This article introduces continuous $H^2$-nonconforming finite elements in two and three space dimensions which satisfy a strong discrete Miranda--Talenti inequality in the sense that the global $L^2$ norm of the piecewise Hessian is bounded by the $L^2$ norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with $C^1$ continuity on the vertices (2D) or edges (3D). As an application, these finite elements are used to approximate uniformly elliptic equations in non-divergence form under the Cordes condition without additional stabilization terms. For the biharmonic equation in three dimensions, the proposed methods has less degrees of freedom than existing nonconforming schemes of the same order. Numerical results in two and three dimensions confirm the practical feasibility of the proposed schemes.

翻译：本文介绍了在二维和三维空间中的连续$H^2$-非协调有限元，其满足强离散Miranda–Talenti不等式，即分片Hessian矩阵的全局$L^2$范数由分片Laplacian的$L^2$范数界定。该构造基于全局连续的有限元函数，且在顶点（二维）或边（三维）处具有$C^1$连续性。作为一个应用，这些有限元被用于在Cordes条件下逼近非散度形式的一致椭圆方程，而无需额外的稳定项。对于三维双调和方程，所提方法比同阶的现有非协调格式具有更少的自由度。二维和三维的数值结果证实了所提方案的实际可行性。