We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method. Numerical examples are provided for the equation $-\Delta_\mathcal{M} u + u = f$ on the 2- and 3-spheres, where $\Delta_\mathcal{M}$ is the Laplace-Beltrami operator.

翻译：针对任意维无边界光滑紧流形上的椭圆微分方程，我们给出基于核的无网格有限差分方法的最小二乘版本的误差界。特别地，我们获得了该方法收敛的充分条件。对于方程 $-\Delta_\mathcal{M} u + u = f$ 在二维和三维球面上的情形，其中 $\Delta_\mathcal{M}$ 为拉普拉斯-贝尔特拉米算子，文中提供了数值算例。