论文标题
固有表面有限元法的收敛分析
Convergence analysis of the intrinsic surface finite element method
论文作者
论文摘要
最近提出了固有的表面有限元法（ISFEM）来求解表面上的部分微分方程（PDE）。 ISFEM通过编写有关固定在表面的局部坐标系的PDE来进行的，并直接使用所得的协变量基础。从表面的形状定型三角剖分开始，每个三角形的局部参数化的存在被利用为局部图表上的近似相关数量。标准的二维FEM技术与表面正交规则结合使用，完成了ISFEM公式，从而实现了一种完全固有的方法，并且仅对基础函数的定义有限地使用表面嵌入。但是，尚未证明理论属性。在这项工作中，我们通过仔细跟踪几何数量在误差不平等的常数中的作用来补充ISFEM的原始推导，并提出对稳定性和误差估计的分析。包括数值实验以支持理论结果。
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.
