Finite element approximations of the Lamé system with penalized ideal contact boundary conditions

We consider finite element approximations of the Lamé system of elasticity with ideal contact boundary conditions imposed with the penalty method. 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 two-dimensional example that convergence may not hold, due to a Babuska’s type paradox. We also propose and test numerically several remedies.