In this article, we developed a computationally efficient multilevel local radial basis function (RBF-FD) mesh-free algorithm. The algorithm provides a new strategy to get good order of accuracy with less computational time, which is most important in the present world. The main idea is the layer-by-layer calculation and then layer-by-layer correction from coarsest level to finest level node points. Numerical experiments are presented to verify the accuracy and efficiency of our developed algorithm with two-dimensional Poisson equation and vorticity–stream function of the incompressible Navier–Stokes equations. The flow inside a lid-driven cavity constitutes a classical benchmark problem, due to its unique boundary conditions that allow comparing any new method’s efficiency for solving Navier–Stokes equations for internal flows. Numerical results are presented through the figures and tables to demonstrate accuracy, efficiency, and convergence of the method. The developed scheme saves at least 60% of CPU time for Poisson equation and 59% of the CPU time for Navier–Stokes equation than the usual local RBF method. The iteration matrix of the proposed local RBF method satisfies the necessary and sufficient condition for convergence.

在本文中，我们开发了一种计算效率高的多层局部径向基函数（RBF-FD）无网格算法。该算法提供了一种新的策略，可以在较少的计算时间下获得良好的精度，这在当前世界中最为重要。主要思想是逐层计算，然后进行从最粗糙级别到最精细级别节点的逐层校正。提出了数值实验，以验证我们使用二维Poisson方程和不可压缩的Navier-Stokes方程的涡流函数产生的算法的准确性和效率。由于其独特的边界条件，盖驱动腔内部的流动构成了一个经典的基准问题，可以比较任何新方法求解内部流动的Navier–Stokes方程的效率。通过图表显示数值结果，以证明该方法的准确性，效率和收敛性。与通常的本地RBF方法相比，开发的方案为Poisson方程节省了至少60％的CPU时间，为Navier–Stokes方程节省了59％的CPU时间。所提出的局部RBF方法的迭代矩阵满足收敛的充要条件。