A nodal integration based two level local projection meshfree stabilization method for convection diffusion problems

Abstract

This paper presents a new two level local projection meshfree stabilization (LPMS) method under the classical stabilized conforming nodal integration (SCNI) framework to solve convection diffusion problems suitable to a linear order approximation. Dropping the higher order terms in the residual based Petrov-Galerkin stabilization methods, usual in the Gauss integration, produces erroneous results if the classical SCNI is used because of the product of a quantity and its smoothed gradient (SCNI divergence operator) in the convective term. LPMS method is an alternative way to achieve a stable solution with the linear order approximation. In the present method, the design of the stabilization of convective terms stems from the fine scales defined on the subcells of Voronoi cells of SCNI. This LPMS with SCNI combination excludes the computational need for derivatives. First order mesh basis functions are sufficient as the SCNI replaces lower order derivatives with smoothed gradients, while local projection stabilization (LPS) excludes the terms with higher order derivatives. The present LMPS method is tested against standard benchmark problems with different distorted grids and compared with three existing stabilization methods. It is found to be in good agreement with the classical numerical methods for problems with thin layers.