A convergent FEM-DG method for the compressible Navier-Stokes equations

TL;DR

This paper introduces a new convergent numerical method combining FEM and DG techniques for the compressible Navier-Stokes equations, ensuring convergence to a global weak solution.

Contribution

The paper develops a novel FEM-DG method for the compressible Navier-Stokes equations with proven convergence to weak solutions, extending the theoretical analysis of Lions and Feireisl.

Findings

Method converges to a global weak solution as discretization parameters tend to zero.

Combines DG discretization for continuity with a Crouzeix-Raviart FEM for momentum.

Provides a numerical convergence analysis aligned with existence theory.

Abstract

This paper presents a new numerical method for the compressible Navier-Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix-Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax-Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.