  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.
