A convergent nonconforming finite element method for compressible Stokes flow

TL;DR

This paper introduces a new nonconforming finite element method for modeling isentropic viscous gas flow, demonstrating convergence to weak solutions and addressing nonlinear pressure challenges.

Contribution

It develops a convergent nonconforming finite element approach for compressible Stokes flow, combining discontinuous Galerkin and div-curl formulations with novel convergence analysis.

Findings

Proves convergence of the finite element method to weak solutions.

Establishes strong convergence of density approximations.

Utilizes higher integrability and effective viscous flux techniques.

Abstract

We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin method. The velocity (momentum) equation is approximated by a finite element method on div-curl form using the nonconforming Crouzeix-Raviart space. Our main result is that the finite element method converges to a weak solution. The main challenge is to demonstrate the strong convergence of the density approximations, which is mandatory in view of the nonlinear pressure function. The analysis makes use of a higher integrability estimate on the density approximations, an equation for the "effective viscous flux", and renormalized versions of the discontinuous Galerkin method.