In this paper, we provide a theoretical analysis for a preconditioned steepest descent (PSD) iterative solver that improves the computational time of a finite difference numerical scheme for the Cahn-Hilliard equation with Flory-Huggins energy potential. In the numerical design, a convex splitting approach is applied to the chemical potential such that the logarithmic and the surface diffusion terms are treated implicitly while the expansive concave term is treated with an explicit update. The nonlinear and singular nature of the logarithmic energy potential makes the numerical implementation very challenging. However, the positivity-preserving property for the logarithmic arguments, unconditional energy stability, and optimal rate error estimates have been established in a recent work and it has been shown that successful solvers ensure a similar positivity-preserving property at each iteration stage. Therefore, in this work, we will show that the PSD solver ensures a positivity-preserving property at each iteration stage. The PSD solver consists of first computing a search direction (involved with solving a Poisson-like equation) and then takes a one-parameter optimization step over the search direction in which the Newton iteration becomes very powerful. A theoretical analysis is applied to the PSD iteration solver and a geometric convergence rate is proved for the iteration. In particular, the strict separation property of the numerical solution, which indicates a uniform distance between the numerical solution and the singular limit values of $\pm 1$ for the phase variable, plays an essential role in the iteration convergence analysis. A few numerical results are presented to demonstrate the robustness and efficiency of the PSD solver.

翻译：本文对一种预条件最速下降迭代求解器进行了理论分析，该求解器提升了采用Flory-Huggins能量势的Cahn-Hilliard方程有限差分数值格式的计算效率。在数值设计中，对化学势采用了凸分裂方法，使得对数项和表面扩散项被隐式处理，而膨胀凹项则采用显式更新。对数能量势的非线性和奇异特性使得数值实现极具挑战。然而，近期研究已证明了对数项的正性保持特性、无条件能量稳定性及最优阶误差估计，并表明成功的求解器需在每一迭代阶段保证类似的正性保持特性。因此，本工作将证明该预条件最速下降求解器在每一迭代阶段均能确保正性保持特性。该求解器首先计算搜索方向（涉及求解类泊松方程），随后沿搜索方向进行单参数优化步，其中牛顿迭代法将发挥强大效力。我们对预条件最速下降迭代求解器进行了理论分析，并证明了迭代的几何收敛速率。特别地，数值解的严格分离性质——即数值解与相变量奇异极限值±1之间保持均匀距离——在迭代收敛分析中起着关键作用。文中给出了若干数值结果，以验证预条件最速下降求解器的鲁棒性与计算效率。