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

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

A convergent stochastic scalar auxiliary variable method

We discuss an extension of the scalar auxiliary variable approach which was originally introduced by Shen et al.~([Shen, Xu, Yang, J.~Comput.~Phys., 2018]) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable, this approach allows to derive a linear scheme, while still maintaining unconditional stability. Our extension augments the approximation of the evolution of this scalar auxiliary variable with higher order terms, which enables its application to stochastic partial differential equations. Using the stochastic Allen--Cahn equation as a prototype for nonlinear stochastic partial differential equations with multiplicative noise, we propose an unconditionally energy stable, linear, fully discrete finite element scheme based on our stochastic scalar auxiliary variable method. Recovering a discrete version of the energy estimate and establishing Nikolskii estimates with respect to time, we are able to prove convergence of appropriate subsequences of discrete solutions towards pathwise unique martingale solutions by applying Jakubowski's generalization of Skorokhod's theorem. A generalization of the Gy\"ongy--Krylov characterization of convergence in probability to quasi-Polish spaces finally provides convergence of fully discrete solutions towards strong solutions of the stochastic Allen--Cahn equation