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.

翻译：本文致力于分析基于有限元方法的数值格式，用于逼近线性弹性椭圆膜壳在限制于指定半空间内的Koiter模型解。首先，我们证明所考虑的障碍问题的解唯一确定，并满足一组由四阶椭圆算子控制的变分不等式，这些不等式定义在适当空间的非空闭凸子集上。其次，我们证明所考虑的障碍问题的解可通过惩罚方法逼近。第三，我们证明相应惩罚问题的解在边界上具有更高的正则性。第四，我们写出所考虑惩罚问题的混合变分公式，并证明混合变分公式的解同样在边界上具有更高正则性。基于这一关于混合惩罚问题解正则性提升的结果，我们能够通过有限元格式逼近该问题的解。最后，我们给出数值实验，验证所获数学结果的有效性。