曲面上收敛的离散Laplace-Beltrami算子
Convergent discrete Laplace-Beltrami operators over surfaces
摘要：
Laplace-Beltrami算子的收敛性问题在一些涉及该算子的重要几何偏微分方程数值模拟的收敛性分析中起着至关重要的作用。本文提出了一种新的计算作用于曲面上函数的离散Laplace-Beltrami算子的有效且收敛的算法。我们证明了我们的离散化的收敛定理。据我们所知，这是曲面上离散Laplace-Beltrami算子对一般曲面上函数的第一个收敛算法。我们的算法是概念简单，易于计算。事实上，我们的新算法的离散Laplace-Beltrami算子在表面上的收敛速度是$O(r)$，其中r表示网格的大小的表面离散。
英文摘要：
The convergence problem of the Laplace-Beltrami operators plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve the operator. In this note we present a new effective and convergent algorithm to compute discrete Laplace-Beltrami operators acting on functions over surfaces. We prove a convergence theorem for our discretization. To our knowledge, this is the first convergent algorithm of discrete Laplace-Beltrami operators over surfaces for functions on general surfaces. Our algorithm is conceptually simple and easy to compute. Indeed, the convergence rate of our new algorithm of discrete Laplace-Beltrami operators over surfaces is $O(r)$ where r represents the size of the mesh of discretization of the surface.
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