Backward-Euler convex splitting for Allen–Cahn and Cahn–Hilliard

Let $X_h$ be either finite-dimensional realization in No-flux finite-volume and conforming realizations for phase fields, with real inner product $⟨\cdot ,\cdot {⟩}_h$, constant vector $\mathbf{1}$, self-adjoint nonnegative $A_h$ with kernel $\mathrm{span}\{\mathbf{1}\}$, and $A_h^{†}$ from Kernel, range, and mean-zero inverse for the no-flux Laplacian. Let $P_{c,h},P_{e,h}:X_h\to \mathbb{R}$ be the differentiable convex energy components specified there; ${\mathrm{\nabla }}_{h}P$ denotes the Riesz gradient relative to $⟨\cdot ,\cdot {⟩}_h$. Put ${\mathcal{E}}_h(u)=\frac{\kappa }{2}⟨A_{h}u,u{⟩}_h+P_{c,h}(u)-P_{e,h}(u),{\mu }_h(u,v)=\kappa A_{h}u+{\mathrm{\nabla }}_h{P}_{c,h}(u)-{\mathrm{\nabla }}_h{P}_{e,h}(v)\mathrm{.}$ Given $u^n\in X_h$, $\tau >0$, and $M>0$, the backward-Euler convex-splitting step is:

1. Allen–Cahn branch. Return the minimizer over $v\in X_h$ of $J_{AC}(v)=\frac{1}{2M\tau }\mathrm{\parallel }v-u^n{\mathrm{\parallel }}_h^2+\frac{\kappa }{2}⟨A_{h}v,v{⟩}_h+P_{c,h}(v)-⟨{\mathrm{\nabla }}_h{P}_{e,h}(u^n),v{⟩}_h,$ and set ${\mu }^{n+1}={\mu }_h(u^{n+1},u^n)$.

2. Cahn–Hilliard branch. Restrict to the affine set $⟨v,\mathbf{1}{⟩}_h=⟨u^n,\mathbf{1}{⟩}_h$ and return the minimizer of $J_{CH}(v)=\frac{1}{2M\tau }⟨A_h^{†}(v-u^n),v-u^n{⟩}_h+\frac{\kappa }{2}⟨A_{h}v,v{⟩}_h+P_{c,h}(v)-⟨{\mathrm{\nabla }}_h{P}_{e,h}(u^n),v{⟩}_h,$ then set ${\mu }^{n+1}={\mu }_h(u^{n+1},u^n)$.

For the finite-volume realization, $P_{c,h}(u)={\sum }_K\mathrm{\mid }K\mathrm{\mid }{F}_c(u_K)$ and $P_{e,h}(u)={\sum }_K\mathrm{\mid }K\mathrm{\mid }{F}_e(u_K)$. For the conforming interval realization, these are the exact integrals of $F_c$ and $F_e$ over the reconstructed piecewise-affine state. The recurrence, minimizer equivalence, uniqueness, conservation, and energy bounds are proved in Existence, uniqueness, mass conservation, and energy decay of a convex-splitting step ↗. This is the selected backward-Euler branch; it asserts no ARS(2,2,2), Lie, or Strang property.