In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method is benefited from a direct discretization approach and is called the `direct RBF partition of unity (D-RBF-PU)' method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, the discontinuous PU weight functions can now be utilized to develop the method in a more efficient and less expensive way. Alternatively, the new method is an RBF-generated finite difference (RBF-FD) method in a PU setting which is much faster and in some situations more accurate than the original RBF-FD. The polyharmonic splines are used for local approximations, and the error and stability issues are considered. Some numerical experiments on irregular 2D and 3D domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method.

中文翻译：

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用于求解偏微分方程的直接径向基函数统一划分 (D-RBF-PU) 方法

在本文中，提出了一种新的基于单位划分（PU）的局部径向基函数（RBF）方法来解决边界和初边界值问题。新方法受益于直接离散化方法，称为“统一的直接 RBF 划分（D-RBF-PU）”方法。由于避免了 PU 权重函数的所有导数以及局部近似的所有低导数，新方法比标准 RBF-PU 方法更快、更简单。此外，现在可以利用不连续的 PU 权重函数以更有效和更便宜的方式开发该方法。或者，新方法是在 PU 设置中的 RBF 生成的有限差分 (RBF-FD) 方法，它比原始 RBF-FD 更快且在某些情况下更准确。多谐样条用于局部逼近，并考虑了误差和稳定性问题。在不规则的 2D 和 3D 域上进行了一些数值实验，以及成本比较测试，以支持理论分析并展示新方法的效率。