Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows

Robert Eymard; Raphaèle Herbin; Jean-Claude Latché; Bruno Piar

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 5, page 889-927
  • ISSN: 0764-583X

Abstract

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We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

How to cite

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Eymard, Robert, et al. "Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows." ESAIM: Mathematical Modelling and Numerical Analysis 43.5 (2009): 889-927. <http://eudml.org/doc/250597>.

@article{Eymard2009,
abstract = { We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm. },
author = {Eymard, Robert, Herbin, Raphaèle, Latché, Jean-Claude, Piar, Bruno},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite volumes; collocated discretizations; Stokes problem; Navier-Stokes equations; incompressible flows; analysis; finite volumes},
language = {eng},
month = {8},
number = {5},
pages = {889-927},
publisher = {EDP Sciences},
title = {Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows},
url = {http://eudml.org/doc/250597},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Eymard, Robert
AU - Herbin, Raphaèle
AU - Latché, Jean-Claude
AU - Piar, Bruno
TI - Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/8//
PB - EDP Sciences
VL - 43
IS - 5
SP - 889
EP - 927
AB - We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm.
LA - eng
KW - Finite volumes; collocated discretizations; Stokes problem; Navier-Stokes equations; incompressible flows; analysis; finite volumes
UR - http://eudml.org/doc/250597
ER -

References

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  1. F. Archambeau, N. Méchitoua and M. Sakiz, Code saturne: A finite volume code for turbulent flows. International Journal of Finite Volumes1 (2004), .  URIhttp://www.latp.univ-mrs.fr/IJFV/
  2. M. Bern, D. Eppstein and J. Gilbert, Provably good mesh generation. J. Comput. System Sci.48 (1994) 384–409.  Zbl0799.65119
  3. F. Boyer and P. Fabrie, Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles, Mathématiques et Applications52. Springer-Verlag (2006).  
  4. F. Brezzi and M. Fortin, A minimal stabilisation procedure for mixed finite element methods. Numer. Math.89 (2001) 457–491.  Zbl1009.65067
  5. E. Chénier, R. Eymard and O. Touazi, Numerical results using a colocated finite-volume scheme on unstructured grids for incompressible fluid flows. Numer. Heat Transf. Part B: Fundam.49 (2006) 259–276.  
  6. E. Chénier, R. Eymard, R. Herbin and O. Touazi, Collocated finite volume schemes for the simulation of natural convective flows on unstructured meshes. Int. J. Num. Methods Fluids56 (2008) 2045–2068.  Zbl1133.76028
  7. Y. Coudière, T. Gallouët and R. Herbin, Discrete Sobolev inequalities and LP error estimates for finite volume solutions of convection diffusion equations. ESAIM: M2AN35 (2001) 767–778.  Zbl0990.65122
  8. K. Deimling, Nonlinear functional analysis. Springer-Verlag (1985).  Zbl0559.47040
  9. R. Eymard and T. Gallouët, H-convergence and numerical schemes for elliptic equations. SIAM J. Numer. Anal.41 (2003) 539–562.  Zbl1049.35015
  10. R. Eymard and R. Herbin, A new colocated finite volume scheme for the incompressible Navier-Stokes equations on general non-matching grids. C. R. Acad. Sci., Sér. I Math.344 (2007) 659–662.  Zbl1114.76047
  11. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handbook of Numerical AnalysisVII. North Holland (2000) 713–1020.  Zbl0981.65095
  12. R. Eymard, T. Gallouët and R. Herbin, A finite volume scheme for anisotropic diffusion problems. C. R. Acad. Sci., Sér. I Math.339 (2004) 299–302.  Zbl1055.65124
  13. R. Eymard, R. Herbin and J.C. Latché, On a stabilized colocated finite volume scheme for the Stokes problem. ESAIM: M2AN40 (2006) 501–528.  Zbl1160.76370
  14. R. Eymard, T. Gallouët, R. Herbin and J.-C. Latché, Analysis tools for finite volume schemes. Acta Mathematica Universitatis Comenianae76 (2007) 111–136.  Zbl1133.65062
  15. R. Eymard, R. Herbin and J.C. Latché, Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes. SIAM J. Numer. Anal.45 (2007) 1–36.  Zbl1173.76028
  16. R. Eymard, R. Herbin, J.C. Latché and B. Piar, On the stability of colocated clustered finite volume simplicial discretizations for the 2D Stokes problem. Calcolo44 (2007) 219–234.  Zbl1137.76062
  17. L.P. Franca and R. Stenberg, Error analysis of some Galerkin Least Squares methods for the elasticity equations. SIAM J. Numer. Anal.28 (1991) 1680–1697.  Zbl0759.73055
  18. T. Gallouët, R. Herbin and M.H. Vignal, Error estimates for the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal.37 (2000) 1935–1972.  Zbl0986.65099
  19. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations – Theory and Algorithms, Springer Series in Computational Mathematics5. Springer-Verlag (1986).  Zbl0585.65077
  20. J. Nečas, Équations aux dérivées partielles. Presses de l'Université de Montréal (1965).  
  21. L.E. Payne and H.F. Weinberger, An optimal Poincaré-inequality for convex domains. Arch. Rational Mech. Anal.5 (1960) 286–292.  Zbl0099.08402
  22. B. Piar, PELICANS : Un outil d'implémentation de solveurs d'équations aux dérivées partielles. Note Technique 2004/33, IRSN/DPAM/SEMIC (2004).  
  23. R. Temam, Navier-Stokes Equations, Studies in mathematics and its applications. North-Holland (1977).  
  24. R. Verfürth, Error estimates for some quasi-interpolation operators. ESAIM: M2AN33 (1999) 695–713.  Zbl0938.65125

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