Radial basis function generated finite-difference (RBF-FD) methods have been gained popularity recently. For the approximation order of RBF-FDs' weights on scattered nodes, one can find mathematical theories in the literature. Many practical problems in numerical analysis, however, do not have a uniform node-distribution. Instead, it would be better suited if a relatively higher node-density is imposed on specific areas of domain where complicated physics neeeded to be resolved. In this paper, we propose a practical adaptive RBF-FD with a user defined convergence order with respect to the total number data points $N$. Our algorithm can output a sparse differentiation matrix system with the desired approximation order. Numerical examples about elliptic and parabolic equations are provided to show that the proposed adaptive RBF-FD method yields the expected convergence order. The proposed method reduces the number of non-zero elements in the linear system without sacrificing the accuracy. Furthermore, we apply our adaptive RBFFD method to the elastic wave models and obtain the desired convergence order.

翻译：径向基函数有限差分（RBF-FD）方法近年来广受关注。现有文献已为散射节点上RBF-FD权重的逼近阶建立了数学理论。然而，数值分析中的许多实际问题并不具备均匀节点分布。相反，在需要求解复杂物理过程的特定区域中施加更高的节点密度更为合适。本文提出一种实用的自适应RBF-FD方法，其收敛阶可由用户针对总数据点数$N$定义。该算法可输出具有预期逼近阶的稀疏差分矩阵系统。通过椭圆型和抛物型方程的数值算例验证，所提出的自适应RBF-FD方法能够达到预期的收敛阶。该方法在保持精度的前提下减少了线性系统中非零元素的数目。此外，我们将所提出的自适应RBF-FD方法应用于弹性波模型，并获得了预期的收敛阶。