Finite element least-squares methods for a compressible stokes system

where the symbols ∆, ∇, and ∇· stand for the Laplacian, gradient, and divergence operators, respectively (∆u is the vector of components ∆ui); the number μ is a viscous constant; f is a given vector function; β = (U,V)t is a given C1 function. The system (1.1) may be obtained by linearizing the steady-state barotropic compressible viscous Navier-Stokes equations without an ambient flow (see [8, 9] for more detail). Since the continuity equation is of hyperbolic type containing a convective derivative of p, we further assume that the boundary condition for the pressure is given on the inlet of the boundary where the characteristic function β points into Ω, that is,