In this paper we derive the discretization of the nonlinear Monge-Ampère equation by means of the explicit formulae of the meshless Generalized Finite Difference Method (GFDM) in both two and three dimensional settings (2D and 3D). To do so we implement the Cascadic iterative algorithm. We provide a rigorous proof of the consistency of the GFDM for this elliptic equation and present several examples in 2D and 3D, where the method shows its potential and accuracy. We provide a discussion on the number of total nodes in the domain and the number of local supporting nodes. Also, we give an example with physical meaning where we find a surface whose Gaussian curvature is given.

在本文中，我们利用二维和二维设置（2D和3D）中的无网格广义有限差分法（GFDM）的显式公式，推导了非线性Monge-Ampère方程的离散化。为此，我们实现了Cascadic迭代算法。我们提供了该椭圆方程的GFDM一致性的严格证明，并提供了2D和3D的几个示例，其中该方法显示了其潜力和准确性。我们讨论了域中的总节点数和本地支持节点数。另外，我们给出一个具有物理意义的示例，在该示例中，我们找到了给出了高斯曲率的曲面。