Finite-Volume Analysis for the Cahn-Hilliard Equation with Dynamic Boundary Conditions

Abstract

This work is devoted to the convergence analysis of a finite-volume approximation of the 2D Cahn-Hilliard equation with dynamic boundary conditions. The method that we propose couples a 2d-finite-volume method in a bounded, smooth domain $\varOmega \subset \mathbb R^2$ and a 1d-finite-volume method on $\partial \varOmega$. We prove convergence of the sequence of approximate solutions. One of the main ingredient is a suitable space translation estimate that gives a limit in $L^\infty \left( 0,T,H^1(\varOmega )\right)$ whose trace is in $L^\infty \left( 0,T,H^1(\partial \varOmega )\right)$.