High-order conforming finite elements for the Cahn-Hilliard equation: Relative-energy stability and energy defects

arXiv:2606.06719v1 Announce Type: new Abstract: We study a semidiscrete single-field Galerkin approximation of the Cahn-Hilliard equation using high-order conforming finite element spaces. More specifically, globally $C^1$ finite elements with $H^2$-conforming trial spaces, including Argyris, Bell, and Bogner-Fox-Schmit elements, allow a direct discretization of the fourth-order formulation and preserve mass exactly. The main structural result is an exact energy balance for the physical Cahn-Hilliard energy. Besides the expected discrete dissipation, the balance contains an explicitly computable energy defect. This defect vanishes for Laplacian-invariant periodic spaces, such as Fourier spaces, but is generally nonzero for classical $C^1$ finite elements. It therefore quantifies the precise loss of a discrete gradient-flow structure. We prove semidiscrete a priori error estimates by a relative-energy argument. The estimate is closed using an augmented relative energy and a discrete elliptic reconstruction bound for the inverse discrete Laplacian. The resulting convergence rates match the expected approximation orders. Numerical experiments with Bell and Argyris elements confirm the rates and demonstrate the defect mechanism by comparison with a Fourier reference discretization.