2000): Primary 35J85, 73J05
  • DOI: https://doi.org/10.1090/S0025-5718-01-01413-2
  • MathSciNet review: 1933319