A convergent mixed method for the Stokes approximation of viscous compressible flow

Abstract

We propose a mixed finite element method for the motion of a strongly viscous, ideal, and isentropic gas. At the boundary we impose a Navier–slip condition such that the velocity equation can be posed in mixed form with the vorticity as an auxiliary variable. In this formulation we design a finite element method, where the velocity and vorticity is approximated with the divand curlconforming Nedelec elements, respectively, of the first order and first kind. The mixed scheme is coupled to a standard piecewise constant upwind discontinuous Galerkin discretization of the continuity equation. For the time discretization, implicit Euler time stepping is used. Our main result is that the numerical solution converges to a weak solution as the discretization parameters go to zero. The convergence analysis is inspired by the continuous analysis of Feireisl and Lions for the compressible Navier–Stokes equations. Tools used in the analysis include an equation for the effective viscous flux and various renormalizations of the density scheme.