Numerical approximations and error analysis of the cahn-hilliard equation with dynamic boundary conditions

We consider the numerical approximations of the Cahn-Hilliard equation with dynamic boundary conditions [C. Liu et. al., Arch. Ration. Mech. Anal., 2019]. We propose a first-order in time, linear and energy-stable numerical scheme, which is based on the stabilized linearly implicit approach. The energy stability of the scheme is proved and the semi-discrete-in-time error estimates are carried out. Numerical experiments, including the comparison with the former work, the accuracy tests with respect to the time step size and the shape deformation of a droplet, are performed to validate the accuracy and the stability of the proposed scheme.