High order finite element calculations for the deterministic Cahn-Hilliard equation

In this work, we propose a numerical method based on high degree continuous nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the finite element method proves to be very efficient and favorably compares with other existing strategies (C1 elements, adaptive mesh refinement, multigrid resolution, etc). Beyond the classical benchmarks, a numerical study has been carried out to investigate the influence of a polynomial approximation of the logarithmic free energy and the bifurcations near the first eigenvalue of the Laplace operator. Introduction We consider an isothermal binary alloy of two species A and B, and denote by u ∈ [−1, 1] the ratio between the two components. By thermodynamic arguments, and under a mass conservation property, Cahn and Hilliard described a fourth-order model for the evolution of an isotropic system of nonuniform composition or density. They introduced a free energy density f̄ to define a chemical potential, and use it in the classical transport equation (see [10], [12] and [13]). The total free energy F of the binary alloy is a volume integral on Ω of this free energy density (bulk free energy): F := ∫ Ω f̄(u,∇u,∇u, . . . ) dV. (0.1) They assumed f̄ to be a function of u and its spatial derivatives. A truncated Taylor expansion of f̄ has thus the following general form: f̄(u) ∼ f(u) + L · ∇u+K1 ⊗∇u+∇u ·K2 · ∇u, (0.2) where ∇ is the Nabla operator. By symmetry arguments, they showed that L = ~0 and K1 and K2 are homothetic operators. Moreover they used Neumann boundary condition to cancel the term in ∇u which yields F := ∫ Ω ( f(u) + κ|∇u| ) dV, (0.3) where κ is a parameter (often denoted ε/2) which is referred to as the gradient coefficient. Then, the chemical potential w is defined by: w := f (u)− 2κ∆u. (0.4) ∆ is the Laplace operator. If we denote by J the flux and by M(u) the mobility, the classical Fick law provide the following equations: ∂tu = −∇ · J and J = −M(u)∇w. (0.5) AMS 2000 subject classifications. 65M60, 35K55, 35K65, 35K45, 82C26