A quasi-optimal a priori error estimate for the two-dimensional Signorini problem approximated by linear finite elements
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 eit...
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| Published in | Comptes rendus. Mathématique Vol. 350; no. 5-6; pp. 325 - 328 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Paris
Elsevier SAS
01.03.2012
Elsevier Académie des sciences (Paris) |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1631-073X 1778-3569 1778-3569 |
| DOI | 10.1016/j.crma.2012.01.024 |
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| Abstract | 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.
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. |
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| AbstractList | 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.
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. On présente dans cette Note une estimation optimale de l'erreur d'approximation par éléments finis affines du problème de Signorini, c'est à dire du problème de l'équilibre d'un 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. |
| Author | Renard, Yves |
| Author_xml | – sequence: 1 givenname: Yves surname: Renard fullname: Renard, Yves email: Yves.Renard@insa-lyon.fr organization: Université de Lyon, CNRS, INSA-Lyon, ICJ UMR5208, 69621 Villeurbanne, France |
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| Cites_doi | 10.1002/mma.921 10.1137/S0036142903436678 10.1137/1.9781611970845 10.1016/S0045-7825(99)00096-1 10.21136/AM.1977.103691 10.1137/S0036142998347966 10.1090/S0025-5718-03-01490-X |
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| Keywords | Finite element method Estimation error Error estimation Signorini problem A priori estimation Two-dimensional calculations Optimal estimation Body Equilibrium Elastic contact Finite element |
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| References | Ben Belgacem (br0010) 2000; 37 Haslinger (br0070) 1977; 22 Hild (br0090) 2000; 184 Laborde, Renard (br0120) 2008; 31 Ern, Guermond (br0050) 2002; vol. 36 Haslinger, Hlaváček, Nečas (br0080) 1996; vol. IV Ciarlet (br0030) 1978 Hüeber, Wohlmuth (br0100) 2005; 43 Grisvard (br0060) 1992 Duvaut, Lions (br0040) 1972 Ben Belgacem, Renard (br0020) 2003; 72 Kikuchi, Oden (br0110) 1988 Hild, P. (key2024021915220364608_9) 2000; 184 Hüeber, S. (key2024021915220364608_10) 2005; 43 Haslinger, J. (key2024021915220364608_7) 1977; 22 Ben Belgacem, F. (key2024021915220364608_2) 2003; 72 Grisvard, P. (key2024021915220364608_6) 1992 Haslinger, J. (key2024021915220364608_8) 1996; vol. IV Ciarlet, P.G. (key2024021915220364608_3) 1978 Ben Belgacem, F. (key2024021915220364608_1) 2000; 37 Kikuchi, N. (key2024021915220364608_11) 1988 Duvaut, G. (key2024021915220364608_4) 1972 Ern, A. (key2024021915220364608_5) 2002; vol. 36 Laborde, P. (key2024021915220364608_12) 2008; 31 |
| References_xml | – volume: 22 start-page: 180 year: 1977 end-page: 188 ident: br0070 article-title: Finite element analysis for unilateral problems with obstacles on the boundary publication-title: Appl. Math. – volume: 72 start-page: 1117 year: 2003 end-page: 1145 ident: br0020 article-title: Hybrid finite element methods for Signoriniʼs problem publication-title: Math. Comp. – volume: 37 start-page: 1198 year: 2000 end-page: 1216 ident: br0010 article-title: Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method publication-title: SIAM J. Numer. Anal. – volume: 43 start-page: 156 year: 2005 end-page: 173 ident: br0100 article-title: An optimal a priori error estimate for nonlinear multibody contact problems publication-title: SIAM J. Numer. Anal. – volume: vol. 36 year: 2002 ident: br0050 article-title: Éléments finis: théorie, applications, mise en œuvre publication-title: Mathématiques et Applications – year: 1988 ident: br0110 article-title: Contact Problems in Elasticity – volume: vol. IV start-page: 313 year: 1996 end-page: 485 ident: br0080 article-title: Numerical Methods for Unilateral Problems in Solids Mechanics publication-title: Handbook of Numerical Analysis – volume: 31 start-page: 415 year: 2008 end-page: 441 ident: br0120 article-title: Fixed point strategies for elastostatic frictional contact problems publication-title: Math. Methods Appl. Sci. – year: 1978 ident: br0030 article-title: The Finite Element Method for Elliptic Problems publication-title: Studies in Mathematics and Its Applications – year: 1972 ident: br0040 article-title: Les inéquations en mécanique et en physique – year: 1992 ident: br0060 article-title: Singularities in Boundary Value Problems – volume: 184 start-page: 99 year: 2000 end-page: 123 ident: br0090 article-title: Numerical implementation of two nonconforming finite element methods for unilateral contact publication-title: Comput. Meth. Appl. Mech. Eng. – volume: 31 start-page: 415 year: 2008 ident: key2024021915220364608_12 article-title: Fixed point strategies for elastostatic frictional contact problems publication-title: Math. Methods Appl. Sci. doi: 10.1002/mma.921 – volume: 43 start-page: 156 issue: 1 year: 2005 ident: key2024021915220364608_10 article-title: An optimal a priori error estimate for nonlinear multibody contact problems publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142903436678 – year: 1988 ident: key2024021915220364608_11 publication-title: Contact Problems in Elasticity doi: 10.1137/1.9781611970845 – volume: 184 start-page: 99 issue: 1 year: 2000 ident: key2024021915220364608_9 article-title: Numerical implementation of two nonconforming finite element methods for unilateral contact publication-title: Comput. Meth. Appl. Mech. Eng. doi: 10.1016/S0045-7825(99)00096-1 – year: 1978 ident: key2024021915220364608_3 publication-title: The Finite Element Method for Elliptic Problems – year: 1972 ident: key2024021915220364608_4 publication-title: Les inéquations en mécanique et en physique – volume: vol. 36 year: 2002 ident: key2024021915220364608_5 publication-title: Éléments finis: théorie, applications, mise en œuvre – volume: vol. IV start-page: 313 year: 1996 ident: key2024021915220364608_8 publication-title: Numerical Methods for Unilateral Problems in Solids Mechanics – volume: 22 start-page: 180 issue: 3 year: 1977 ident: key2024021915220364608_7 article-title: Finite element analysis for unilateral problems with obstacles on the boundary publication-title: Appl. Math. doi: 10.21136/AM.1977.103691 – volume: 37 start-page: 1198 issue: 4 year: 2000 ident: key2024021915220364608_1 article-title: Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142998347966 – year: 1992 ident: key2024021915220364608_6 publication-title: Singularities in Boundary Value Problems – volume: 72 start-page: 1117 year: 2003 ident: key2024021915220364608_2 article-title: Hybrid finite element methods for Signoriniʼs problem publication-title: Math. Comp. doi: 10.1090/S0025-5718-03-01490-X |
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| Title | A quasi-optimal a priori error estimate for the two-dimensional Signorini problem approximated by linear finite elements |
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