An evolving surface finite element method for the Cahn-Hilliard equation with a logarithmic potential

TL;DR

This paper develops and analyzes evolving surface finite element methods for the Cahn-Hilliard equation with a logarithmic potential, providing error bounds and numerical validation.

Contribution

It introduces semi-discrete and fully discrete schemes for the Cahn-Hilliard equation on evolving surfaces with rigorous error analysis and numerical results.

Findings

Established $L^2_{H^1}$ error bounds for the schemes

Validated the methods with numerical experiments

Provided geometric assumptions for surface evolution

Abstract

In this paper we study semi-discrete and fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation with a logarithmic potential. Specifically we consider linear finite elements discretising space and backward Euler time discretisation. Our analysis relies on a specific geometric assumption on the evolution of the surface. Our main results are $L^2_{H^1}$ error bounds for both the semi-discrete and fully discrete schemes, and we provide some numerical results.