On the approximability and the selection of particle shape functions

Particle methods, also known as meshless or meshfree methods, have become popular in approximating solutions of partial differential equations, especially in the engineering community. These methods do not employ a mesh, or use it minimally, in the construction of shape functions. There is a wide variety of classes of shape functions that can be used in particle methods. In this paper, we primarily address the issue of selecting a class of shape functions, among this wide variety, that would yield efficient approximation of the unknown solution. We have also made several comments and observations on the order of convergence of the interpolation error, when these shape functions are used; specifically, we have shown that the interpolation error estimate, for certain classes of shape functions, may not indicate the actual order of convergence of the approximation error.