A fully discrete evolving surface finite element method for the Cahn-Hilliard equation with a regular potential

TL;DR

This paper develops and analyzes two fully discrete finite element schemes for solving the Cahn-Hilliard equation on evolving surfaces, providing optimal error bounds for both implicit and semi-implicit time discretizations.

Contribution

It introduces and rigorously analyzes two novel fully discrete evolving surface finite element methods for the Cahn-Hilliard equation with polynomial growth potential.

Findings

Optimal order error bounds established for backward Euler scheme

Optimal error bounds proven for implicit-explicit scheme

Effective discretization of evolving surface Cahn-Hilliard problem

Abstract

We study two fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation on an evolving surface, given a smooth potential with polynomial growth. In particular we establish optimal order error bounds for a (fully implicit) backward Euler time-discretisation, and an implicit-explicit time-discretisation, with isoparametric surface finite elements discretising space.