A stabilized and coupled meshfree/meshbased method for the incompressible Navier–Stokes equations—Part I: Stabilization

A stabilized meshfree Galerkin method is employed for the approximation of the incompressible Navier–Stokes equations in Eulerian or arbitrary Lagrangian–Eulerian (ALE) formulation. Equal-order interpolations for velocities and pressure are used. It is well-known from the meshbased context, i.e. from finite volume and finite element methods, that in convection-dominated flow problems in Eulerian or ALE formulation, stabilization is a crucial requirement for reliable solutions. Also, stabilization is needed in order to enable equal-order interpolations of the incompressible Navier–Stokes equations. Standard stabilization techniques, developed in a meshbased context, are extended to meshfree methods. It is found that the same structure of the stabilization schemes may be used, however the aspect of the stabilization parameter, weighting the stabilization terms, has to be reconsidered. In Part II of this work, the resulting stabilized meshfree Galerkin method is coupled with a stabilized finite element method. The resulting coupled method employs the comparatively costly meshfree Galerkin method only where it is needed—i.e. in areas of the domain, where a mesh is difficult to maintain—and the efficient meshbased finite element method is used in the rest of the domain. The fluid solver resulting from this technique is able to solve complex flow problems, involving large deformations of the physical domain and/or moving and rotating obstacles.