Least squares FEM approximation and subgrid extraction for convection dominated problems

As it is well known, the Galerkin FEM gives an oscillating solution in convection dominated problems. The oscillations grow in magnitude with the Peclet number and may even totally hide the true solution. The cure commonly used is to modify a-priori the formulation of the problem, by adding a stabilizing term to avoid an oscillating solution. This is called a stabilized method. Here, instead, we analyze these oscillations from a least squares perspective and propose a post-processing technique that both stabilizes the solution and partially resolve the sub grid scales.