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 case 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 is often referred to as the global, Kansa, or pseudospectral method, while the latter is referred to as the local radial basis function (RBF) finite difference (RBF-FD) method. Both techniques are used extensively for numerically solving certain partial differential equations (PDEs) on spheres (and other domains). For time-dependent PDEs on spheres like the (surface) 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 (i.e., it is Hurwitz stable). 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$X$上配置法生成的正定和条件正定核微分算子的谱性质。我们重点分析两类情形：基于所有$X$点构建的全局配置矩阵，以及基于局部子集（模板）构建的局部配置矩阵。前者通常称为全局法、Kansa法或伪谱法，后者则称为局部径向基函数有限差分法（RBF-FD）。这两种方法广泛应用于球面（及其他区域）上偏微分方程（PDE）的数值求解。对于球面上依赖时间的偏微分方程（如表面扩散方程），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，我们给出了与Laplace-Beltrami算子可交换的算子离散化微分矩阵的扰动估计。由此证明：若此类算子谱为负（非正），则微分矩阵谱也具有相同性质（即满足Hurwitz稳定性）。对于条件正定核，由于微分矩阵未必可对角化，该结论的证明尤为困难。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有效性。