论文标题
具有动态边界条件的CAHN- hilliard方程的收敛SAV方案
A convergent SAV scheme for Cahn--Hilliard equations with dynamic boundary conditions
论文作者
论文摘要
Cahn-Hilliard方程是描述两种材料混合物中相位分离过程的最常见模型之一。为了更好地描述材料与边界之间的短距离相互作用，已经提出了该方程的各种动态边界条件。最近，分析了使用域边界上使用Cahn-Hilliard型方程的模型家族来描述吸附过程（参见Knopf，Lam，Liu，Metzger，Esaim，Esaim：Math。Model。Numer。Numer。Anal。，2021年）。这个模型家族包括戈德斯坦，米兰维尔和奇伯纳研究的瞬时吸附过程（Physica d，2011年），以及消失的吸附速率的案例，这是由Liu and Wu（Arch。Cration。Mech。Mech。Anal。，2019年）进行了研究。在本文中，我们对这些模型的数值处理感兴趣，并提出了基于标量辅助变量方法的无条件稳定，线性，完全离散的有限元方案。此外，我们建立了离散解决方案朝着原始模型的合适弱解决方案的收敛性。因此，当传递到限制时，我们能够完全删除离散设置中引入的辅助变量。最后，我们根据提出的线性方案提出模拟，并将其与使用稳定的非线性方案获得的结果进行比较，以强调我们方案的实用性。
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.
