发表机构
National University of Singapore; Universität Tübingen（新加坡国立大学; 蒂宾根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于局部拉回流映射的任意高阶保结构参数有限元方法，用于表面扩散和保体积平均曲率流等几何PDE，通过共形和等分布两种表述实现面积/体积精确保持及周长/表面积耗散。
AI 中文摘要
我们基于局部拉回流映射，为几何PDE（包括表面扩散和保体积平均曲率流）发展了任意高阶保结构参数有限元方法。核心思想是在任意中间超曲面上重新表述几何PDE，并通过局部拉回流映射表示后续演化。利用可容许的局部拉回，我们从拉普拉斯-贝尔特拉米算子的相应拉回恒等式推导出两种表述，分别称为共形表述和等分布表述。我们使用任意次等参有限元和具有正权重的龙格-库塔配点方法对两种表述进行离散化。对于共形表述，法向雅可比向量的加权平均值确保封闭面积或体积的精确保持，而龙格-库塔方法的代数稳定性额外保证周长或表面积的耗散。对于等分布表述，路径平均法向雅可比向量和几何离散梯度保证封闭面积或体积的精确保持以及周长或表面积的耗散，而无需代数稳定性。数值实验证实了预期的高阶精度和保结构性质。
英文摘要
We develop arbitrarily high-order structure-preserving parametric finite element methods for geometric PDEs including surface diffusion and volume-preserving mean curvature flow based on local pullback flow maps. The central idea is to reformulate the geometric PDEs on an arbitrary intermediate hypersurface and represent the subsequent evolution through a local pullback flow map. Using admissible local pullbacks, we derive two formulations from the corresponding pullback identities for the Laplace-Beltrami operator, referred to as the conformal and equidistribution formulations. We discretize both formulations using arbitrary-degree isoparametric finite elements and Runge-Kutta collocation methods with positive weights. For the conformal formulation, weighted averages of the normal-Jacobian vector ensure exact preservation of the enclosed area or volume, while algebraic stability of the Runge-Kutta method additionally guarantees dissipation of the perimeter or surface area. For the equidistribution formulation, a path-averaged normal-Jacobian vector and a geometric discrete gradient guarantee both exact preservation of the enclosed area or volume and dissipation of the perimeter or surface area, without requiring algebraic stability. Numerical experiments confirm the expected high-order accuracy and structure-preserving properties.
