Stability and error estimates for non-linear Cahn-Hilliard-type equations on evolving surfaces

TL;DR

This paper develops stability and error estimates for a non-linear Cahn-Hilliard-type equation on evolving surfaces, using high-order finite elements and a novel stability proof to ensure optimal accuracy over time.

Contribution

It introduces a new stability analysis for non-linear surface PDEs with evolving geometries, preserving anti-symmetric structure and providing uniform error bounds.

Findings

Optimal-order error estimates are proven.

Numerical experiments confirm theoretical results.

The method maintains structure-preserving discretization.

Abstract

In this paper, we consider a non-linear fourth-order evolution equation of Cahn-Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix-vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.