The Intrinsic Surface Finite Element Method (ISFEM) was recently proposed to solve Partial Differential Equations (PDEs) on surfaces. ISFEM proceeds by writing the PDE with respect to a local coordinate system anchored to the surface and makes direct use of the resulting covariant basis. Starting from a shape-regular triangulation of the surface, existence of a local parametrization for each triangle is exploited to approximate relevant quantities on the local chart. Standard two-dimensional FEM techniques in combination with surface quadrature rules complete the ISFEM formulation thus achieving a method that is fully intrinsic to the surface and makes limited use of the surface embedding only for the definition of basis functions. However, theoretical properties have not yet been proved. In this work we complement the original derivation of ISFEM with its complete convergence theory and propose the analysis of the stability and error estimates by carefully tracking the role of the geometric quantities in the constants of the error inequalities. Numerical experiments are included to support the theoretical results.

翻译：内禀曲面有限元法（ISFEM）是近期提出的用于求解曲面上偏微分方程（PDEs）的数值方法。ISFEM 通过将 PDE 相对于锚定在曲面上的局部坐标系进行表述，并直接利用由此产生的协变基来实现。该方法从曲面的形状正则三角剖分出发，利用每个三角形存在局部参数化的特性，在局部坐标卡上近似相关物理量。结合曲面求积法则的标准二维有限元技术完善了 ISFEM 的表述，从而形成了一种完全内禀于曲面的方法——仅在对基函数进行定义时有限地利用曲面的嵌入信息。然而，其理论性质尚未得到证明。本文在 ISFEM 原始推导的基础上，补充建立了完整的收敛理论，并通过细致追踪几何量在误差不等式常数中的作用，提出了稳定性分析与误差估计的论证框架。数值实验被纳入以支撑理论结果。