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Abstract. We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L 2 -stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential. The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes. Numerical tests with an exact smooth solution show the convergence of the scheme.Mathematics Subject Classification. 35Q30, 65N12, 65N30, 76M25.

International audienceIn this paper, we prove an adaptation of the classical compactness Aubin-Simon lemma to sequences of functions obtained through a sequence of discretizations of a parabolic problem. The main difficulty tackled here is to generalize the classical proof to handle the dependency of the norms controlling each function u (n) of the sequence with respect to n. This compactness result is then used to prove the convergence of a numerical scheme combining finite volumes and finite elements for the solution of a reduced turbulence problem. 1. Introduction. Let us suppose given a sequence of approximations of a parabolic problem (P), which, for instance, may be though of as discretizations of (P) by a numerical scheme, or resulting from the combination of a Faedo-Galerkin technique with a time discretization. In both cases, we are in presence of a family of finite-dimensional systems, and their solution, let us say (u (n)) n∈N , may be considered as a sequence of functions of time, taking their values in a finite dimensional subspace, let us say B (n) of a Banach space (usually a Lebesgue L q space, q ≥ 1). To show the convergence of such a process, a common path is to follow the following steps: 1. first, for each approximate problem, prove the existence of a solution, and derive estimates satisfied by any solution, 2. then use compactness arguments to show (possibly up to the extraction of a subsequence) the existence of a limit, 3. and, finally, prove that this limit satisfies the initial problem (P). Let us now focus on item 2, which is the issue addressed in this paper. The problem here is to prove a compactness result for a sequence of functions of time taking their value in a sequence of discrete spaces and controlled (themselves and their discrete time-derivative) in discrete norms; in the general case, both B (n) and the space part of the norms depend on n. The compactness result given in this paper relies on this particular structure, and consists in a generalization of the classical Aubin-Simon lemma to this specific case

SUMMARYWe develop in this paper a discretization for the convection term in variable density unstationary Navier-Stokes equations, which applies to low-order non-conforming finite element approximations (the so-called Crouzeix-Raviart or Rannacher-Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L 2 stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L 2 norm, second-order space convergence for the velocity and first-order space convergence for the pressure are observed.

Abstract. 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 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.