Stabilized Galerkin and Collocation Meshfree Methods

Meshfree methods have been formulated based on Galerkin type weak formulation and collocation type strong formulation. The approximation functions commonly used in the Galerkin based meshfree methods are the moving least-squares (MLS) and reproducing kernel (RK) approximations, while the radial basis functions (RBFs) are usually employed in the strong form collocation method. Galerkin type formulation in conjunction with approximation functions with polynomial reproducibility yields algebraic convergence. Alternatively, strong form collocation method with RBF approximation offers exponential convergence, however the method is suffered from ill-conditioning due to its "nonlocal" approximation. In this work, we discuss stability issues related to nodal integration of Galerkin type meshfree method and ill-conditioning of the radial basis collocation method. We show how to combine the advantages of RBF and RK approximations 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 that that of Galerkin type reproducing kernel method.