Augmented Lagrangian finite element methods for contact problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 173-195.

We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2018047
Classification : 65N12, 65N30, 74M15, 74S05
Mots-clés : Signorini problem, obstacle problem, finite element method, Lagrange mutlipliers, augmented Lagrangian, error estimates
Burman, Erik 1 ; Hansbo, Peter 1 ; Larson, Mats G. 1

1 
@article{M2AN_2019__53_1_173_0,
     author = {Burman, Erik and Hansbo, Peter and Larson, Mats G.},
     title = {Augmented {Lagrangian} finite element methods for contact problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {173--195},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {1},
     year = {2019},
     doi = {10.1051/m2an/2018047},
     mrnumber = {3937350},
     zbl = {1422.65374},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2018047/}
}
TY  - JOUR
AU  - Burman, Erik
AU  - Hansbo, Peter
AU  - Larson, Mats G.
TI  - Augmented Lagrangian finite element methods for contact problems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2019
SP  - 173
EP  - 195
VL  - 53
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2018047/
DO  - 10.1051/m2an/2018047
LA  - en
ID  - M2AN_2019__53_1_173_0
ER  - 
%0 Journal Article
%A Burman, Erik
%A Hansbo, Peter
%A Larson, Mats G.
%T Augmented Lagrangian finite element methods for contact problems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2019
%P 173-195
%V 53
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2018047/
%R 10.1051/m2an/2018047
%G en
%F M2AN_2019__53_1_173_0
Burman, Erik; Hansbo, Peter; Larson, Mats G. Augmented Lagrangian finite element methods for contact problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 173-195. doi : 10.1051/m2an/2018047. http://www.numdam.org/articles/10.1051/m2an/2018047/

P. Alart, A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods, Comput. Methods Appl. Mech. Eng. 92 (1991) 353–375. | DOI | MR | Zbl

D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982) 742–760. | MR | Zbl

I. Athanasopoulos, L.A. Caffarelli, Optimal regularity of lower dimensional obstacle problems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004) 49–66. | MR | Zbl

H.J.C. Barbosa, T.J.R. Hughes, Boundary Lagrange multipliers in finite element methods: error analysis in natural norms, Numer. Math. 62 (1992) 1–15. | MR | Zbl

Z. Belhachmi, F. Ben Belgacem, Quadratic finite element approximation of the Signorini problem, Math. Comp. 72 (2003) 83–104. | DOI | MR | Zbl

F. Ben Belgacem, Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods, SIAM J. Numer. Anal. 37 (2000) 1198–1216. | MR | Zbl

F. Ben Belgacem, Y. Renard, Hybrid finite element methods for the Signorini problem, Math. Comp. 72 (2003) 1117–1145. | DOI | MR | Zbl

D. Braess, C. Carstensen, R.H.W. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007) 455–471. | DOI | MR | Zbl

F. Brezzi, W.W. Hager, P.-A. Raviart, Error estimates for the finite element solution of variational inequalities. II. Mixed methods, Numer. Math. 31 (1978/1979) 1–16. | DOI | MR | Zbl

E. Burman, A. Ern, Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations, Math. Comp. 76 (2007) 1119–1140. | MR | Zbl

E. Burman, P. Hansbo, Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method, Comput. Methods Appl. Mech. Eng. 199 (2010) 2680–2686. | DOI | MR | Zbl

E. Burman, P. Hansbo, Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems, IMA J. Numer. Anal. 30 (2010) 870–885. | DOI | MR | Zbl

E. Burman, P. Hansbo, Deriving robust unfitted finite element methods from augmented Lagrangian formulations, edited by S. Bordas, E. Burman, M. Larson, M. Olshanskii. In: Geometrically Unfitted Finite Element Methods and Applications.In Vol. 121 of Lecture Notes in Computational Science and Engineering. Springer, Cham. (2017) | MR

E. Burman, F. Schieweck, Local CIP stabilization for composite finite elements, SIAM J. Numer. Anal. 54 (2016) 1967–1992. | DOI | MR | Zbl

