A convergent mixed method for the Stokes approximation of viscous compressible flow
This work provides a theoretical convergence guarantee for a numerical method in compressible flow, which is important for computational fluid dynamics but is an incremental theoretical contribution.
The authors propose a mixed finite element method for viscous compressible flow with Navier-slip boundary conditions, proving convergence to a weak solution as discretization parameters approach zero.
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 div- and curl- conforming 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.