论文标题
对应用于Helmholtz问题的全球径向基函数搭配方法的研究
An investigation of global radial basis function collocation methods applied to Helmholtz problems
论文作者
论文摘要
具有针对部分微分方程（PDES）在一般几何形状中起作用的无限平滑基函数的全局径向基函数（RBF）搭配方法，并且可以具有用于平滑溶液函数的指数收敛属性。同时，出现的线性系统是密集的，并且针对大量未知数和形状参数的小值严重不良，以决定基本函数的平坦程度。我们使用Helmholtz方程作为理论分析和数值实验的应用问题。我们分析和表征收敛属性是未知数和不同形状参数范围的函数。我们为PDE溶液的平坦极限提供了理论结果，并研究了何时非对称搭配矩阵变得单数。我们还提供了选择方法参数的实用策略，并在弯曲的波导几何形状中评估有关Helmholtz问题的结果。
Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution functions. At the same time, the linear systems that arise are dense and severely ill-conditioned for large numbers of unknowns and small values of the shape parameter that determines how flat the basis functions are. We use Helmholtz equation as an application problem for the theoretical analysis and numerical experiments. We analyse and characterise the convergence properties as a function of the number of unknowns and for different shape parameter ranges. We provide theoretical results for the flat limit of the PDE solutions and investigate when the non-symmetric collocation matrices become singular. We also provide practical strategies for choosing the method parameters and evaluate the results on Helmholtz problems in a curved waveguide geometry.
