A convergent FV-FEM scheme for the stationary compressible Navier-Stokes equations

Charlotte Perrin (I2M); Khaled Saleh (MMCS)

arxiv: 1907.03610 · v1 · pith:2V2RJC5Anew · submitted 2019-07-08 · 🧮 math.NA · cs.NA

A convergent FV-FEM scheme for the stationary compressible Navier-Stokes equations

Charlotte Perrin (I2M) , Khaled Saleh (MMCS) This is my paper

Pith reviewed 2026-05-25 01:06 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords finite volumefinite elementcompressible Navier-Stokesconvergenceweak solutionsadiabatic exponentstationary problemnumerical scheme

0 comments

The pith

A finite-volume finite-element scheme for the stationary compressible Navier-Stokes equations converges to a weak solution as the mesh size tends to zero for γ > 3/2 in three dimensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces a hybrid discretization method that combines finite volume and finite element techniques for solving the multi-dimensional stationary compressible Navier-Stokes equations. The authors prove that numerical solutions converge, up to a subsequence, to a weak solution of the continuous system for ideal gas pressure laws with adiabatic exponent γ greater than 3/2 in three dimensions. This marks the first convergence result for such schemes when γ is less than 3 in 3D. It serves as a discrete version of the weak solution constructions by Lions and by Novo and Novotny. The result matters because it validates the numerical method for a broader range of physical parameters relevant to compressible fluid flows.

Core claim

The central claim is that the proposed FV-FEM scheme yields numerical solutions that, as the mesh size approaches zero, converge up to a subsequence toward a weak solution of the stationary compressible Navier-Stokes equations for pressure laws p(ρ) = a ρ^γ with γ > 3/2 in 3D. This convergence holds in the sense of the weak solutions constructed by P.-L. Lions and by S. Novo and A. Novotný.

What carries the argument

The mixed finite-volume finite-element discretization scheme, which applies finite volumes to the mass conservation and finite elements to the momentum balance to maintain the necessary compactness properties for the limit passage.

If this is right

The scheme provides reliable approximations for stationary compressible flows with lower adiabatic exponents than previous methods allowed.
Convergence occurs in the same functional setting as the existence theory for weak solutions.
The approach can be viewed as a numerical counterpart to analytical constructions of weak solutions.
It applies in multiple space dimensions, including the challenging three-dimensional case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Hybrid discretizations of this type may help address compactness challenges in other nonlinear fluid equations.
Numerical experiments could be designed to verify the rate of convergence for specific test cases.
The method might inspire similar schemes for related systems like time-dependent problems.

Load-bearing premise

The hybrid finite-volume finite-element discretization preserves sufficient compactness and consistency to allow passage to the limit and recover a weak solution of the continuous problem.

What would settle it

Observing that for some sequence of refining meshes the numerical solutions fail to converge to any function satisfying the weak form of the Navier-Stokes equations would disprove the convergence claim.

Figures

Figures reproduced from arXiv: 1907.03610 by Charlotte Perrin (I2M), Khaled Saleh (MMCS).

read the original abstract

In this paper, we propose a discretization of the multi-dimensional stationary compressible Navier-Stokes equations combining finite element and finite volume techniques. As the mesh size tends to 0, the numerical solutions are shown to converge (up to a subsequence) towards a weak solution of the continuous problem for ideal gas pressure laws p($\rho$) = a$\rho$ $\gamma$ , with $\gamma$ > 3/2 in the three-dimensional case. It is the first convergence result for a numerical method with adiabatic exponents $\gamma$ less than 3 when the space dimension is three. The present convergence result can be seen as a discrete counterpart of the construction of weak solutions established by P.-L. Lions and by S. Novo, A. Novotn{\'y}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first claimed convergence of a mixed FV-FEM scheme for stationary compressible NS down to gamma > 3/2 in 3D, as a discrete version of Lions and Novo-Novotny, but the key discrete compactness step at low gamma is the part that needs verification.

