A finite element method for Allen-Cahn equation on deforming surface

TL;DR

This paper develops a finite element method for solving the Allen-Cahn equation on deforming surfaces, analyzing its stability and convergence, and linking the solution behavior to geometric flows.

Contribution

It introduces a trace finite element method for Allen-Cahn equations on moving surfaces, with comprehensive stability and convergence analysis.

Findings

The method accurately captures phase separation on evolving surfaces.

The solution converges to a geodesic mean curvature flow in the limit.

The approach handles geometry approximation errors effectively.

Abstract

The paper studies an Allen-Cahn-type equation defined on a time-dependent surface as a model of phase separation with order-disorder transition in a thin material layer. By a formal inner-outer expansion, it is shown that the limiting behavior of the solution is a geodesic mean curvature type flow in reference coordinates. A geometrically unfitted finite element method, known as a trace FEM, is considered for the numerical solution of the equation. The paper provides full stability analysis and convergence analysis that accounts for interpolation errors and an approximate recovery of the geometry.