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：我们认为,正式匹配的随机学表明,一个退化的区域在保护Allen-Cahn-Cahn-Cayn等方程式上与保持大地测量曲线流的区域的尖锐界面界限相融合。这种退化性来自一个Gennes-Cahn-Hilliard的表面能量,对于从数字上解决对地表几何特性的依赖至关重要。我们实验性地展示了数字算法的趋同性,该算法考虑到一个图形的配方、适应性有限的元素和时间上的半隐含的离散,并且使用了尖锐界面界限的数字解决方案,也在图表的配方中被考虑,作为基准解决方案。</s>