Meshfree methods can be collectively classified as Galerkin type and collocation type.
Galerkin type formulation in conjunction with approximation functions with polynomial 
reproductivity yields algebraic convergence. Through three classes of problems at continuum 
macro-scale, meso-scale, and quantum-scale, we demonstrate the convergence properties of 
Galerkin meshfree approach and how it can be constructed to alleviate the numerical 
difficulties associated with the standard finite element methods. The examples include large 
deformation and fragment impact problems, modeling of microstructure evolution, and solution 
of Schrodinger equation in quantum mechanics. Alternatively, radial basis collocation method 
offers exponential convergence. However, the method is suffered from the large condition 
numbers due to its "nonlocal" approximation, and the nonlocality of radial basis function 
limits its application to small scale problems. In the second part of this talk, we show how
to combine the advantages of radial basis function and reproducing kernel function to yield 
a local approximation that is better conditioned than that of the radial basis collocation 
method, while at the same time offers a higher rate of convergence than that of Galerkin type
reproducing kernel method.