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  • Convergence Analysis of a Preconditioned Steepest Descent Solver for the Cahn-Hilliard Equation with Logarithmic Potential

This content will become publicly available on June 1, 2026

Title: Convergence Analysis of a Preconditioned Steepest Descent Solver for the Cahn-Hilliard Equation with Logarithmic Potential
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 (which requires solving a constant-coefficient 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 ±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.  more » « less
Award ID(s):
2110768
PAR ID:
10647144
Author(s) / Creator(s):
; ;
Publisher / Repository:
Institute for Scientific Computing and Information
Date Published:
Journal Name:
International Journal of Numerical Analysis and Modeling
Volume:
22
Issue:
4
ISSN:
1705-5105
Page Range / eLocation ID:
483 to 509
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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