This paper is concerned with the unilateral contact problem in linear elasticity. We define two a posteriori error estimators of residual type to evaluate the accuracy of the mixed finite element approximation of the contact problem. Upper and lower bounds of the discretization error are proved for both estimators and several computations are performed to illustrate the theoretical results.
Mots clés : mixed finite element method, a posteriori error estimates, residuals, unilateral contact
@article{M2AN_2007__41_5_897_0,
author = {Hild, Patrick and Nicaise, Serge},
title = {Residual a posteriori error estimators for contact problems in elasticity},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {897--923},
publisher = {EDP-Sciences},
volume = {41},
number = {5},
year = {2007},
doi = {10.1051/m2an:2007045},
mrnumber = {2363888},
zbl = {1140.74024},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007045/}
}
TY - JOUR AU - Hild, Patrick AU - Nicaise, Serge TI - Residual a posteriori error estimators for contact problems in elasticity JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 897 EP - 923 VL - 41 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007045/ DO - 10.1051/m2an:2007045 LA - en ID - M2AN_2007__41_5_897_0 ER -
%0 Journal Article %A Hild, Patrick %A Nicaise, Serge %T Residual a posteriori error estimators for contact problems in elasticity %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 897-923 %V 41 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007045/ %R 10.1051/m2an:2007045 %G en %F M2AN_2007__41_5_897_0
Hild, Patrick; Nicaise, Serge. Residual a posteriori error estimators for contact problems in elasticity. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 5, pp. 897-923. doi : 10.1051/m2an:2007045. http://www.numdam.org/articles/10.1051/m2an:2007045/
[1] , Sobolev spaces. Academic Press (1975). | MR | Zbl
[2] and, Theoretical numerical analysis: a functional analysis framework, in Texts in Applied Mathematics 39, Springer, New-York (2001); (second edition 2005). | MR | Zbl
[3] , Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method. SIAM J. Numer. Anal. 37 (2000) 1198-1216. | Zbl
[4] and, Hybrid finite element methods for the Signorini problem. Math. Comp. 72 (2003) 1117-1145. | Zbl
[5] , and, A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind. Appl. Numer. Math. 52 (2005) 13-38. | Zbl
[6] , A posteriori error estimators for obstacle problems - another look. Numer. Math. 101 (2005) 415-421. | Zbl
[7] , and, Adaptive finite elements for elastic bodies in contact. SIAM J. Sci. Comput. 20 (1999) 1605-1626. | Zbl
[8] and, Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000) 527-548. | Zbl
[9] , The finite element method for elliptic problems, in Handbook of Numerical Analysis, Volume II, Part 1, P.G. Ciarlet and J.-L. Lions Eds., North Holland (1991) 17-352. | Zbl
[10] , and, A posteriori error estimation for unilateral contact with matching and nonmatching meshes. Comput. Methods Appl. Mech. Engrg. 186 (2000) 65-83. | Zbl
[11] , and, A posteriori error control of finite element approximations for Coulomb's frictional contact. SIAM J. Sci. Comput. 23 (2001) 976-999. | Zbl
[12] ,, and, Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. Math. Comp. 71 (2002) 1-25. | Zbl
[13] and, Les inéquations en mécanique et en physique. Dunod (1972). | MR | Zbl
[14] and, A residual-based error estimator for BEM-discretizations of contact problems. Numer. Math. 95 (2003) 253-282. | Zbl
[15] , Problemi elastici con vincoli unilaterali il problema di Signorini con ambigue condizioni al contorno. Mem. Accad. Naz. Lincei. 8 (1964) 91-140. | Zbl
[16] , Existence theorems in elasticity, in Handbuch der Physik, Band VIa/2, Springer (1972) 347-389.
[17] , Lectures on numerical methods for nonlinear variational problems, in Lectures on Mathematics and Physics 65, Notes by M. G. Vijayasundaram and M. Adimurthi, Tata Institute of Fundamental Research, Bombay; Springer-Verlag, Berlin-New York (1980). | MR | Zbl
[18] and, Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society (2002). | MR | Zbl
[19] , and, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Volume IV, Part 2, P.G. Ciarlet and J.-L. Lions Eds., North Holland (1996) 313-485. | Zbl
[20] , A priori error analysis of a sign preserving mixed finite element method for contact problems. Preprint 2006/33 of the Laboratoire de Mathématiques de Besançon, submitted.
[21] and, A posteriori error estimations of residual type for Signorini's problem. Numer. Math. 101 (2005) 523-549. | Zbl
[22] and, Convex analysis and minimization algorithms I. Springer (1993). | MR
[23] and, An optimal error estimate for nonlinear contact problems. SIAM J. Numer. Anal. 43 (2005) 156-173. | Zbl
[24] and, Contact problems in elasticity. SIAM (1988). | MR | Zbl
[25] , Computational contact and impact mechanics. Springer (2002). | MR | Zbl
[26] and, A posteriori error estimation of - finite element approximations of frictional contact problems. Comput. Methods Appl. Mech. Engrg. 113 (1994) 11-45. | Zbl
[27] , Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39 (2001) 146-167. | Zbl
[28] , A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner (1996). | Zbl
[29] , A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Engrg. 176 (1999) 419-440. | Zbl
[30] , Computational Contact Mechanics. Wiley (2002).
[31] and, Different a posteriori error estimators and indicators for contact problems. Mathl. Comput. Modelling 28 (1998) 437-447. | Zbl
Cité par Sources :
