A Least Square Residual version of the Modified Finite Particle Method to solve saddle point problems: Application to stationary Stokes and Navier–Stokes equations

The Modified Finite Particle Method (MFPM) is a meshless method belonging to the class of meshless method. Given the absence of a pre-determined connectivity among nodes, such methods are mainly used in applications where large deformations are involved, such as fluid-dynamics. In the present work we combine the Modified Finite Particle Method with a Weighted Least Square Residual Method, and use the combined version of the method for the solution of saddle point problems, such as the Stokes and Navier–Stokes equations for incompressible fluid flow simulations. The use of MFPM derivative approximation techniques in the framework of Least Square Residual Method permits a simple handling of the numerical difficulties typical of incompressible fluid simulations, avoiding problems such as pressure spurious oscillations, and preserving the convergence properties of the Modified Finite Particle Method also in presence of random collocation point distributions.