Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Calcolo
Published in: Calcolo 1/2015

01-03-2015

An extension of the MAC scheme to locally refined meshes: convergence analysis for the full tensor time-dependent Navier–Stokes equations

Authors: Eric Chénier, Robert Eymard, Thierry Gallouët, Raphaèle Herbin

Published in: Calcolo | Issue 1/2015

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

A variational formulation of the standard marker-and-cell scheme for the approximation of the Navier–Stokes problem yields an extension of the scheme to general 2D and 3D domains and more general meshes. An original discretization of the trilinear form of the nonlinear convection term is proposed; it is designed so as to vanish for discrete divergence free functions. This property allows us to give a mathematical proof of the convergence of the resulting approximate solutions, for the nonlinear Navier–Stokes equations in both steady-state and time-dependent regimes, without any small data condition. Numerical examples (analytical steady and time-dependent ones, inclined driven cavity) confirm the robustness and the accuracy of this method.
previous article Mimetic finite difference approximation of quasilinear elliptic problems
next article Lower and upper bounds of Stokes eigenvalue problem based on stabilized finite element methods

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
Appendix
Available only for authorised users
Literature
1.
Blanc, P.: Convergence of a finite volume scheme on a MAC mesh for the Stokes problem with right-hand side in \(H^{-1}\). In: Finite volumes for complex applications IV, pp. 133–142. ISTE, London (2005)
2.
Bramble, J.H.: A proof of the inf-sup condition for the Stokes equations on Lipschitz domains. Math. Models Meth. Appl. Sci. 13(3), 361–371 (2003). (Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday)
3.
Chénier, R., Eymard, R., Herbin, R.: An Extension of the Mac Scheme to Some Unstructured Meshes. In: Finite Volumes for Complex Applications VI, vol. 1, pp. 253–261. Springer, London, Finite Volumes for Complex Applications VI (FVCA VI). Czech Republic, June, Prague (2011)
4.
Chou, S.H., Kwak, D.Y.: A covolume method based on rotated bilinears for the generalized Stokes problem. SIAM J. Numer. Anal. 35(2), 494–507 (1998). (electronic)CrossRefMATHMathSciNet
5.
Demirdzic, I., Lilek, Z., Peric, M.: Fluid flow and heat transfer test problems for non-orthogonal grids: Bench-mark solutions. Int. J. Numer. Meth. Fluids 15, 329–354 (1992)CrossRef
6.
Eymard, R., Fuhrmann, J., Linke, A.: MAC Schemes on Triangular Meshes. In: Finite Volumes for Complex Applications VI, vol. 1, pp. 399–407. Springer, London. Finite Volumes for Complex Applications VI (FVCA VI), Czech Republic, June, Prague (2011)
7.
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.-L. (eds.) Techniques of Scientific Computing. Part III, Handbook of Numerical Analysis, VII, pp. 713–1020. North-Holland, Amsterdam (2000)
8.
Eymard, R., Gallouët, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30(4), 1009–1043 (2010)CrossRefMATHMathSciNet
9.
Eymard, R., Herbin, R., Latché, J.-C.: Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3d meshes. SIAM J. Numer. Anal. 45(1), 1–36 (2007)CrossRefMATHMathSciNet
10.
Fabrie, P., Gallouët, T.: Modeling wells in porous media flow. Math. Models Meth. Appl. Sci. 10(5), 673–709 (2000)CrossRefMATH
11.
Gallouët, T., Latché, J.-C.: Compactness of discrete approximate solutions to parabolic pdes, application to a turbulence model. Commun. Pure Appl. Anal. 11(6), 2371–2391 (2012)CrossRefMATHMathSciNet
12.
Girault, V., Lopez, H.: Finite-element error estimates for the MAC scheme. IMA J. Numer. Anal. 16(3), 247–379 (1996)CrossRefMathSciNet
13.
Han, H., Wu, X.: A new mixed finite element formulation and the MAC method for the Stokes equations. SIAM J. Numer. Anal. 35(2), 560–571 (1998). (electronic)CrossRefMATHMathSciNet
14.
Harlow, F., Welch, J.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 8, 2182–2189 (1965)CrossRefMATH
15.
Kanschat, G.: Divergence-free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme. Internat. J. Numer. Meth. Fluids 56(7), 941–950 (2008)CrossRefMATHMathSciNet
16.
Nečas, J.: Les méthodes Directes en théorie des équations Elliptiques. Masson et Cie, Éditeurs (1967)MATH
17.
18.
19.
Nicolaides, R.A.: The Covolume Approach to Computing Incompressible Flows. In: Incompressible Computational Fluid Dynamics: Trends and Advances, pp. 295–333. Cambridge University Press, Cambridge (2008)
20.
Patankar, S.: Numerical Heat Transfer and Fluid Flow. Series in Computational Methods in Mechanics and Thermal Sciences, vol. XIII. Hemisphere Publishing Corporation, New York. McGraw-Hill Book Company, New York (1980)
21.
Porsching, T.A.: Error estimates for MAC-like approximations to the linear Navier-Stokes equations. Numer. Math. 29(3), 291–306 (1978)CrossRefMATHMathSciNet
22.
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)CrossRefMATHMathSciNet
23.
Simon, J.: Compact sets in the space lp(0, t;b). Annali di Matematica Pura ed Applicata 146, 65–96 (1987)CrossRefMATH
24.
Temam, R.: Navier-Stokes Equations, volume 2 of Studies in Mathematics and its Applications, 3rd (edn.). Theory and Numerical Analysis, with an Appendix by F, Thomasset. North-Holland Publishing Co., Amsterdam (1984)
25.
Vidović, D., Segal, A., Wesseling, P.: A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids. J. Comput. Phys. 217(2), 277–294 (2006)CrossRefMATHMathSciNet
26.
Wesseling, P.: Principles of Computational Fluid Dynamics. Springer, New York (2001)CrossRef
Metadata
Title
An extension of the MAC scheme to locally refined meshes: convergence analysis for the full tensor time-dependent Navier–Stokes equations
Authors
Eric Chénier
Robert Eymard
Thierry Gallouët
Raphaèle Herbin
Publication date
01-03-2015
Publisher
Springer Milan
Published in
Calcolo / Issue 1/2015
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-014-0108-x

Other articles of this Issue 1/2015

Calcolo 1/2015 Go to the issue

OriginalPaper

Lower and upper bounds of Stokes eigenvalue problem based on stabilized finite element methods

OriginalPaper

The palindromic cyclic reduction and related algorithms

OriginalPaper

Positive blending Hermite rational cubic spline fractal interpolation surfaces

OriginalPaper

Mimetic finite difference approximation of quasilinear elliptic problems

Premium Partner

    Image Credits