作为数值求解偏微分方程的方法之一，径向基函数无网格方法不仅避免了在计算区域生成网格，而且更加适合求解高维问题。但是采用该方法离散偏微分方程时，导出的离散代数系统有很坏的条件数。尽管紧支集径向基函数有稀疏的离散矩阵，但是存在支集大小与逼近精度之间的矛盾，即所谓的‘trade-off’难题。该问题一直没有得到很好解决。为了既能保持好的逼近精度同时又能拥有稀疏的代数系统，本研究将在散乱数据集上使用快速多水平算法。通过分裂再生核Hilbert空间，建立等级的径向基函数，得到一族嵌套的函数空间。我们将估计等级径向基函数离散矩阵的条件数，以及分析嵌套函数空间上的瀑布型无网格算法的逼近性质。瀑布型无网格算法的好处在于能够避免网格生成的巨大工作量和冗繁的插值算子的构造，加速径向基函数无网格离散系统的代数求解，从而提升用径向基函数无网格方法数值求解偏微分方程的计算效率。

As one of the numerical methods for partial differential equations, the radial basis functions meshfree method not only avoids the mesh generation in computing domain, but also is more suitable for solving high-dimensional problems. However, the discrete algebraic system produced by it for partial differential equations is ill-conditioned. Although the discrete matrix produced by compactly supported radial basis functions is sparse, there still exists contradiction between the size of support and the accuracy of approximation, which is so called ‘trade-off’ dilemma. This problem has not been well resolved up to now. In order to maintain better approximation accuracy with sparse algebraic system, in this project we will use the fast multilevel algorithms on the scattered data set. We will construct the hierarchical radial basis functions by splitting the reproducing kernel Hilbert spaces, and get a nested family of function spaces. We will estimate the condition number of the discrete matrix produced by hierarchical radial basis functions, and analyze the approximation properties of the cascadic meshfree algorithm on nested function spaces. The advantage of the cascadic meshfree algorithm is that it can avoid the huge amount of work of mesh generation and the tedious work to construct interpolation operators. It will speed up the solution of discrete system produced by radial basis functions, thereby enhance the computational efficiency for solving partial differential equations by radial basis functions meshfree method.