Construction of stabilization operators for Galerkin least-squares discretizations of compressible and incompressible flows

The design and analysis of a class of stabilization operators suitable for space-time Galerkin least-squares finite element discretizations of the symmetrized compressible Navier-Stokes equations is discussed. The obtained stabilization matrix is well defined in the incompressible limit and reduces to the matrix described in [M. Polner, J.J.W. van der Vegt and R.M.J. van Damme, Analysis of stabilization operators for Galerkin least-squares discretizations of the incompressible Navier-Stokes equations, Comput. Meth. Appl. Mech. Eng., 195(2006):982--1006]. The stabilization matrix is investigated to give necessary and sufficient conditions for its positive definiteness. Under these conditions on the stabilization matrix, the Galerkin least-squares method for the symmetrized compressible Navier-Stokes equations satisfies the entropy condition, which ensures global nonlinear stability of the discretization.