This paper is delicate to the development of high-order numerical algorithms for solving partial differential equations on point clouds. We introduce a new geometric error analysis framework which requires neither global continuity of surface patches nor exact geometric information, e.g., embedding maps, tangent spaces. The new framework provides us a fundamental tool to analyze discontinuous Galerkin (DG) methods for the Laplace-Beltrami operator on point clouds. It is illustrated using examples of an interior penalty DG method for solving the Laplace-Beltrami equation and the corresponding eigenvalue problem with numerical verification.
翻译:本文对于制定解决点云部分差异方程式的高序数字算法十分微妙。 我们引入了新的几何错误分析框架,既不要求地表补丁的全球连续性,也不要求精确的几何信息,例如嵌入地图、正切空间。新框架为我们提供了一个基本工具,用于分析点云上Laplace-Beltrami操作员的不连续的Galerkin(DG)方法。我们用内部惩罚DG方法解决Laplace-Beltrami方程式和相应的数字核查等离子值问题的例子来说明。