  The Cahn-Hilliard equation is one of the most common models to describe phase
separation processes in mixtures of two materials. For a better description of
short-range interactions between the material and the boundary, various dynamic
boundary conditions for this equation have been proposed. Recently, a family of
models using Cahn-Hilliard-type equations on the boundary of the domain to
describe adsorption processes was analysed (cf. Knopf, Lam, Liu, Metzger,
ESAIM: Math. Model. Numer. Anal., 2021). This family of models includes the
case of instantaneous adsorption processes studied by Goldstein, Miranville,
and Schimperna (Physica D, 2011) as well as the case of vanishing adsorption
rates which was investigated by Liu and Wu (Arch. Ration. Mech. Anal., 2019).
In this paper, we are interested in the numerical treatment of these models and
propose an unconditionally stable, linear, fully discrete finite element scheme
based on the scalar auxiliary variable approach. Furthermore, we establish the
convergence of discrete solutions towards suitable weak solutions of the
original model. Thereby, when passing to the limit, we are able to remove the
auxiliary variables introduced in the discrete setting completely. Finally, we
present simulations based on the proposed linear scheme and compare them to
results obtained using a stable, non-linear scheme to underline the
practicality of our scheme.
