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

In this paper we discuss the design and analysis of a class of stabilization operators for space–time Galerkin least-squares finite element discretizations suitable for the incompressible limit of the symmetrized Navier–Stokes equations given in [G. Hauke, T.J.R. Hughes, A comparative study of different sets of variables for solving compressible and incompressible flows, Comput. Methods Appl. Mech. Engrg. 153 (1998) 1–44]. This set of equations consists of the incompressible Navier–Stokes equations in combination with the heat equation. The analysis results in stabilization operators which are positive definite and dimensionally consistent. In addition, a detailed proof is given that the space–time Galerkin least squares discretization together with the proposed stabilization operators satisfies a coercivity condition for the linearized form of the equations. This ensures that necessary conditions for uniqueness and stability of the numerical solution are satisfied by the finite element discretization.