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Conférence externe
Penalty/finite element approximations of slip boundary conditions and Babuska's paradox
I. Dione
Département de mathématiques et de statistique, Université Laval

The penalty method is a classical and widespread method for the numerical treatment of constrained problems such as unilateral contact problems and problems with Dirichlet boundary conditions. It provides an  alternative approach to constrained optimization problems which avoids the necessity of introducing 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 their efficiency and to their mathematical analysis is that a Babuska's type paradox occurs.

Observed first by Sapondzyan  and Babuska-Pitkaranta on the plate equation in a disk with simple support boundary conditions, Babuska's paradox can be stated as follows:{\it{ 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 polygonal or
polyhedral boundary, we prove convergence estimates in terms of both the penalty and discretization parameters. In the case of a smooth curved boundary, we show through a numerical example that convergence  may not hold due to a Babuska'type paradox. Finally, we propose and test numerically several remedies.


 

Date: 2012-12-09 à 09:00
Endroit: Rencontre d'hiver de la SMC