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Error estimates for Stokes problem with Tresca frictionconditions

Published online by Cambridge University Press:  13 August 2014

Mekki Ayadi ,
Leonardo Baffico ,
Mohamed Khaled Gdoura  and
Taoufik Sassi
Mekki Ayadi
Affiliation:
Université Tunis El Manar, Laboratoire de Modélisation Mathématiques et Numérique dans les Sciences de l’Ingénieur, Ecole Nationale d’Ingénieurs de Tunis, B.P. 32, 1002 Tunis, Tunisie.. [email protected]; [email protected]
Leonardo Baffico
Affiliation:
Université de Caen Basse-Normandie, Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, UFR sciences Campus II, Bd Maréchal JUIN, 14032 Caen Cedex, France.; [email protected]; [email protected]
Mohamed Khaled Gdoura
Affiliation:
Université Tunis El Manar, Laboratoire de Modélisation Mathématiques et Numérique dans les Sciences de l’Ingénieur, Ecole Nationale d’Ingénieurs de Tunis, B.P. 32, 1002 Tunis, Tunisie.. [email protected]; [email protected] Université de Caen Basse-Normandie, Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, UFR sciences Campus II, Bd Maréchal JUIN, 14032 Caen Cedex, France.; [email protected]; [email protected]
Taoufik Sassi
Affiliation:
Université de Caen Basse-Normandie, Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, UFR sciences Campus II, Bd Maréchal JUIN, 14032 Caen Cedex, France.; [email protected]; [email protected]
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Abstract

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 bypolymer melt, which can slip on a solid wall. The mixed formulation is based on adualization of the non-differentiable term which define the slip conditions. Existence anduniqueness of both continuous and discrete solutions of these problems is guaranteed bymeans of continuous and discrete inf-sup conditions that are proved. Velocity and pressureare approximated by P1 bubble-P1 finite element and piecewise linearelements are used to discretize the Lagrange multiplier associated to the shear stress onthe friction boundary. Optimal a priori error estimates are derived usingclassical tools of finite element analysis and two uncoupled discrete inf-sup conditionsfor the pressure and the Lagrange multiplier associated to the fluid shear stress.

Keywords

Stokes problemTresca frictionvariational inequalitymixed finite elementerror estimates
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1413 - 1429
Copyright
© EDP Sciences, SMAI 2014

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References

Arnold, D., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344. Google Scholar
Ayadi, M., Gdoura, M. K. and Sassi, T., Mixed formulation for Stokes problem with Tresca friction. C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1069–1072. Google Scholar
Baillet, L. and Sassi, T., Mixed finite element methods for the Signorini problem with friction. Numer. Methods Partial Differ. Eq. 22 (2006) 1489–1508. Google Scholar
Ben Belgacem, F. and Renard, Y., Hybrid finite element method for the Signorini problem. Math. Comput. 72 (2003) 1117–1145. Google Scholar
Boukrouche, M. and Saidi, F., Non-isothermal lubrication problem with Tresca fluid-solid interface law. Part I. Nonlinear Analysis: Real World Appl. 7 (2006) 1145–1166. Google Scholar
Brezzi, F., Hager, W. and Raviart, P.A., Error estimates for the finite element solution of variational inequalities, part II. Mixed methods. Numer. Math. 31 (1978) 1–16. Google Scholar
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15. Series Comput. Math. Springer, New York (1991).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Studies Math. Appl. North Holland, Netherland (1980).
Clément, Ph., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84. Google Scholar
Coorevits, P., Hild, P., Lhalouani, K. and Sassi, T., Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. Math. Comput. 71 (2001) 1–25. Google Scholar
Crouzeix, M. and Thomée, V., The stability in L p and W 1,p of the L 2-projection on finite element function spaces. Math. Comput. 48 (1987) 521–532. Google Scholar
Doltsinis, J.St., Luginsland, J. and Nölting, S., Some developments in the numerical simulation of metal forming processes. Eng. Comput. 4 (1987) 266–280. Google Scholar
A. Ern and J.-L. Guermond, Éléments Finis: Théorie, Application, Mise en Oeuvre. Math. Appl. SMAI, Springer 36 (2001).
Fortin, M. and Côté, D., On the imposition of friction boundary conditions for the numerical simulation of Bingham fluid flows. Comput. Methods Appl. Mech. Engrg. 88 (1991) 97–109. Google Scholar
H. Fujita, Flow Problems with Unilateral Boundary Conditions. Leçons, Collège de France (1993).
H. Fujita, A Mathematical analysis of motions of viscous incompressible fluid under leak and slip boundary conditions. Math. Fluid Mech. Model. \hbox{$S\overline{u}rikaisekikenky\overline{u}sho \, K\overline{o}ky\overline{u}roko$}Surikaisekikenkyusho Kokyuroko 888 (1994) 199–216.
Fujita, H., A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149 (2002) 57–69. Google Scholar
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).
Geymonat, G. and Krasucki, F., On the existence of the Airy function in Lipschitz domains. Application to the traces of H 2 C. R. Acad. Sci. Paris, Série I 330 (2000) 355–360. Google Scholar
V. Girault and P.A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin (1979).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monogr. Studies Math. Pitman (Advanced Publishing Program), Boston, MA 24 (1985).
Hatzikiriakos, S.G. and Dealy, J.M., Wall slip of molten high density polyethylene. I. Sliding plate rheometer studies. J. Rheology 3 (1991) 497–523. Google Scholar
Haslinger, J. and Sassi, T., Mixed finite element approximation of 3D contact problem with given friction: Error analysis and numerical realisation, ESAIM: M2AN 38 (2004) 563–578. Google Scholar
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).
Li, Y. and Li, K., Penalty finite element method for Stokes problem with nonlinear slip boundary conditions. Appl. Math. Comput. 204 (2008) 216–226. Google Scholar
J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, New York (1972).
Magnin, A. and Piau, J.M., Shear rheometry of fluids with a yield stress. J. Non-Newtonian Fluid Mech. 23 (1987) 91–106. Google Scholar
Marini, L. and Quarteroni, A., A relaxation procedure for domain decomposition method using finite elements. Numer. Math. 55 (1989) 575–598. Google Scholar
Rao, I.J. and Rajagopal, K.R., The effect of the slip boundary condition on the flow of fluids in a channel. Acta Mechanica 135 (1999) 113–126. Google Scholar
Saito, N. and Fujita, H., 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. Google Scholar
Saito, N., 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. CrossRefGoogle Scholar
Santanach Carreras, E., El Kissi, N. and Piau, J.-M., 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. Google Scholar