Asymptotic Preserving Error Estimates for Numerical Solutions of Compressible Navier-Stokes Equations in the Low Mach Number Regime

We study the convergence of numerical solutions of the compressible Navier--Stokes system to its incompressible limit. The numerical solution is obtained by a combined finite element--finite volume method based on the linear Crouzeix--Raviart finite element for the velocity and piecewise constant approximation for the density. The convective terms are approximated using upwinding. The distance between a numerical solution of the compressible problem and the strong solution of the incompressible Navier--Stokes equations is measured by means of a relative energy functional. For barotropic pressure exponent $\gamma \geq 3/2$ and for well-prepared initial data we obtain uniform convergence of order ${\cal O}(\sqrt{\Delta t}, h^a, \varepsilon)$, $a = \min \{ \frac{2 \gamma - 3 }{ \gamma}, 1\}$. Extensive numerical simulations confirm that the numerical solution of the compressible problem converges to the solution of the incompressible Navier--Stokes equations as the discretization parameters $\Delta t$, $h$ a...