Error estimates for Stokes problem with Tresca friction conditions
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1413-1429.

In this paper, we present and study a mixed variational method in order to approximate, with the finite element method, a Stokes problem with Tresca friction boundary conditions. These non-linear boundary conditions arise in the modeling of mold filling process by polymer melt, which can slip on a solid wall. The mixed formulation is based on a dualization of the non-differentiable term which define the slip conditions. Existence and uniqueness of both continuous and discrete solutions of these problems is guaranteed by means of continuous and discrete inf-sup conditions that are proved. Velocity and pressure are approximated by P1 bubble-P1 finite element and piecewise linear elements are used to discretize the Lagrange multiplier associated to the shear stress on the friction boundary. Optimal a priori error estimates are derived using classical tools of finite element analysis and two uncoupled discrete inf-sup conditions for the pressure and the Lagrange multiplier associated to the fluid shear stress.

DOI : 10.1051/m2an/2014001
Classification : 45N30, 76D07, 35J87, 35M87
Mots clés : Stokes problem, Tresca friction, variational inequality, mixed finite element, error estimates
@article{M2AN_2014__48_5_1413_0,
     author = {Ayadi, Mekki and Baffico, Leonardo and Gdoura, Mohamed Khaled and Sassi, Taoufik},
     title = {Error estimates for {Stokes} problem with {Tresca} friction conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1413--1429},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {5},
     year = {2014},
     doi = {10.1051/m2an/2014001},
     mrnumber = {3264359},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2014001/}
}
TY  - JOUR
AU  - Ayadi, Mekki
AU  - Baffico, Leonardo
AU  - Gdoura, Mohamed Khaled
AU  - Sassi, Taoufik
TI  - Error estimates for Stokes problem with Tresca friction conditions
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2014
SP  - 1413
EP  - 1429
VL  - 48
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2014001/
DO  - 10.1051/m2an/2014001
LA  - en
ID  - M2AN_2014__48_5_1413_0
ER  - 
%0 Journal Article
%A Ayadi, Mekki
%A Baffico, Leonardo
%A Gdoura, Mohamed Khaled
%A Sassi, Taoufik
%T Error estimates for Stokes problem with Tresca friction conditions
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2014
%P 1413-1429
%V 48
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2014001/
%R 10.1051/m2an/2014001
%G en
%F M2AN_2014__48_5_1413_0
Ayadi, Mekki; Baffico, Leonardo; Gdoura, Mohamed Khaled; Sassi, Taoufik. Error estimates for Stokes problem with Tresca friction conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1413-1429. doi : 10.1051/m2an/2014001. http://www.numdam.org/articles/10.1051/m2an/2014001/

[1] D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344. | MR | Zbl

[2] M. Ayadi, M. K. Gdoura and T. Sassi, Mixed formulation for Stokes problem with Tresca friction. C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1069-1072. | MR | Zbl

[3] L. Baillet and T. Sassi, Mixed finite element methods for the Signorini problem with friction. Numer. Methods Partial Differ. Eq. 22 (2006) 1489-1508. | MR | Zbl

[4] F. Ben Belgacem and Y. Renard, Hybrid finite element method for the Signorini problem. Math. Comput. 72 (2003) 1117-1145. | MR | Zbl

[5] M. Boukrouche and F. Saidi, Non-isothermal lubrication problem with Tresca fluid-solid interface law. Part I. Nonlinear Analysis: Real World Appl. 7 (2006) 1145-1166. | MR | Zbl

[6] F. Brezzi, W. Hager and P.A. Raviart, Error estimates for the finite element solution of variational inequalities, part II. Mixed methods. Numer. Math. 31 (1978) 1-16. | MR | Zbl

[7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15. Series Comput. Math. Springer, New York (1991). | MR | Zbl

[8] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Studies Math. Appl. North Holland, Netherland (1980). | MR | Zbl

[9] Ph. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | MR | Zbl

[10] P. Coorevits, P. Hild, K. Lhalouani and T. Sassi, Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. Math. Comput. 71 (2001) 1-25. | MR | Zbl

[11] M. Crouzeix and V. Thomée, The stability in Lp and W1,p of the L2-projection on finite element function spaces. Math. Comput. 48 (1987) 521-532. | MR | Zbl

[12] J.St. Doltsinis, J. Luginsland and S. Nölting, Some developments in the numerical simulation of metal forming processes. Eng. Comput. 4 (1987) 266-280.