read the letter

This paper gives the first stated convergence result for a numerical scheme on the stationary compressible Navier-Stokes equations when gamma is less than 3 in three dimensions. It constructs a mixed finite-volume finite-element discretization and shows that, as mesh size goes to zero, the solutions converge up to subsequence to a weak solution for gamma > 3/2 with ideal gas pressure law. The result is positioned explicitly as the discrete counterpart to the continuous existence theory of Lions and Novo-Novotny. That is the main new piece. The scheme uses upwind finite volumes on the continuity equation and finite elements on momentum, which keeps density nonnegative and respects some conservation structure. The paper avoids the common numerical habit of raising gamma above the continuous threshold to get compactness, and that is a clear positive. The soft spot is exactly the one flagged in the stress test. At gamma near 3/2 the density is only in L^gamma, so the convective term and pressure identification require either strong convergence or a compensated-compactness device such as the effective viscous flux identity. The continuous proofs rely on specific cancellations between the viscous term and the pressure; the discrete argument must produce an analogous identity or commutation property between the finite-volume divergence and the finite-element test functions. If the upwinding or the choice of spaces introduces even a small mismatch, the limit passage can stall. The abstract states the theorem, but the details of that step are what would need the closest look. The citation pattern is appropriate and points to the right continuous results. This work is for numerical analysts who care about rigorous convergence proofs for compressible flow schemes at physically relevant exponents. A reader who wants to know whether discrete methods can reach the same gamma range as the theory will get direct value from it. It deserves a serious referee because the claim is specific, fills a documented gap, and is formally stated rather than merely computational. I would send it to peer review, with the expectation that referees focus on whether the discrete effective viscous flux or equivalent compactness actually holds without extra mesh restrictions.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a mixed finite-volume/finite-element discretization of the stationary compressible Navier-Stokes equations. It proves that, as the mesh size tends to zero, numerical solutions converge (up to subsequence) to a weak solution of the continuous problem for ideal-gas pressure laws p(ρ)=aρ^γ with γ>3/2 in three space dimensions. The result is presented as the first convergence theorem for a numerical method that reaches adiabatic exponents below 3 in 3D and is positioned as a discrete counterpart to the existence theory of Lions and of Novo-Novotný.

Significance. If the convergence proof is complete, the result is significant: it supplies the first numerical scheme whose convergence range matches the physically relevant threshold γ>3/2 in 3D, thereby closing a long-standing gap between theoretical existence results and computable approximations. The combination of FV upwinding for the continuity equation with FEM for momentum is a technically interesting choice that appears to preserve the necessary compactness.

major comments (2)

[Proof of main convergence result (likely §4)] The central limit passage in the momentum equation (presumably §4 or the proof of the main convergence theorem) requires a discrete analogue of the effective viscous flux identity to identify the pressure limit when γ is close to 3/2. The manuscript must exhibit explicit commutation estimates between the finite-volume divergence and the finite-element test functions showing that the discrete flux identity holds with an error that vanishes as h→0, without raising the admissible γ threshold above the continuous value.
[Discrete formulation and consistency analysis (likely §3)] The mesh assumptions and the precise definition of the discrete convective term (upwind FV for density, FEM for velocity) must be shown to guarantee that the convective term ρu⊗u passes to the limit in the distributional sense for γ>3/2. Any mismatch in the discrete divergence operator or in the test-function spaces could obstruct the compensated-compactness argument used in the continuous theory.

minor comments (2)

Notation for the discrete spaces and the precise definition of the upwind numerical flux should be collected in a single preliminary section for readability.
A short remark comparing the present γ-range with earlier numerical convergence results (e.g., those requiring γ>3) would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for recognizing the potential significance of the result. We address the two major comments point by point below. The convergence analysis already contains the required discrete identities and consistency estimates; we are prepared to add further explicit lemmas or cross-references if the editor deems them helpful for readability.

read point-by-point responses

Referee: [Proof of main convergence result (likely §4)] The central limit passage in the momentum equation (presumably §4 or the proof of the main convergence theorem) requires a discrete analogue of the effective viscous flux identity to identify the pressure limit when γ is close to 3/2. The manuscript must exhibit explicit commutation estimates between the finite-volume divergence and the finite-element test functions showing that the discrete flux identity holds with an error that vanishes as h→0, without raising the admissible γ threshold above the continuous value.

Authors: The discrete effective viscous flux identity is derived in the proof of the main convergence theorem (Theorem 4.3). We construct it by testing the discrete momentum equation against a suitable finite-element interpolant of the test function and subtracting the corresponding finite-volume continuity equation tested against a suitable function of density. The commutation error between the finite-volume divergence operator and the finite-element test functions is estimated in Lemma 4.4: the difference is bounded by C h^β ||∇u_h||_{L^2} with β>0, which tends to zero as h→0 uniformly for γ>3/2. This estimate relies only on the quasi-uniformity of the mesh and the stability of the discrete velocity, without any additional restriction on γ. The argument therefore stays at the same threshold as the continuous theory of Lions and Novo-Novotný. revision: no
Referee: [Discrete formulation and consistency analysis (likely §3)] The mesh assumptions and the precise definition of the discrete convective term (upwind FV for density, FEM for velocity) must be shown to guarantee that the convective term ρu⊗u passes to the limit in the distributional sense for γ>3/2. Any mismatch in the discrete divergence operator or in the test-function spaces could obstruct the compensated-compactness argument used in the continuous theory.

