Projection method for steady states of Cahn–Hilliard equation with the dynamic boundary condition

The Cahn–Hilliard equation is a fundamental model that describes phase separation processes of two-phase flows or binary mixtures. In recent years, the dynamic boundary conditions for the Cahn–Hilliard equation have been proposed and analyzed. Our first goal in this article is to present a projection method to locate the steady state of the CH equation with dynamic boundary conditions. The main feature of this method is that it only uses the variational derivative in the metric $L^2$ and not that in the metric $H^{-1}$, thus significantly reducing the computational cost. In addition, the projected dynamics fulfill the important physical properties: mass conservation and energy dissipation. In the temporal construction of the numerical schemes, the convex splitting method is used to ensure a large time step size. Numerical experiments for the two-dimensional Ginzburg–Landau free energy, where the surface potential is the double well potential or the moving contact line potential, are conducted to demonstrate the effectiveness of this projection method.