A Penalized Finite-Element Method for a Compressible Stokes System

The linearized steady-state compressible viscous Navier--Stokes equations with constant coefficients are considered in this paper. A penalty method has been applied to the continuity equation. A finite-element formulation is constructed by applying a streamline diffusion method to the penalized continuity equation and applying the usual Galerkin method to the momentum equations. The resulting finite-element formulation is uniquely solvable without requiring any compatibility condition on the subspaces for velocity and pressure. An error estimate is obtained.