A New Mixed Finite Element Method For The Cahn-Hilliard Equation

This paper presents a new mixed finite element method for the Cahn-Hilliard equation. The well-posedness of the mixed formulation is established and the error estimates for its linearized fully discrete scheme are provided. The new mixed finite element method provides a unified construction in two and three dimensions allowing for arbitrary polynomial degrees. Numerical experiments are given to validate the efficiency and accuracy of the theoretical results.

💡 Research Summary

The paper introduces a novel mixed finite element method (FEM) for solving the Cahn‑Hilliard equation, a fourth‑order nonlinear PDE that models phase separation phenomena. Traditional conforming FEMs require C¹ continuity, which is difficult to achieve in two and three dimensions, especially for higher‑order elements. To avoid this restriction, the authors reformulate the problem by introducing an auxiliary symmetric tensor variable σ = ∇²u − ε⁻²f(u)I, where f(u)=u³−u derives from the double‑well potential. This transformation splits the original fourth‑order equation into a system of two second‑order equations: a time‑dependent mass balance ∂ₜu + divDiv σ = 0 and a constitutive relation σ = ∇²u − ε⁻²f(u)I.

The mixed variational formulation (3.4) seeks (σ,u) in the product space Σ × L⁶(Ω), where Σ is a subspace of H(divDiv,Ω;S) consisting of symmetric tensor fields whose divDiv belongs to L²(Ω) and which satisfy appropriate boundary conditions. The authors rigorously prove the equivalence between this mixed formulation and the primal H²‑based formulation (3.1) via Theorem 3.4, using Green’s identities, Nečas’ inequality, and careful trace analysis. Two sets of boundary conditions are discussed; the more “sufficient” set (Π_F(σ n_F)=0 and n_FᵀDiv σ=0 on each face) is shown to be consistent with the exact solution and easier to enforce in practice.

For temporal discretisation, a uniform partition of