A stabilized MLPG method for steady state incompressible fluid flow simulation

In this paper, the meshless local Petrov–Galerkin (MLPG) method is extended to solve the incompressible fluid flow problems. The streamline upwind Petrov–Galerkin (SUPG) method is applied to overcome oscillations in convection-dominated problems, and the pressure-stabilizing Petrov–Galerkin (PSPG) method is applied to satisfy the so-called Babuška–Brezzi condition. The same stabilization parameter τ(τSUPG = τPSPG) is used in the present method. The circle domain of support, linear basis, and fourth-order spline weight function are applied to compute the shape function, and Bubnov–Galerkin method is applied to discretize the PDEs. The lid-driven cavity flow, backward facing step flow and natural convection in the square cavity are applied to validate the accuracy and feasibility of the present method. The results show that the stability of the present method is very good and convergent solutions can be obtained at high Reynolds number. The results of the present method are in good agreement with the classical results. It also seems that the present method (which is a truly meshless) is very promising in dealing with the convection- dominated problems.