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Conférence externe
Penalty - finite element approximation of Stokes equations with slip boundary conditions
Dione, Ibrahima and José Urquiza
GIREF, Département de mathématiques et de statistique, Université Laval

The penalty method is a classical and widespread method for the numerical treatment of boundary conditions such as Dirichlet boundary conditions or unilateral contact boundary conditions. This approach leads to an unconstrained problems and avoids the introduction of additional unknowns in the form of Lagrange multipliers.

In the case of slip boundary conditions for fluid flows or elastic deformations, one of the main obstacle to the efficiency of the finite element method is that a Babuška’s type paradox occurs. Observed first by Sapondzyan [2] and Babuška [1] on the plate equation in a disk with simple support boundary conditions, Babuška’s paradox can be stated as follows: on a sequence of polygonal domains converging to the domain with a smooth boundary, the solutions of the corresponding problems do not converge to the solution of the problem on the limit domain.

Our presentation will focus on the finite element approximation of Stokes equations with slip boundary conditions imposed with the penalty method in two and three space dimensions. For a smooth curved boundary, we prove convergence estimates in terms of both the penalty and discretization parameters. A numerical example is presented confirming this analysis.

[1] Babuška I. Stability of domains with respect to basic problems in the theory of partial differential equations, mainly the theory of elasticity I. (Russian) Czechoslovak Math. J. 11:76–105, 1961.
[2] Sapondzyan O.M. Bending of a simply supported polygonal plate (Russian). Akad. Nauk Armyan. SSR. Izv. Fiz.-Mat. Estest. Tehn. Nauki, 5:29–46, 1952.

Date: 2013-06-19 à 07:00
Endroit: Rencontre annuelle de la SCMAI, Québec