A multigrid method for the Cahn–Hilliard equation with obstacle potential

摘要

We present a multigrid finite element method for the deep quench obstacle Cahn–Hilliard equation. The non-smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging. The method exhibits mesh-independent convergence properties in practice for arbitrary time step sizes. In addition, numerical evidence shows that this behaviour extends to small values of the interfacial parameter γ. Several numerical examples are given, including comparisons with existing alternative solution methods for the Cahn–Hilliard equation.