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

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.

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Cited by in corpus (5)

• Well-posedness of a bulk-surface convective Cahn--Hilliard system with dynamic boundary conditions
• Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn-Hilliard systems with bounded mass source
• Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials
• Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise
• A convergent stochastic scalar auxiliary variable method