We investigate the spectrum of differentiation matrices for certain operators on the sphere that are generated from collocation at a set of scattered points $X$ with positive definite and conditionally positive definite kernels. We focus on the cases where these matrices are constructed from collocation using all the points in $X$ and from local subsets of points (or stencils) in $X$. The former case are called global methods (e.g., the Kansa or radial basis function (RBF) pseudospectral method), while the latter are referred to as local methods (e.g., the RBF finite difference (RBF-FD) method). Both techniques are used extensively for numerically solving certain partial differential equations on spheres, as well as other domains. For time-dependent PDEs like the diffusion equation, the spectrum of the differentiation matrices and their stability under perturbations are central to understanding the temporal stability of the underlying numerical schemes. In the global case, we present a perturbation estimate for differentiation matrices which discretize operators that commute with the Laplace-Beltrami operator. In doing so, we demonstrate that if such an operator has negative (non-positive) spectrum, then the differentiation matrix does, too. For conditionally positive definite kernels this is particularly challenging since the differentiation matrices are not necessarily diagonalizable. This perturbation theory is then used to obtain bounds on the spectra of the local RBF-FD differentiation matrices based on the conditionally positive definite surface spline kernels. Numerical results are presented to confirm the theoretical estimates.

翻译：本文研究球面上由散乱点集$X$处配点法生成的若干算子微分矩阵的谱性质，其中核函数为正定核与条件正定核。我们重点分析基于$X$中全部点的全局配点构造（如Kansa方法或径向基函数伪谱法）与基于局部子集（即模板）的局部构造（如径向基函数有限差分法）。这两种方法广泛用于球面及其他区域上偏微分方程的数值求解。对于扩散方程等含时偏微分方程，微分矩阵的谱及其扰动稳定性是理解数值格式时间稳定性的核心。在全局情形中，我们给出了离散化与拉普拉斯-贝尔特拉米算子可交换算子的微分矩阵的扰动估计，并证明若此类算子谱为负（非正），则微分矩阵亦然。对于条件正定核，由于微分矩阵未必可对角化，该分析尤为困难。进而利用该扰动理论，基于条件正定曲面样条核，获得了局部RBF-FD微分矩阵谱的界。数值实验验证了理论估计的准确性。