Stabilized semi-implicit phase-field step

On either discrete realization of No-flux finite-volume and conforming realizations for phase fields, write $X_h={\mathbb{R}}^{\mathcal{K}}$ for the finite-volume state space and $X_h=V_h$ for the conforming interval state space; use its inner product $⟨\cdot ,\cdot {⟩}_h$ and the energy components $E_{c,h},E_{e,h}$ defined there. Let $g_h(u)={\mathrm{\nabla }}_h(E_{c,h}-E_{e,h})(u)$ be the Riesz gradient of the full potential energy. In the finite-volume realization, $(g_h(u){)}_K=f_{\epsilon }(u_K)$. In the conforming interval realization, $g_h(u)\in V_h$ is defined by $⟨g_h(u),v_h{⟩}_h={\int }_0^L{f}_{\epsilon }(u_h(x))v_h(x)dx(v_h\in V_h),$ where $⟨\cdot ,\cdot {⟩}_h$ is the exact $L^2$ inner product. Given a previous state $u^n\in X_h$, parameters $\epsilon ,\kappa ,M,\tau >0$, and a stabilization parameter $S\ge 0$, the stabilized semi-implicit Allen–Cahn step is the linear equation $\frac{u^{n+1}-u^n}{\tau }=-M(\kappa A_h{u}^{n+1}+g_h(u^n)+S(u^{n+1}-u^n))\mathrm{.}$ For the same input, the stabilized semi-implicit Cahn–Hilliard step is $\frac{u^{n+1}-u^n}{\tau }=-M{A}_h{\mu }^{n+1},{\mu }^{n+1}=\kappa A_h{u}^{n+1}+g_h(u^n)+S(u^{n+1}-u^n)\mathrm{.}$ These formulas define alternative updates with an explicitly lagged nonlinear potential and an added increment term. No energy law, unconditional solvability, maximum principle, or error estimate is asserted here for the quartic double well; any such property requires its own hypotheses and proof. In particular, no stability estimate proved for the convex-splitting update is silently transferred to this variant.