Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: The isothermal case
HTML articles powered by AMS MathViewer

by T. Gallouët, R. Herbin and J.-C. Latché PDF
Math. Comp. 78 (2009), 1333-1352 Request permission


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 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.
  • Pavel Bochev, Sang Dong Kim, and Byeong-Chun Shin, Analysis and computation of least-squares methods for a compressible Stokes problem, Numer. Methods Partial Differential Equations 22 (2006), no. 4, 867–883. MR 2230276, DOI 10.1002/num.20126
  • F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, Efficient solutions of elliptic systems (Kiel, 1984) Notes Numer. Fluid Mech., vol. 10, Friedr. Vieweg, Braunschweig, 1984, pp. 11–19. MR 804083
  • P. G. Ciarlet, Handbook of numerical analysis volume II: Finite elements methods – Basic error estimates for elliptic problems, Handbook of Numerical Analysis, Volume II (P. Ciarlet and J.L. Lions, eds.), North Holland, 1991, pp. 17–351.
  • M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
  • Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
  • Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
  • R. Eymard and R. Herbin, Entropy estimate for the approximation of the compressible barotropic Navier-Stokes equations using a collocated finite volume scheme, in preparation (2007).
  • Robert Eymard, Raphaèle Herbin, and Jean Claude Latché, On a stabilized colocated finite volume scheme for the Stokes problem, M2AN Math. Model. Numer. Anal. 40 (2006), no. 3, 501–527. MR 2245319, DOI 10.1051/m2an:2006024
  • Eduard Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 26, Oxford University Press, Oxford, 2004. MR 2040667
  • Thierry Gallouët, Laura Gastaldo, Raphaele Herbin, and Jean-Claude Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations, M2AN Math. Model. Numer. Anal. 42 (2008), no. 2, 303–331. MR 2405150, DOI 10.1051/m2an:2008005
  • L. Gastaldo, R. Herbin, and J.-C. Latché, An entropy-preserving finite element–finite volume pressure correction scheme for the drift-flux model, submitted (2008).
  • R. Bruce Kellogg and Biyue Liu, A finite element method for the compressible Stokes equations, SIAM J. Numer. Anal. 33 (1996), no. 2, 780–788. MR 1388498, DOI 10.1137/0733039
  • R. Bruce Kellogg and Biyue Liu, A penalized finite-element method for a compressible Stokes system, SIAM J. Numer. Anal. 34 (1997), no. 3, 1093–1105. MR 1451115, DOI 10.1137/S0036142994273276
  • Jae Ryong Kweon, An optimal order convergence for a weak formulation of the compressible Stokes system with inflow boundary condition, Numer. Math. 86 (2000), no. 2, 305–318. MR 1777491, DOI 10.1007/PL00005408
  • Jae Ryong Kweon, Optimal error estimate for a mixed finite element method for compressible Navier-Stokes system, Appl. Numer. Math. 45 (2003), no. 2-3, 275–292. MR 1967577, DOI 10.1016/S0168-9274(02)00213-1
  • Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. Compressible models; Oxford Science Publications. MR 1637634
  • A. Novotný and I. Straškraba, Introduction to the mathematical theory of compressible flow, Oxford Lecture Series in Mathematics and its Applications, vol. 27, Oxford University Press, Oxford, 2004. MR 2084891
  • L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
  • Roger Temam, Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 0609732
  • Alberto Valli, On the existence of stationary solutions to compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 1, 99–113 (English, with French summary). MR 877992
  • Rüdiger Verfürth, Error estimates for some quasi-interpolation operators, M2AN Math. Model. Numer. Anal. 33 (1999), no. 4, 695–713 (English, with English and French summaries). MR 1726480, DOI 10.1051/m2an:1999158
Similar Articles
Additional Information
  • T. Gallouët
  • Affiliation: Université de Provence, France
  • Email:
  • R. Herbin
  • Affiliation: Université de Provence, France
  • Email:
  • J.-C. Latché
  • Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN)
  • MR Author ID: 715367
  • Email:
  • Received by editor(s): December 7, 2007
  • Published electronically: January 30, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 78 (2009), 1333-1352
  • MSC (2000): Primary 35Q30, 65N12, 65N30, 76N15, 76M10, 76M12
  • DOI:
  • MathSciNet review: 2501053