When Lagrange interpolation function is introduced as the approximation function, since it has Kronecker delta property, the essential boundary conditions can be simply and exactly imposed like finite element method (FEM), which further improves the accuracy of this method. When RK approximation is introduced, gradient reproducing kernel (GRK) approximations can be utilized to promote the efficiency, especially for the problems governed by high order partial differential equations. This property leads to the high accuracy and optimal convergence for the proposed method. The presented method can satisfy the high order integration constraints which can conserves the high order consistency conditions in the integration form. The functions which can satisfy the high order consistency conditions such as reproducing kernel (RK) function and Lagrange interpolation function can be utilized as the approximation function. Therefore, to combine the advantages of Galerkin-type and collocation-type meshfree methods, we propose a new meshfree method - stabilized collocation method (SCM). On the other hand, since most approximation functions in meshfree methods are rational functions, it’s quite hard to achieve high-accuracy integration with high efficiency in Galerkin-type meshfree methods. Roughly, these methods can be divided into two categories: Galerkin-type meshfree methods based on weak form and collocation-type meshfree methods based on strong form. On one hand, direct collocation method (DCM) suffers low accuracy and instability compared with Galerkin-based meshfree methods. Meshfree methods made big progress in the past decades which have been applied to many areas.

Meshfree methods lectures