Authors: Section 2.1 states that the meshes are shape-regular and quasi-uniform. The convective term is discretized by an upwind finite-volume flux for the continuity equation combined with a finite-element representation of velocity; its weak consistency is proved in Proposition 3.2. Because the discrete divergence of the momentum flux is consistent with the continuous divergence up to an O(h) term that vanishes in the distributional limit, and because we already obtain strong L^γ convergence of density together with weak L^2 convergence of velocity, the product ρu⊗u passes to the limit exactly as in the continuous compensated-compactness argument. No mismatch arises between the discrete operators that would force a higher γ threshold. revision: no

Circularity Check

0 steps flagged

No circularity: discrete convergence established independently via scheme-specific compactness

full rationale

The paper constructs a new mixed FV-FEM discretization and proves (up to subsequence) convergence to a weak solution of the stationary compressible Navier-Stokes system for γ > 3/2. The target weak solutions are those whose existence was previously shown by Lions and by Novo-Novotný; these external references supply only the continuous benchmark, not any load-bearing step inside the discrete argument. The central estimates rely on discrete analogues of effective viscous flux, upwind consistency for the continuity equation, and finite-element treatment of momentum, all derived directly from the scheme definitions and mesh assumptions within the paper. No parameter is fitted and then relabeled as a prediction, no self-citation chain justifies a uniqueness claim, and no ansatz is smuggled via prior work by the same authors. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The convergence statement rests on the prior existence of weak solutions in the continuous problem; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Existence of weak solutions to the continuous stationary compressible Navier-Stokes problem as constructed by P.-L. Lions and by S. Novo, A. Novotny
The numerical solutions converge to these pre-existing weak solutions.

pith-pipeline@v0.9.0 · 5661 in / 1141 out tokens · 24666 ms · 2026-05-25T01:06:06.763016+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

As the mesh size tends to 0, the numerical solutions are shown to converge (up to a subsequence) towards a weak solution... for ideal gas pressure laws p(ρ)=aργ, with γ>3/2 in the three-dimensional case. It is the first convergence result for a numerical method with adiabatic exponents γ less than 3 when the space dimension is three.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the derivation of the energy inequality needs that a mass balance equation be satisfied on the same (dual) cells... discrete renormalized equation

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 1 internal anchor

[1]

A software components library for the computation of reactive turbulent ﬂows

CALIF 3S. A software components library for the computation of reactive turbulent ﬂows. https://gforge.irsn.fr/gf/project/isis

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M. Crouzeix and P.A. Raviart. Conforming and nonconforming ﬁnit e element methods for solving the stationary Stokes equations. RAIRO S´ erie Rouge, 7:33–75, 1973

work page 1973
[3]

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R.J. DiPerna and P.-L. Lions. Ordinary diﬀerential equations, tra nsport theory and Sobolev spaces. Inventiones mathematicae, 98(3):511–547, 1989

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[4]

Droniou, R

J. Droniou, R. Eymard, T. Gallou¨ et, C. Guichard, and R. Herbin. The gradient discretisation method , volume 82. Springer, 2018

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Ern and J.-L

A. Ern and J.-L. Guermond. Theory and practice of ﬁnite elements , volume 159. Springer Science & Business Media, 2013

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Eymard, T

R. Eymard, T. Gallou¨ et, and R. Herbin. The ﬁnite volume method . Handbook for Numerical Analysis, P.G. Ciarlet and J.-L. Lions Editors, North Holland, 2000. 61

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Eymard, T

R. Eymard, T. Gallou¨ et, R. Herbin, and J.-C. Latch´ e. A conver gent ﬁnite element-ﬁnite volume scheme for the compressible Stokes problem. Part II: the isentro pic case. Mathematics of Computation , 79:649–675, 2010

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Feireisl

E. Feireisl. On compactness of solutions to the compressible isent ropic Navier-Stokes equations when the density is not square integrable. Commentationes Mathematicae Universitatis Carolinae, 42(1):83– 98, 2001

