A convergent FV-FEM scheme for the stationary compressible Navier-Stokes equations

TL;DR

This paper introduces a new finite element-finite volume discretization for the stationary compressible Navier-Stokes equations, proving convergence to weak solutions for ideal gas laws with specific adiabatic exponents in three dimensions.

Contribution

It presents the first convergence proof for a numerical scheme with adiabatic exponents less than 3 in three-dimensional space.

Findings

Numerical solutions converge to weak solutions as mesh size tends to zero.

The scheme is applicable to ideal gas pressure laws with γ > 3/2 in 3D.

First convergence result for such schemes with low adiabatic exponents in 3D.

Abstract

In this paper, we propose a discretization of the multi-dimensional stationary compressible Navier-Stokes equations combining finite element and finite volume techniques. As the mesh size tends to 0, the numerical solutions are shown to converge (up to a subsequence) towards a weak solution of the continuous problem for ideal gas pressure laws p($\rho$) = a$\rho$ $\gamma$ , with $\gamma$ > 3/2 in the three-dimensional case. It is the first convergence result for a numerical method with adiabatic exponents $\gamma$ less than 3 when the space dimension is three. The present convergence result can be seen as a discrete counterpart of the construction of weak solutions established by P.-L. Lions and by S. Novo, A. Novotn{\'y}.