Degenerate area preserving surface Allen-Cahn equation and its sharp interface limit

• Main Authors: Benes, Michal, Kolar, Miroslav, Sischka, Jan M, Voigt, Axel
• Format: Journal Article
• Language: English
• Published: 07-03-2023
• Subjects: Computer Science - Numerical Analysis; Mathematics - Analysis of PDEs; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.2303.04018

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.