Numerical solutions of meshless methods

Meshless method is a recently-developed new numerical approach. Many different meshless methods have been constructed based on different approximation function methods and different discretized schemes of partial differential equations (PDEs). In this paper, its basic principle was introduced and the construction methods for the approximation function and the discretization of the partial differential equations were presented. The characteristics of weight function and the shape function were analyzed in detail taking the moving least-squares (MLS) method as an example. Results show that radial basis function (RBF) and point interpolation methods possess Kronecker Delta function property (δ function), but the robustness is poor in some cases; the MLS approximation function does not possess Kronecker Delta function property, but it has good robustness. Differences among the three discretization schemes of meshless method are as follows: the collocation method requires no numerical integration and very little computational time while its robustness is poor; Galerkin method is not a truly meshless method due to the background meshes required for integration; the Petrov-Galerkin method is a truly meshless method and need numerical integration in each sub-domain, so it needs more computational time.