[13] A. Ern and J.-L. Guermond, Éléments Finis: Théorie, Application, Mise en Oeuvre. Math. Appl. SMAI, Springer 36 (2001). | Zbl

[14] M. Fortin and D. Côté, On the imposition of friction boundary conditions for the numerical simulation of Bingham fluid flows. Comput. Methods Appl. Mech. Engrg. 88 (1991) 97-109. | Zbl

[15] H. Fujita, Flow Problems with Unilateral Boundary Conditions. Leçons, Collège de France (1993).

[16] H. Fujita, A Mathematical analysis of motions of viscous incompressible fluid under leak and slip boundary conditions | MR | Zbl

[17] H. Fujita, A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149 (2002) 57-69. | MR | Zbl

[18] M.K. Gdoura, Problème de Stokes avec des conditions aux limites non-linéaires: analyse numérique et algorithmes de résolution, Thèse en co-tutelle, Université Tunis El Manar et Université de Caen Basse Normandie (2011).

[19] G. Geymonat and F. Krasucki, On the existence of the Airy function in Lipschitz domains. Application to the traces of H2 C. R. Acad. Sci. Paris, Série I 330 (2000) 355-360. | MR | Zbl

[20] V. Girault and P.A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin (1979). | MR | Zbl

[21] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monogr. Studies Math. Pitman (Advanced Publishing Program), Boston, MA 24 (1985). | MR | Zbl

[22] S.G. Hatzikiriakos and J.M. Dealy, Wall slip of molten high density polyethylene. I. Sliding plate rheometer studies. J. Rheology 3 (1991) 497-523.

[23] J. Haslinger and T. Sassi, Mixed finite element approximation of 3D contact problem with given friction: Error analysis and numerical realisation, ESAIM: M2AN 38 (2004) 563-578. | Numdam | MR | Zbl

[24] N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM Studies in Appl. Math. Philadelphia (1988). | MR | Zbl

[25] Y. Li and K. Li, Penalty finite element method for Stokes problem with nonlinear slip boundary conditions. Appl. Math. Comput. 204 (2008) 216-226. | MR | Zbl

[26] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, New York (1972). | Zbl

[27] A. Magnin and J.M. Piau, Shear rheometry of fluids with a yield stress. J. Non-Newtonian Fluid Mech. 23 (1987) 91-106.

[28] L. Marini and A. Quarteroni, A relaxation procedure for domain decomposition method using finite elements. Numer. Math. 55 (1989) 575-598. | MR | Zbl

[29] I.J. Rao and K.R. Rajagopal, The effect of the slip boundary condition on the flow of fluids in a channel. Acta Mechanica 135 (1999) 113-126. | MR | Zbl

[30] N. Saito and H. Fujita, Regularity of solutions to the Stokes equations under a certain nonlinear boundary condition, The Navier-Stokes Equations. Lect. Notes Pure Appl. Math. 223 (2001) 73-86. | MR | Zbl

[31] N. Saito, On the stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions. Pub. RIMS. Kyoto University 40 (2004) 345-383. | MR | Zbl

[32] E. Santanach Carreras, N. El Kissi and J.-M. Piau, Block copolymer extrusion distortions: Exit delayed transversal primary cracks and longitudinal secondary cracks: Extrudate splitting and continuous peeling. J. Non-Newt. Fluid Mech. 131 (2005) 1-21.

Cité par Sources :