Error estimates for Stokes problem with Tresca friction conditions
Mekki Ayadi; Leonardo Baffico; Mohamed Khaled Gdoura; Taoufik Sassi
- Volume: 48, Issue: 5, page 1413-1429
- ISSN: 0764-583X
Access Full Article
topAbstract
topHow to cite
topAyadi, Mekki, et al. "Error estimates for Stokes problem with Tresca friction conditions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.5 (2014): 1413-1429. <http://eudml.org/doc/273283>.
@article{Ayadi2014,
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 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.},
author = {Ayadi, Mekki, Baffico, Leonardo, Gdoura, Mohamed Khaled, Sassi, Taoufik},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Stokes problem; Tresca friction; variational inequality; mixed finite element; error estimates},
language = {eng},
number = {5},
pages = {1413-1429},
publisher = {EDP-Sciences},
title = {Error estimates for Stokes problem with Tresca friction conditions},
url = {http://eudml.org/doc/273283},
volume = {48},
year = {2014},
}
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 - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 5
SP - 1413
EP - 1429
AB - 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.
LA - eng
KW - Stokes problem; Tresca friction; variational inequality; mixed finite element; error estimates
UR - http://eudml.org/doc/273283
ER -
References
top- [1] D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo21 (1984) 337-344. Zbl0593.76039MR799997
- [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. Zbl1205.35183MR2735009
- [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. Zbl1105.74041MR2257645
- [4] F. Ben Belgacem and Y. Renard, Hybrid finite element method for the Signorini problem. Math. Comput.72 (2003) 1117–1145. Zbl1023.74043MR1972730
- [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. Zbl1122.35155MR2260905
- [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. Zbl0427.65077MR508584
- [7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15. Series Comput. Math. Springer, New York (1991). Zbl0788.73002MR1115205
- [8] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Studies Math. Appl. North Holland, Netherland (1980). Zbl0511.65078MR608971
- [9] Ph. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.9 (1975) 77–84. Zbl0368.65008MR400739
- [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. Zbl1013.74062MR1862986
- [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. Zbl0637.41034MR878688
- [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). Zbl0993.65123
- [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. Zbl0745.76067
- [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. Math. Fluid Mech. Model. Surikaisekikenkyusho Kokyuroko 888 (1994) 199–216. Zbl0939.76527MR1338892
- [17] H. Fujita, A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math.149 (2002) 57–69. Zbl1058.76023MR1952966
- [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. Zbl0945.35065MR1751670
- [20] V. Girault and P.A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin (1979). Zbl0441.65081MR548867
- [21] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monogr. Studies Math. Pitman (Advanced Publishing Program), Boston, MA 24 (1985). Zbl0695.35060MR775683
- [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. Zbl1080.74046MR2075760
- [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). Zbl0685.73002MR961258
- [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. Zbl1173.76023MR2458358
- [26] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, New York (1972). Zbl0223.35039
- [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. Zbl0661.65111MR998911
- [29] I.J. Rao and K.R. Rajagopal, The effect of the slip boundary condition on the flow of fluids in a channel. Acta Mechanica135 (1999) 113–126. Zbl0936.76013MR1690164
- [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. Zbl0995.35048MR1864500
- [31] N. Saito, On the stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions. Pub. RIMS. Kyoto University40 (2004) 345–383. Zbl1050.35029MR2049639
- [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.
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.