Abstract
We develop efficient energy stable numerical methods for solving isotropic and strongly anisotropic Cahn–Hilliard systems with the Willmore regularization. The scheme, which involves adaptive mesh refinement and a nonlinear multigrid finite difference method, is constructed based on a convex splitting approach. We prove that, for the isotropic Cahn–Hilliard system with the Willmore regularization, the total free energy of the system is non-increasing for any time step and mesh sizes. A straightforward modification of the scheme is then used to solve the regularized strongly anisotropic Cahn–Hilliard system, and it is numerically verified that the discrete energy of the anisotropic system is also non-increasing, and can be efficiently solved by using the modified stable method. We present numerical results in both two and three dimensions that are in good agreement with those in earlier work on the topics. Numerical simulations are presented to demonstrate the accuracy and efficiency of the proposed methods.
•We develop efficient stable schemes for solving isotropic and strongly anisotropic Cahn–Hilliard–Willmore systems.•The scheme is constructed based on a convex splitting approach, and is proven to be unconditionally energy stable for the isotropic system.•Numerical results suggest the anisotropic scheme is also energy stable.•This is a successful effort in solving the regularized isotropic/anisotropic Cahn–Hilliard systems using a convex splitting technique and nonlinear multigrid method.