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In this paper we establish the convergence of a numerical scheme based, onthe Finite Element Method, for a time-independent problem modelling thedeformation of a linearly elastic elliptic membrane shell subjected toremaining confined in a half space. Instead of approximating the originalvariational inequalities governing this obstacle problem, we approximate thepenalized version of the problem under consideration. A suitable couplingbetween the penalty parameter and the mesh size will then lead us to establishthe convergence of the solution of the discrete penalized problem to thesolution of the original variational inequalities. We also establish theconvergence of the Brezis-Sibony scheme for the problem under consideration.Thanks to this iterative method, we can approximate the solution of thediscrete penalized problem without having to resort to nonlinear optimizationtools. Finally, we present numerical simulations validating our new theoreticalresults.

Numerical approximation of the solution of an obstacle problem modelling the displacement of elliptic membrane shells via the penalty method