Continuous finite elements satisfying a strong discrete Miranda--Talenti identity

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.