Overcoming logarithmic singularities in the Cahn-Hilliard equation with Flory-Huggins potential: An unconditionally convergent ADMM approach

The Cahn-Hilliard equation with Flory-Huggins potential serves as a fundamental phase field model for describing phase separation phenomena. Due to the presence of logarithmic singularities at $u=\pm 1$, the solution $u$ is constrained within the interval $(-1,1)$. While convex splitting schemes are commonly employed to preserve this bound and guarantee unconditional unique solvability, their practical implementation requires solving nonlinear systems containing singular logarithmic terms at each time step. This introduces significant challenges in both ensuring convergence of iterative solvers and maintaining the solution bounds throughout the iterations. Existing solvers often rely on restrictive conditions -- such as the strict separation property or small time step sizes -- to ensure convergence, which can limit their applicability. In this work, we introduce a novel iterative solver that is specifically designed for singular nonlinear systems, with the use of a variant of the alternating direction method of multipliers (ADMM). By developing a tailored variable splitting strategy within the ADMM framework, our method efficiently decouples the challenging logarithmic nonlinearity, enabling effective handling of singularities. Crucially, we rigorously prove the unconditional convergence of our ADMM-based solver, which removes the need for time step constraints or strict separation conditions. This allows us to fully leverage the unconditional solvability offered by convex splitting schemes. Comprehensive numerical experiments demonstrate the superior efficiency and robustness of our ADMM variant, strongly validating both our algorithmic design and theoretical results.