Recent developments in stabilized Galerkin and collocation meshfree methods

Abstract

Meshfree methods have been developed based on Galerkin type weak formulation and strong formulation with collocation. Galerkin type formulation in conjunction with the compactly supported approximation functions and polynomial reproducibility yields algebraic convergence, while strong form collocation method with nonlocal approximation such as radial basis functions offers exponential convergence. In this work, we discuss rank instability resulting from the nodal integration of Galerkin type meshfree method as well as the ill-conditioning type instability in the radial basis collocation method. We present the recent advances in resolving these difficulties in meshfree methods, and demonstrate how meshfree methods can be applied to problems difficult to be modeled by the conventional finite element methods due to their intrinsic regularity constraints.