# An efficient and convergent finite element scheme for Cahn--Hilliard equations with dynamic boundary conditions

## TL;DR

This paper introduces an efficient, unconditionally energy-stable finite element scheme for the Cahn--Hilliard equation with dynamic boundary conditions, demonstrating convergence and practical effectiveness through simulations.

## Contribution

It proposes a novel finite element method for a recent Cahn--Hilliard model with boundary dynamics, establishing convergence and energy stability.

## Key findings

- Scheme is unconditionally energy stable.
- Convergence towards weak solutions is proven.
- Numerical simulations confirm practicality and convergence order.

## Abstract

The Cahn--Hilliard equation is a widely used model that describes amongst others phase separation processes of binary mixtures or two-phase flows. In the recent years, different types of boundary conditions for the Cahn--Hilliard equation were proposed and analyzed. In this publication, we are concerned with the numerical treatment of a recent model which introduces an additional Cahn--Hilliard type equation on the boundary as closure for the Cahn--Hilliard equation in the domain [C. Liu, H. Wu, Arch. Ration. Mech. An., 2019]. By identifying a mapping between the phase-field parameter and the chemical potential inside of the domain, we are able to postulate an efficient, unconditionally energy stable finite element scheme. Furthermore, we establish the convergence of discrete solutions towards suitable weak solutions of the original model. This serves also as an additional pathway to establish existence of weak solutions. Furthermore, we present simulations underlining the practicality of the proposed scheme and investigate its experimental order of convergence.