We prove the convergence of meshfree method for solving the elliptic Monge-Ampere equation with Dirichlet boundary on the bounded domain. L2 error is obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. We obtain the convergence result when the testing discretization is finer than the trial discretization. The convergence rate depend on the regularity of the solution, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces. The presented convergence theory covers a wide range of kernel-based trial spaces including stationary approximation and non-stationary approximation. An extension to non-Dirichlet boundary condition is in a forthcoming paper.

翻译：我们证明了在有界区域上，采用无网格方法求解带Dirichlet边界的椭圆型Monge-Ampère方程的收敛性。基于紧支径向基函数生成的核函数试验空间，获得了L2误差估计。当测试离散化精细于试验离散化时，我们得到了收敛结果。收敛速度依赖于解的正则性、计算域的光滑性以及缩放核函数空间的逼近特性。本文提出的收敛理论涵盖了包括平稳逼近与非平稳逼近在内的广泛核函数试验空间。关于非Dirichlet边界的推广将在后续论文中给出。