A convergent Finite Element-Finite Volume scheme for the compressible Stokes problem Part I -- the isothermal case
This work provides a rigorous numerical analysis for a finite element-finite volume scheme for compressible Stokes flows, addressing a known bottleneck in stability and convergence.
The paper proposes a discretization for the compressible Stokes problem using Crouzeix-Raviart elements, proving existence and convergence of the scheme to a continuous solution via a priori estimates and topological degree argument.
In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state $ρ=p$, based on Crouzeix-Raviart elements. The approximation of the momentum balance is obtained by usual finite element techniques. Since the pressure is piecewise constant, the discrete mass balance takes the form of a finite volume scheme, in which we introduce an upwinding of the density, together with two additional stabilization terms. We prove {\em a priori} estimates for the discrete solution, which yields its existence by a topological degree argument, and then the convergence of the scheme to a solution of the continuous problem.