This paper is devoted to the study of a novel mixed Finite Element Method for approximating the solutions of fourth order variational problems subjected to a constraint. The first problem we consider consists in establishing the convergence of the error of the numerical approximation of the solution of a biharmonic obstacle problem. The contents of this section are meant to generalise the approach originally proposed by Ciarlet \& Raviart, and then complemented by Ciarlet \& Glowinski. The second problem we consider amounts to studying a two-dimensional variational problem for linearly elastic shallow shells subjected to remaining confined in a prescribed half-space. We first study the case where the parametrisation of the middle surface for the linearly elastic shallow shell under consideration has non-zero curvature, and we observe that the numerical approximation of this model via a mixed Finite Element Method based on conforming elements requires the implementation of the additional constraint according to which the gradient matrix of the dual variable has to be symmetric. However, differently from the biharmonic obstacle problem previously studied, we show that the numerical implementation of this result cannot be implemented by solely resorting to Courant triangles. Finally, we show that if the middle surface of the linearly elastic shallow shell under consideration is flat, the symmetry constraint required for formulating the constrained mixed variational problem announced in the second part of the paper is not required, and the solution can thus be approximated by solely resorting to Courant triangles. The theoretical results we derived are complemented by a series of numerical experiments.

翻译：本文致力于研究一种新颖的混合有限元方法，用于近似求解受约束的四阶变分问题解。我们考虑的第一个问题在于建立双调和障碍问题解数值近似的误差收敛性。本节内容旨在推广最初由Ciarlet & Raviart提出、后经Ciarlet & Glowinski完善的经典方法。我们考虑的第二个问题在于研究受限于预定半空间的线性弹性浅壳二维变分问题。我们首先研究所述线性弹性浅壳中曲面参数化具有非零曲率的情形，并观察到基于协调元的混合有限元法对该模型进行数值近似时，需要施加对偶变量梯度矩阵必须对称的附加约束。然而，与先前研究的双调和障碍问题不同，我们证明该结果的数值实现不能仅通过Courant三角形单元完成。最后我们证明，若所述线性弹性浅壳的中曲面为平面，则本文第二部分提出的约束混合变分问题无需对称性约束，其解可仅通过Courant三角形单元进行近似。我们推导的理论结果通过一系列数值实验得到了验证。