  This paper is devoted to the analysis of a numerical scheme based on the
Finite Element Method for approximating the solution of Koiter's model for a
linearly elastic elliptic membrane shell subjected to remaining confined in a
prescribed half-space. First, we show that the solution of the obstacle problem
under consideration is uniquely determined and satisfies a set of variational
inequalities which are governed by a fourth order elliptic operator, and which
are posed over a non-empty, closed, and convex subset of a suitable space.
Second, we show that the solution of the obstacle problem under consideration
can be approximated by means of the penalty method. Third, we show that the
solution of the corresponding penalised problem is more regular up to the
boundary. Fourth, we write down the mixed variational formulation corresponding
to the penalised problem under consideration, and we show that the solution of
the mixed variational formulation is more regular up to the boundary as well.
In view of this result concerning the augmentation of the regularity of the
solution of the mixed penalised problem, we are able to approximate the
solution of the one such problem by means of a Finite Element scheme. Finally,
we present numerical experiments corroborating the validity of the mathematical
results we obtained.
