Theoretical analysis of the generalized finite difference method

The generalized finite difference method (GFDM) is a typical meshless collocation method based on the Taylor series expansion and the moving least squares technique. In this paper, we first provide theoretical results of the meshless function approximation in the GFDM. Properties, stability and error estimation of the approximation are studied theoretically, and a stabilized approximation is proposed by revising the computational formulas of the original approximation. Then, we provide theoretical results consisting of error bound and condition number of the GFDM. Numerical results are finally provided to confirm these theoretical results.