A convergent Finite Element-Finite Volume scheme for the compressible Stokes problem Part I -- the isothermal case

TL;DR

This paper introduces a convergent finite element-finite volume scheme for the isothermal compressible Stokes problem using Crouzeix-Raviart elements, combining finite element and finite volume techniques with stabilization.

Contribution

It presents a novel discretization method that integrates finite element and finite volume approaches for the nonlinear compressible Stokes problem with proven convergence.

Findings

Proved a priori estimates for the discrete solution.

Established convergence of the scheme to the continuous problem.

Introduced an upwinding and stabilization in the finite volume scheme.

Abstract

In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state $\rho=p$, 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.