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

TL;DR

This paper develops new continuous finite elements with strong discrete inequalities for solving elliptic PDEs, demonstrating their effectiveness through theoretical properties and numerical tests.

Contribution

Introduction of continuous $H^2$-nonconforming finite elements satisfying a strong discrete Miranda--Talenti inequality in 2D and 3D, with applications to elliptic equations.

Findings

Finite elements satisfy a strong discrete Miranda--Talenti inequality.

Effective approximation of elliptic equations in non-divergence form.

Numerical results confirm practical feasibility and efficiency.

Abstract

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.