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.
Keywords: 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},
year = {2014},
publisher = {EDP Sciences},
volume = {48},
number = {5},
doi = {10.1051/m2an/2014001},
mrnumber = {3264359},
language = {en},
url = {https://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 - https://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 https://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
[1] , and , A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344. | Zbl | MR
[2] , and , Mixed formulation for Stokes problem with Tresca friction. C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1069-1072. | Zbl | MR
[3] and , Mixed finite element methods for the Signorini problem with friction. Numer. Methods Partial Differ. Eq. 22 (2006) 1489-1508. | Zbl | MR
[4] and , Hybrid finite element method for the Signorini problem. Math. Comput. 72 (2003) 1117-1145. | Zbl | MR
[5] and , Non-isothermal lubrication problem with Tresca fluid-solid interface law. Part I. Nonlinear Analysis: Real World Appl. 7 (2006) 1145-1166. | Zbl | MR
[6] , and , Error estimates for the finite element solution of variational inequalities, part II. Mixed methods. Numer. Math. 31 (1978) 1-16. | Zbl | MR
[7] and , Mixed and Hybrid Finite Element Methods, vol. 15. Series Comput. Math. Springer, New York (1991). | Zbl | MR
[8] , The Finite Element Method for Elliptic Problems. Studies Math. Appl. North Holland, Netherland (1980). | Zbl | MR
[9] , Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Zbl | MR | Numdam
[10] , , and , Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. Math. Comput. 71 (2001) 1-25. | Zbl | MR
[11] and , The stability in Lp and W1,p of the L2-projection on finite element function spaces. Math. Comput. 48 (1987) 521-532. | Zbl | MR
[12] , and , Some developments in the numerical simulation of metal forming processes. Eng. Comput. 4 (1987) 266-280.
[13] and , Éléments Finis: Théorie, Application, Mise en Oeuvre. Math. Appl. SMAI, Springer 36 (2001). | Zbl
[14] and , 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] , Flow Problems with Unilateral Boundary Conditions. Leçons, Collège de France (1993).
[16] , A Mathematical analysis of motions of viscous incompressible fluid under leak and slip boundary conditions | Zbl | MR
[17] , A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149 (2002) 57-69. | Zbl | MR
[18] , 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] and , 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. | Zbl | MR
[20] and , Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin (1979). | Zbl | MR
[21] , Elliptic Problems in Nonsmooth Domains. Monogr. Studies Math. Pitman (Advanced Publishing Program), Boston, MA 24 (1985). | Zbl | MR
[22] and , Wall slip of molten high density polyethylene. I. Sliding plate rheometer studies. J. Rheology 3 (1991) 497-523.
[23] and , Mixed finite element approximation of 3D contact problem with given friction: Error analysis and numerical realisation, ESAIM: M2AN 38 (2004) 563-578. | Zbl | MR | Numdam
[24] and , Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM Studies in Appl. Math. Philadelphia (1988). | Zbl | MR
[25] and , Penalty finite element method for Stokes problem with nonlinear slip boundary conditions. Appl. Math. Comput. 204 (2008) 216-226. | Zbl | MR
[26] and , Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, New York (1972). | Zbl
[27] and , Shear rheometry of fluids with a yield stress. J. Non-Newtonian Fluid Mech. 23 (1987) 91-106.
[28] and , A relaxation procedure for domain decomposition method using finite elements. Numer. Math. 55 (1989) 575-598. | Zbl | MR
[29] and , The effect of the slip boundary condition on the flow of fluids in a channel. Acta Mechanica 135 (1999) 113-126. | Zbl | MR
[30] and , 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. | Zbl | MR
[31] , 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. | Zbl | MR
[32] , and , 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 :
