Analysis of stabilization operators for Galerkin least-squares discretizations of the incompressible Navier–Stokes equations

A simple stabilized finite element method for solving two phase compressible–incompressible interface flows

“…Since it belongs to unified scheme category, the main advantage of stabilized finite element method is its capability to solve both compressible and incompressible Navier-Stokes equations (15) in the same formalism [15,16,29,30]. That is to say: the same mesh, the same unknowns, the same approximation spaces and the same finite element formulation.…”

“…Moreover the density has its material derivative equal to zero and is only a function of the initial state (constant along characteristics). This is also equivalent to use a p and b T as small parameters to derive the incompressible model [15,16,29,30]. Within this context, it has been shown that the matrices for incompressible formulation A I i are the same as the one for weakly compressible model A C i where v is set to zero [15,16] i.e.…”

“…Nevertheless, it should be interesting to use unified stabilization operator for the numerical scheme. For this perspective we refer to recent work done in [29,30].…”

“…Contrary to non-unified schemes, unified ones are those designed for solving both compressible and incompressible flows. Among these methods we can distinguish the low Mach schemes [13,38], Generalized Projection Method [39][40][41][42], stabilized finite element (SFE) [15,16,29,30]. More recently, we have the Marker and Cell (MAC) scheme [22] generalized to Euler equations and the Discontinuous Galerkin (DG) scheme [28].…”

“…The proposed model relies on the unified Navier-Stokes equations for each phase parameterized by a level-set function to easily capture the interface motion. For stability, robustness and computational efficiency this model is expressed in terms of pressure primitive unknowns (pressure, velocity, temperature) which are suitable for both compressible and incompressible limits [29,30]. This not only allows easily working with high order approximation on unstructured meshes (needed for complex geometries).…”