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

TL;DR

This paper introduces a convergent mixed finite element method for simulating viscous, compressible flow with Navier-slip boundary conditions, combining vorticity and velocity approximations with a discontinuous Galerkin scheme.

Contribution

It develops a novel mixed finite element approach for viscous compressible flow that ensures convergence to a weak solution, inspired by continuous analysis techniques.

Findings

The numerical scheme converges to a weak solution as discretization parameters tend to zero.

The method effectively couples Nedelec elements for velocity and vorticity with a discontinuous Galerkin scheme for density.

Convergence is proven using tools like effective viscous flux equations and density renormalizations.

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 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.