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.

翻译：我们采用形式匹配渐近分析方法，证明退化保面积曲面Allen-Cahn方程收敛至其尖锐界面极限——保面积测地曲率流。该退化现象源于表面de Gennes-Cahn-Hilliard能量，且对数值求解曲面几何特性对解的影响至关重要。我们通过实验验证了数值算法的收敛性，该算法采用图形式、自适应有限元方法和半隐式时间离散格式，并以同样基于图形式的尖锐界面极限数值解作为基准解。