• Galerkin least squares
• GLS
• LSIC
• PSPG
• streamline upwind Petrov Galerkin
• SUPG
• CFD

The present work has the objective to build a finite element method able to solve the Navier-Stokes equations for unsteady, incompressible flow. Approaching this problem with the standard Galerkin method presents two sources of instabilities, the first due to the convective term and appearing at high Reynolds numbers, the second linked to the presence of the pressure term and arising if the so-called “inf-sup condition” for the velocity and pressure approximation spaces is not satisfied.
These two problems will be separately considered, in order to isolate and overcome the sources of instability: thus, at the beginning the pressure term will not be taken into account, obtaining the Advection-Diffusion equation; then the neglect of the convective term will lead to the Stokes problem.
The general approach used is called Variational Multiscale Method, which is based on the decomposition of the solution in coarse and fine scales: in fact, both problems arising with the standard Galerkin method are viewed as a matter of missing scales, and they thus may be overcome if more scales are included in the model. Actually, this approach allows one to obtain a stable solution of the problem without refining the mesh employed, but rather taking into account the effect of the fine scales, which are usually neglected, in the coarse-scale equation. Moreover, this method is able to theoretically legitimate the so-called stabilized methods, which historically appeared as a generalization of one-dimensional problems.
In the present work both stabilize and variational multiscale methods will be analyzed, showing their analogies and pointing out their differences; it will also be obtained that their performances are strictly linked to the value set for the stabilization parameters, which may be chosen after considerations based on the experience or referring to a stability and error analysis of the method.