Abstract error analysis for Cahn--Hilliard type equations with dynamic boundary conditions

TL;DR

This paper develops an abstract formulation and finite element error analysis for the numerical solution of Cahn-Hilliard equations with dynamic boundary conditions, extending the variational framework to a broader class of problems.

Contribution

It introduces a new variational formulation for Cahn-Hilliard equations with dynamic boundary conditions and proves error bounds for finite element discretizations.

Findings

Error bounds established for finite element semidiscretization.

Variational formulation applicable to a broader class of problems.

Spaces adapted for dynamic boundary conditions.

Abstract

This work addresses the problem of solving the Cahn-Hilliard equation numerically. For that we introduce an abstract formulation for Cahn-Hilliard type equations with dynamic boundary conditions, we conduct the spatial semidiscretization via finite elements and prove error bounds based on the technique of energy estimates. The variational formulation for Cahn-Hilliard/Cahn-Hilliard coupling, will apply to a larger abstract class of problems and is similar to the usual weak formulation of parabolic problems. In contrast to problems with non dynamic boundary conditions, the Hilbert spaces $L^2(\Omega)$ and $H^1(\Omega)$ are exchanged with the spaces $L^2(\Omega)\times L^2(\Gamma)$ and $\lbrace v\in H^1(\Omega): \gamma v \in H^1(\Gamma)\rbrace$, respectively. Because we are considering a fourth-order differential equation, which will be described by a system of two second-order differential equations, the variational formulation also consists of a system of two equations.