Error estimates for Stokes problem with Tresca friction conditions
1 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.
2 Université de Caen Basse-Normandie, Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, UFR sciences Campus II, Bd Maréchal JUIN, 14032 Caen Cedex, France.
Revised: 18 November 2013
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.
Mathematics Subject Classification: 45N30 / 76D07 / 35J87 / 35M87
Key words: Stokes problem / Tresca friction / variational inequality / mixed finite element / error estimates
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