In recent years, a variety of meshless methods have been developed to solve partial differential equations in complex domains. Meshless methods discretize the partial differential equations over scattered points instead of grids. Radial basis functions (RBFs) have been popularly used as high accuracy interpolants of function values at scattered locations. In this paper, we apply the polyharmonic splines (PHS) as the RBF together with appended polynomial and solve the heat conduction equation in several geometries using a collocation procedure. We demonstrate the expected exponential convergence of the numerical solution as the degree of the appended polynomial is increased. The method holds promise to solve several different governing equations in thermal sciences.

中文翻译：

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高阶精确无网格方法在复杂几何中的热传导求解中的应用

近年来，已经开发了多种无网格方法来求解复杂域中的偏微分方程。无网格方法在散点而不是网格上离散偏微分方程。径向基函数 (RBF) 已广泛用作分散位置处函数值的高精度插值。在本文中，我们将多调和样条 (PHS) 作为 RBF 与附加多项式一起应用，并使用搭配程序求解多个几何形状的热传导方程。我们证明了随着附加多项式的次数增加，数值解的预期指数收敛。该方法有望解决热科学中的几个不同的控制方程。