论文标题
求解PDE的Unity（D-RBF-PU）方法的直接径向基函数分区
The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs
论文作者
论文摘要
在本文中，提出了一种基于Unity（PU）分配的新的局部径向基函数（RBF）方法，以解决边界和初始有限值问题。新方法得益于直接离散方法，被称为“统一直接RBF分区（D-RBF-PU）”方法。由于避免了PU重量函数的所有衍生物以及局部近似值的所有下衍生物，因此新方法比标准RBF-PU方法更快，更简单。此外，现在可以利用不连续的PU重量功能以更高效，更便宜的方式开发该方法。另外，新方法是PU设置中的RBF生成的有限差（RBF-FD）方法，该方法比原始RBF-FD更快，在某些情况下更准确。多谐波花纹用于局部近似，并且考虑了误差和稳定性问题。对不规则2D和3D域以及成本比较测试进行了一些数值实验，以支持理论分析并显示新方法的效率。
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.
