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法的数值格式，用于逼近线性弹性椭圆膜壳在保持于预设半空间内约束下的Koiter模型解。首先，我们证明所考虑的障碍问题解唯一存在，并满足由四阶椭圆算子控制、定义在适当空间非空闭凸子集上的变分不等式组。其次，我们证明该障碍问题的解可通过罚函数法进行逼近。第三，证明对应惩罚问题解在边界处具有更高正则性。第四，建立所考虑惩罚问题的混合变分公式，并证明其解同样在边界处具有更高正则性。基于混合惩罚问题正则性增强这一结论，我们得以通过有限元格式逼近该问题的解。最后，通过数值实验验证所获数学结果的有效性。