Error Bounds for a Least Squares Meshless Finite Difference Method on Closed Manifolds

Abstract: 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.