TY - JOUR
T1 - A Preconditioned Steepest Descent Solver for the Cahn-Hilliard Equation with Variable Mobility
AU - Chen , Xiaochun
AU - Wang , Cheng
AU - Wise , Steven M.
JO - International Journal of Numerical Analysis and Modeling
VL - 6
SP - 839
EP - 863
PY - 2022
DA - 2022/09
SN - 19
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/21036.html
KW - Cahn-Hilliard equation, variable mobility function, convex splitting numerical scheme, energy stability, preconditioned steepest descent iteration solver, iteration convergence analysis.
AB - <p style="text-align: justify;">In this paper we provide a detailed analysis of the preconditioned steepest descent
(PSD) iteration solver for a convex splitting numerical scheme to the Cahn-Hilliard equation
with variable mobility function. In more details, the convex-concave decomposition is applied to
the energy functional, which in turn leads to an implicit treatment for the nonlinear term and
the surface diffusion term, combined with an explicit update for the expansive concave term.
In addition, the mobility function, which is solution-dependent, is explicitly computed, which
ensures the elliptic property of the operator associated with the temporal derivative. The unique
solvability of the numerical scheme is derived following the standard convexity analysis, and the
energy stability analysis could also be carefully established. On the other hand, an efficient
implementation of the numerical scheme turns out to be challenging, due to the coupling of the
nonlinear term, the surface diffusion part, and a variable-dependent mobility elliptic operator.
Since the implicit parts of the numerical scheme are associated with a strictly convex energy,
we propose a preconditioned steepest descent iteration solver for the numerical implementation.
Such an iteration solver consists of a computation of the search direction (involved with a Poisson-like equation), and a one-parameter optimization over the search direction, in which the Newton’s
iteration becomes very powerful. In addition, a theoretical analysis is applied to the PSD iteration
solver, and a geometric convergence rate is proved for the iteration. A few numerical examples
are presented to demonstrate the robustness and efficiency of the PSD solver.</p>