In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis–Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.

在本文中，我们建立了基于有限元方法的数值方案的收敛性，用于模拟半空间内剩余约束下的线弹性椭圆膜壳的变形的与时间无关的问题。我们不是近似控制这个障碍问题的原始变分不等式，而是近似所考虑问题的惩罚版本。惩罚参数和网格大小之间的适当耦合将引导我们建立离散惩罚问题的解与原始变分不等式的解的收敛性。我们还针对所考虑的问题建立了 Brezis-Sibony 方案的收敛性。由于这种迭代方法，我们可以近似求解离散惩罚问题，而不必求助于非线性优化工具。最后，我们提出了数值模拟来验证我们的新理论结果。