L.A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998) 383–402. | DOI | MR | Zbl

F. Chouly, An adaptation of Nitsche’s method to the Tresca friction problem, J. Math. Anal. Appl. 411 (2014) 329–339. | MR | Zbl

F. Chouly, M. Fabre, P. Hild, R. Mlika, J. Pousin, Y. Renard, An overview of recent results on Nitsche’s method for contact problems, edited by S. Bordas, E. Burman, M. Larson, M. Olshanskii, Geometrically Unfitted Finite Element Methods and Applications. In Vol. 121 of Lecture Notes in Computational Science and Engineering. Springer, Cham. (2017) | MR

F. Chouly, P. Hild, A Nitsche-based method for unilateral contact problems: numerical analysis, SIAM J. Numer. Anal. 51 (2013) 1295–1307. | MR | Zbl

F. Chouly, P. Hild, On convergence of the penalty method for unilateral contact problems, Appl. Numer. Math. 65 (2013) 27–40. | MR | Zbl

F. Chouly, P. Hild, Y. Renard, Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments, Math. Comp. 84 (2015) 1089–1112. | DOI | MR | Zbl

G. Drouet, P. Hild, Optimal convergence for discrete variational inequalities modelling Signorini contact in 2D and 3D without additional assumptions on the unknown contact set, SIAM J. Numer. Anal. 53 (2015) 1488–1507. | MR | Zbl

R. Glowinski, P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, In Vol. 9 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1989). | DOI | MR | Zbl

P. Hansbo, A. Rashid, K. Salomonsson, Least-squares stabilized augmented Lagrangian multiplier method for elastic contact, Finite Elem. Anal. Des. 116 (2016) 32–37. | MR

J. Haslinger, I. Hlaváček, J. Nečas, Numerical methods for unilateral problems in solid mechanics. In Handbook of Numerical Analysis. IV. North-Holland, Amsterdam (1996) 313–485. | MR | Zbl

P. Hild, Y. Renard, A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics, Numer. Math. 115 (2010) 101–129. | DOI | MR | Zbl

P. Hild, Y. Renard, An improved a priori error analysis for finite element approximations of Signorini’s problem, SIAM J. Numer. Anal. 50 (2012) 2400–2419. | DOI | MR | Zbl

K. Ito, K. Kunisch, An augmented Lagrangian technique for variational inequalities, Appl. Math. Optim. 21 (1990) 223–241. | MR | Zbl

N. Kikuchi, J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods. In: Vol 8 of Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1988) | MR | Zbl

R. Mlika, Y. Renard, F. Chouly, An unbiased Nitsche’s formulation of large deformation frictional contact and self-contact, Comput. Methods Appl. Mech. Eng. 325 (2017) 265–288. | DOI | MR | Zbl

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. | MR | Zbl

R.H. Nochetto, K.G. Siebert, A. Veeser, Pointwise a posteriori error control for elliptic obstacle problems, Numer. Math. 95 (2003) 163–195. | DOI | MR | Zbl

R. Stenberg, On some techniques for approximating boundary conditions in the finite element method, J. Comput. Appl. Math. 63 (1995) 139–148. | MR | Zbl

M.A. Taylor, B.A. Wingate, L.P. Bos, Several new quadrature formulas for polynomial integration in the triangle, Preprint ArXiv: math/0501496 (2005).

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Reprint of the 1984 edn. AMS Chelsea Publishing, Providence, RI, (2001). | MR | Zbl

B.I. Wohlmuth, A. Popp, M.W. Gee, W.A. Wall, An abstract framework for a priori estimates for contact problems in 3D with quadratic finite elements, Comput. Mech. 49 (2012) 735–747. | MR | Zbl

P. Wriggers, G. Zavarise, A formulation for frictionless contact problems using a weak form introduced by Nitsche, Comput. Mech. 41 (2007) 407–420. | Zbl

L. Zhang, T. Cui, H. Liu, A set of symmetric quadrature rules on triangles and tetrahedra, J. Comput. Math. 27 (2009) 89–96. | MR | Zbl

Cité par Sources :