A convergent finite element-finite volume scheme for the compressible Stokes problem. Part II: the isentropic case

Abstract

In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with an equation of state of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals rho Superscript gamma"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi> ρ </mml:mi> <mml:mi> γ </mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">p=\rho ^\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stands for the pressure and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi> ρ </mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the density). This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element technique. The discrete mass balance is obtained by a finite volume scheme, with an upwinding of the density, and two additional stabilization terms. We prove <italic>a priori</italic> estimates for the discrete solution, which yield its existence. Then the convergence of the scheme to a solution of the continuous problem is established. The passage to the limit in the equation of state requires the a.e. convergence of the density. It is obtained by adapting at the discrete level the “effective viscous pressure lemma” of the theory of compressible Navier-Stokes equations.