Degenerate Area Preserving Surface Allen-Cahn Equation and Its Sharp Interface Limit

Abstract

We consider formal matched asymptotics to show the convergence of a degenerate area preserving surface Allen-Cahn equation to its sharp interface limit of area preserving geodesic curvature flow. The degeneracy results from a surface de Gennes-Cahn-Hilliard energy and turns out to be essential to numerically resolve the dependency of the solution on geometric properties of the surface. We experimentally demonstrate convergence of the numerical algorithm, which considers a graph formulation, adaptive finite elements and a semi-implicit discretization in time, and uses numerical solutions of the sharp interface limit, also considered in a graph formulation, as benchmark solutions. The results provide the mathematical basis to explore the impact of curvature on cells and their collective behaviour. This is essential to understand the physical processes underlying morphogenesis.