A convergent SAV scheme for Cahn--Hilliard equations with dynamic boundary conditions

TL;DR

This paper introduces a stable, linear finite element scheme for Cahn-Hilliard equations with dynamic boundary conditions, ensuring convergence and demonstrating practical effectiveness through simulations.

Contribution

It develops a new unconditionally stable, linear, fully discrete finite element scheme for Cahn-Hilliard equations with dynamic boundary conditions, with proven convergence.

Findings

The scheme is unconditionally stable and linear.

Convergence to weak solutions is established.

Simulations validate the scheme's practicality.

Abstract

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.