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.
mekki.ayadi@enis.rnu.tn; mohamedkhaled.gdoura@lamsin.rnu.tn
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.
leonardo.baffico@unicaen.fr; taoufik.sassi@unicaen.fr

Received: 2 February 2012
Revised: 18 November 2013

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.