[Une estimation dʼerreur quasi-optimale pour lʼapproximation par éléments finis du problème de Signorini bidimensionnel]
On présente dans cette Note une estimation optimale de lʼerreur dʼapproximation par éléments finis affines du problème de Signorini, cʼest à dire du problème de lʼéquilibre dʼun corps élastique en contact avec une fondation rigide. Les travaux précédents sur ce sujet donnent soit des résultats non optimaux, soit avec des conditions supplémentaires plus contraignantes sur la solution.
The aim of this Note is to present a quasi-optimal a priori error estimate for the linear finite element approximation of the so-called two-dimensional Signorini problem, i.e. the equilibrium of a plane linearly elastic body in contact with a rigid foundation. Previous works on that subject give either non-optimal estimates or with a more restrictive supplementary condition on the solution.
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@article{CRMATH_2012__350_5-6_325_0,
author = {Yves Renard},
title = {A quasi-optimal a priori error estimate for the two-dimensional {Signorini} problem approximated by linear finite elements},
journal = {Comptes Rendus. Math\'ematique},
pages = {325--328},
publisher = {Elsevier},
volume = {350},
number = {5-6},
year = {2012},
doi = {10.1016/j.crma.2012.01.024},
language = {en},
}
TY - JOUR AU - Yves Renard TI - A quasi-optimal a priori error estimate for the two-dimensional Signorini problem approximated by linear finite elements JO - Comptes Rendus. Mathématique PY - 2012 SP - 325 EP - 328 VL - 350 IS - 5-6 PB - Elsevier DO - 10.1016/j.crma.2012.01.024 LA - en ID - CRMATH_2012__350_5-6_325_0 ER -
%0 Journal Article %A Yves Renard %T A quasi-optimal a priori error estimate for the two-dimensional Signorini problem approximated by linear finite elements %J Comptes Rendus. Mathématique %D 2012 %P 325-328 %V 350 %N 5-6 %I Elsevier %R 10.1016/j.crma.2012.01.024 %G en %F CRMATH_2012__350_5-6_325_0
Yves Renard. A quasi-optimal a priori error estimate for the two-dimensional Signorini problem approximated by linear finite elements. Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 325-328. doi : 10.1016/j.crma.2012.01.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.01.024/
[1] Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method, SIAM J. Numer. Anal., Volume 37 (2000) no. 4, pp. 1198-1216
[2] Hybrid finite element methods for Signoriniʼs problem, Math. Comp., Volume 72 (2003), pp. 1117-1145
[3] The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications, North-Holland Publishing, 1978
[4] Les inéquations en mécanique et en physique, Dunod, Paris, 1972
[5] Éléments finis: théorie, applications, mise en œuvre, Mathématiques et Applications, vol. 36, SMAI, Springer-Verlag, 2002
[6] Singularities in Boundary Value Problems, Masson, 1992
[7] Finite element analysis for unilateral problems with obstacles on the boundary, Appl. Math., Volume 22 (1977) no. 3, pp. 180-188
[8] Numerical Methods for Unilateral Problems in Solids Mechanics, Handbook of Numerical Analysis, vol. IV, Elsevier Science, 1996, pp. 313-485
[9] Numerical implementation of two nonconforming finite element methods for unilateral contact, Comput. Meth. Appl. Mech. Eng., Volume 184 (2000) no. 1, pp. 99-123
[10] An optimal a priori error estimate for nonlinear multibody contact problems, SIAM J. Numer. Anal., Volume 43 (2005) no. 1, pp. 156-173
[11] Contact Problems in Elasticity, SIAM, 1988
[12] Fixed point strategies for elastostatic frictional contact problems, Math. Methods Appl. Sci., Volume 31 (2008), pp. 415-441
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