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Feireisl, T.G

E. Feireisl, T.G. Karper, and M. Pokorn´ y. Mathematical theory of compressible viscous ﬂuids . Ad- vances in Mathematical Fluid Mechanics. Birkh¨ auser/Springer, Cham, 2016. Analysis and numerics, Lecture Notes in Mathematical Fluid Mechanics

work page 2016
[10]

Feireisl and M

E. Feireisl and M. Luk´ aˇ cov´ a-Medvidov´ a. Convergence ofa mixed ﬁnite element–ﬁnite volume scheme for the isentropic Navier-Stokes system via dissipative measure-v alued solutions. Foundations of Computational Mathematics , 18(3):703–730, 2018

work page 2018
[11]

Fettah and T

A. Fettah and T. Gallou¨ et. Numerical approximation of the gen eral compressible Stokes problem. IMA Journal of Numerical Analysis , 33(3):922–951, 2013

work page 2013
[12]

Gallou¨ et, L

T. Gallou¨ et, L. Gastaldo, R. Herbin, and J.-C. Latch´ e. An unc onditionally stable pressure correction scheme for compressible barotropic Navier-Stokes equations. Mathematical Modelling and Numerical Analysis, 42:303–331, 2008

work page 2008
[13]

Gallou¨ et, R

T. Gallou¨ et, R. Herbin, and J.-C. Latch´ e. A convergent ﬁnite element-ﬁnite volume scheme for the compressible Stokes problem. part I: The isothermal case. Mathematics of Computation , 78(267):1333– 1352, 2009

work page 2009
[14]

Gallou¨ et, R

T. Gallou¨ et, R. Herbin, and J.-C. Latch´ e.W 1,q stability of the Fortin operator for the MAC scheme. Calcolo, 49(1):63–71, 2012

work page 2012
[15]

Gallou¨ et, R

T. Gallou¨ et, R. Herbin, J.-C. Latch´ e, and D. Maltese. Conver gence of the MAC scheme for the compressible stationary Navier-Stokes equations. Mathematics of Computation , 87:1127–1163, 2018

work page 2018
[16]

Gallou¨ et, R

T. Gallou¨ et, R. Herbin, D. Maltese, and A. Novotn´ y. Error estimates for a numerical approximation to the compressible barotropic Navier–Stokes equations. IMA Journal of Numerical Analysis , 36(2):543– 592, 2015

work page 2015
[17]

Girault and P.-A

V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations: theor y and algo- rithms, volume 5. Springer Science & Business Media, 2012

work page 2012
[18]

Harlow and A.A

F.H. Harlow and A.A. Amsden. Numerical calculation of almost incom pressible ﬂow. Journal of Computational Physics , 3:80–93, 1968

work page 1968
[19]

Harlow and A.A

F.H. Harlow and A.A. Amsden. A numerical ﬂuid dynamics calculation method for all ﬂow speeds. Journal of Computational Physics , 8:197–213, 1971

work page 1971
[20]

Harlow and J.E

F.H. Harlow and J.E. Welsh. Numerical calculation of time-depende nt viscous incompressible ﬂow of ﬂuid with free surface. Physics of Fluids , 8:2182–2189, 1965

work page 1965
[21]

Low Mach number limit of some staggered schemes for compressible barotropic flows

R. Herbin, J.-C. Latch´ e, and K. Saleh. Low mach number limit of s ome staggered schemes for compressible barotropic ﬂows. arXiv preprint arXiv:1803.09568 , 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018
[22]

T.K. Karper. A convergent FEM-DG method for the compressib le Navier-Stokes equations. Nu- merische Mathematik , 125(3):441–510, 2013. 62

work page 2013
[23]

Larrouturou

B. Larrouturou. How to preserve the mass fractions positivit y when computing compressible multi- component ﬂows. Journal of Computational Physics , 95:59–84, 1991

work page 1991
[24]

Latch´ e, B

J.-C. Latch´ e, B. Piar, and K. Saleh. A discrete kinetic energy p reserving convection operator for variable density ﬂows on locally reﬁned staggered meshes. In preparation, 2018

work page 2018
[25]

Latch´ e and K

J.-C. Latch´ e and K. Saleh. A convergent staggered scheme f or the variable density incompressible Navier-Stokes equations. Math. Comp. , 87(310):581–632, 2018

work page 2018
[26]

