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(PSD)迭代求解器进行了理论分析，该求解器可提升Flory-Huggins势能Cahn-Hilliard方程有限差分数值格式的计算效率。在数值设计中，对化学势采用凸分裂方法，使对数项和表面扩散项隐式处理，而将扩张性凹项进行显式更新。由于对数能量势的非线性和奇异特性，数值实现极具挑战性。然而，近期研究已确立了对数参量保持正定性、无条件能量稳定性及最优阶误差估计，并证明成功求解器需在每个迭代阶段保持类似的正定性。因此，本文将证明PSD求解器能在各迭代阶段保持正定性。该求解器先计算搜索方向（需求解类泊松方程），再沿搜索方向执行单参数优化步骤，其中牛顿迭代性能卓越。对PSD迭代求解器进行理论分析，证明其具有几何收敛速率。特别地，数值解的严格分离性质（即相位变量数值解与奇异极限值±1之间存在均匀距离）在迭代收敛性分析中起关键作用。最后给出数值结果以验证PSD求解器的鲁棒性和高效性。