A Review of Petrov-Galerkin Stabilization Approaches and an Extension to Meshfree Methods

This paper gives a detailed review of popular stabilization approaches that have developed to be standard tools in the numerical world. The need for stabilization is outlined and stabilization ideas based on the Petrov-Galerkin concept are discussed. Stabilization methods are explained on the one hand in an illustrative approach with help of the artificial diffusion idea and on the other hand in a theoretical approach by outlining the mathematical background of stabilization. A generalization to meshfree methods is investigated. We find that the structure of stabilizing perturbation terms can be applied in the same manner to meshfree methods. However, the weighting of the stabilizing terms, defined with the stabilization parameter, requires special attention. Using standard formulas for the stabilization parameter, raised in the meshbased finite element context, is only suitable for meshfree shape functions with small dilatation parameters. This is confirmed with numerical experiments for the advection-diffusion equation and incompressible Navier-Stokes equations.