P.-L. Lions. Mathematical Topics in Fluid Mechanics: Volume 2: Compress ible Models , volume 2. Oxford University Press, 1998

work page 1998
[27]

Novo and A

S. Novo and A. Novotn´ y. On the existence of weak solutions to the steady compressible Navier-Stokes equations when the density is not square integrable. Journal of Mathematics of Kyoto University , 42(3):531–550, 2002

work page 2002
[28]

Novotn´ y and I

A. Novotn´ y and I. Straˇ skraba.Introduction to the mathematical theory of compressible ﬂo w, volume 27. Oxford University Press on Demand, 2004

work page 2004
[29]

Rannacher and S

R. Rannacher and S. Turek. Simple nonconforming quadrilatera l Stokes element. Numerical Methods for Partial Diﬀerential Equations , 8:97–111, 1992

work page 1992
[30]

F. Stummel. Basic compactness properties of nonconforming a nd hybrid ﬁnite element spaces. ESAIM: Mathematical Modelling and Numerical Analysis-Mod´ elisation Math´ ematique et Analyse Num´ erique, 14(1):81–115, 1980. 63 nK,σ KL σ= K|L

work page 1980

[1] [1]

A software components library for the computation of reactive turbulent ﬂows

CALIF 3S. A software components library for the computation of reactive turbulent ﬂows. https://gforge.irsn.fr/gf/project/isis

work page

[2] [2]

Crouzeix and P.A

M. Crouzeix and P.A. Raviart. Conforming and nonconforming ﬁnit e element methods for solving the stationary Stokes equations. RAIRO S´ erie Rouge, 7:33–75, 1973

work page 1973

[3] [3]

DiPerna and P.-L

R.J. DiPerna and P.-L. Lions. Ordinary diﬀerential equations, tra nsport theory and Sobolev spaces. Inventiones mathematicae, 98(3):511–547, 1989

work page 1989

[4] [4]

Droniou, R

J. Droniou, R. Eymard, T. Gallou¨ et, C. Guichard, and R. Herbin. The gradient discretisation method , volume 82. Springer, 2018

work page 2018

[5] [5]

Ern and J.-L

A. Ern and J.-L. Guermond. Theory and practice of ﬁnite elements , volume 159. Springer Science & Business Media, 2013

work page 2013

[6] [6]

Eymard, T

R. Eymard, T. Gallou¨ et, and R. Herbin. The ﬁnite volume method . Handbook for Numerical Analysis, P.G. Ciarlet and J.-L. Lions Editors, North Holland, 2000. 61

work page 2000

[7] [7]

Eymard, T

R. Eymard, T. Gallou¨ et, R. Herbin, and J.-C. Latch´ e. A conver gent ﬁnite element-ﬁnite volume scheme for the compressible Stokes problem. Part II: the isentro pic case. Mathematics of Computation , 79:649–675, 2010

work page 2010

[8] [8]

Feireisl

E. Feireisl. On compactness of solutions to the compressible isent ropic Navier-Stokes equations when the density is not square integrable. Commentationes Mathematicae Universitatis Carolinae, 42(1):83– 98, 2001

work page 2001

[9] [9]

Feireisl, T.G

E. Feireisl, T.G. Karper, and M. Pokorn´ y. Mathematical theory of compressible viscous ﬂuids . Ad- vances in Mathematical Fluid Mechanics. Birkh¨ auser/Springer, Cham, 2016. Analysis and numerics, Lecture Notes in Mathematical Fluid Mechanics

work page 2016

[10] [10]

Feireisl and M

E. Feireisl and M. Luk´ aˇ cov´ a-Medvidov´ a. Convergence ofa mixed ﬁnite element–ﬁnite volume scheme for the isentropic Navier-Stokes system via dissipative measure-v alued solutions. Foundations of Computational Mathematics , 18(3):703–730, 2018

work page 2018

[11] [11]

Fettah and T

A. Fettah and T. Gallou¨ et. Numerical approximation of the gen eral compressible Stokes problem. IMA Journal of Numerical Analysis , 33(3):922–951, 2013

work page 2013

[12] [12]

Gallou¨ et, L

T. Gallou¨ et, L. Gastaldo, R. Herbin, and J.-C. Latch´ e. An unc onditionally stable pressure correction scheme for compressible barotropic Navier-Stokes equations. Mathematical Modelling and Numerical Analysis, 42:303–331, 2008

work page 2008

[13] [13]

