This paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov–Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation and the use of local weak forms instead of a global variational formulation. The local Petrov–Galerkin formulation allows to compute the integrals on small independent spherical caps without any dependence on a connected background mesh. Experimental results show the accuracy and the efficiency of the new method.

中文翻译：

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球面上扩散方程的Petrov-Galerkin RBF方法

本文涉及单位球上扩散方程的数值解。给定的方法基于球基函数逼近和Petrov-Galerkin测试离散化。该方法是无网格的，因为既不需要球面三角剖分，也不需要数值积分。此功能是通过球基函数逼近和使用局部弱形式而不是全局变分公式来实现的。本地Petrov-Galerkin公式允许在独立的小球形帽上计算积分，而无需依赖于连接的背景网格。实验结果表明了该方法的准确性和有效性。