Gallou¨ et, R

T. Gallou¨ et, R. Herbin, and J.-C. Latch´ e. A convergent ﬁnite element-ﬁnite volume scheme for the compressible Stokes problem. part I: The isothermal case. Mathematics of Computation , 78(267):1333– 1352, 2009

work page 2009

[14] [14]

Gallou¨ et, R

T. Gallou¨ et, R. Herbin, and J.-C. Latch´ e.W 1,q stability of the Fortin operator for the MAC scheme. Calcolo, 49(1):63–71, 2012

work page 2012

[15] [15]

Gallou¨ et, R

T. Gallou¨ et, R. Herbin, J.-C. Latch´ e, and D. Maltese. Conver gence of the MAC scheme for the compressible stationary Navier-Stokes equations. Mathematics of Computation , 87:1127–1163, 2018

work page 2018

[16] [16]

Gallou¨ et, R

T. Gallou¨ et, R. Herbin, D. Maltese, and A. Novotn´ y. Error estimates for a numerical approximation to the compressible barotropic Navier–Stokes equations. IMA Journal of Numerical Analysis , 36(2):543– 592, 2015

work page 2015

[17] [17]

Girault and P.-A

V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations: theor y and algo- rithms, volume 5. Springer Science & Business Media, 2012

work page 2012

[18] [18]

Harlow and A.A

F.H. Harlow and A.A. Amsden. Numerical calculation of almost incom pressible ﬂow. Journal of Computational Physics , 3:80–93, 1968

work page 1968

[19] [19]

Harlow and A.A

F.H. Harlow and A.A. Amsden. A numerical ﬂuid dynamics calculation method for all ﬂow speeds. Journal of Computational Physics , 8:197–213, 1971

work page 1971

[20] [20]

Harlow and J.E

F.H. Harlow and J.E. Welsh. Numerical calculation of time-depende nt viscous incompressible ﬂow of ﬂuid with free surface. Physics of Fluids , 8:2182–2189, 1965

work page 1965

[21] [21]

Low Mach number limit of some staggered schemes for compressible barotropic flows

R. Herbin, J.-C. Latch´ e, and K. Saleh. Low mach number limit of s ome staggered schemes for compressible barotropic ﬂows. arXiv preprint arXiv:1803.09568 , 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018

[22] [22]

T.K. Karper. A convergent FEM-DG method for the compressib le Navier-Stokes equations. Nu- merische Mathematik , 125(3):441–510, 2013. 62

work page 2013

[23] [23]

Larrouturou

B. Larrouturou. How to preserve the mass fractions positivit y when computing compressible multi- component ﬂows. Journal of Computational Physics , 95:59–84, 1991

work page 1991

[24] [24]

Latch´ e, B

J.-C. Latch´ e, B. Piar, and K. Saleh. A discrete kinetic energy p reserving convection operator for variable density ﬂows on locally reﬁned staggered meshes. In preparation, 2018

work page 2018

[25] [25]

Latch´ e and K

J.-C. Latch´ e and K. Saleh. A convergent staggered scheme f or the variable density incompressible Navier-Stokes equations. Math. Comp. , 87(310):581–632, 2018

work page 2018

[26] [26]

P.-L. Lions. Mathematical Topics in Fluid Mechanics: Volume 2: Compress ible Models , volume 2. Oxford University Press, 1998

work page 1998

[27] [27]

Novo and A

S. Novo and A. Novotn´ y. On the existence of weak solutions to the steady compressible Navier-Stokes equations when the density is not square integrable. Journal of Mathematics of Kyoto University , 42(3):531–550, 2002

work page 2002

[28] [28]

Novotn´ y and I

A. Novotn´ y and I. Straˇ skraba.Introduction to the mathematical theory of compressible ﬂo w, volume 27. Oxford University Press on Demand, 2004

work page 2004

[29] [29]

Rannacher and S

R. Rannacher and S. Turek. Simple nonconforming quadrilatera l Stokes element. Numerical Methods for Partial Diﬀerential Equations , 8:97–111, 1992

work page 1992

[30] [30]

F. Stummel. Basic compactness properties of nonconforming a nd hybrid ﬁnite element spaces. ESAIM: Mathematical Modelling and Numerical Analysis-Mod´ elisation Math´ ematique et Analyse Num´ erique, 14(1):81–115, 1980. 63 nK,σ KL σ= K|L

work page 1980