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Inf-Sup stability of the discrete duality finite volume method for the 2D Stokes problem

Authors: Franck Boyer, Stella Krell and Flore Nabet
Journal: Math. Comp. 84 (2015), 2705-2742
MSC (2010): Primary 65N08, 65N12, 76D07, 76M12
Published electronically: April 29, 2015
MathSciNet review: 3378845
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Abstract: ``Discrete Duality Finite Volume'' schemes (DDFV for short) on general 2D meshes, in particular, non-conforming ones, are studied for the Stokes problem with Dirichlet boundary conditions. The DDFV method belongs to the class of staggered schemes since the components of the velocity and the pressure are approximated on different meshes. In this paper, we investigate from a numerical and theoretical point of view, whether or not the stability condition holds in this framework for various kinds of mesh families. We obtain that different behaviors may occur depending on the geometry of the meshes.

For instance, for conforming acute triangle meshes, we prove the unconditional Inf-Sup stability of the scheme, whereas for some conforming or non-conforming Cartesian meshes we prove that Inf-Sup stability holds up to a single unstable pressure mode. In any case, the DDFV method appears to be very robust.

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  • [1] Boris Andreianov, Mostafa Bendahmane, Florence Hubert, and Stella Krell, On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality, IMA J. Numer. Anal. 32 (2012), no. 4, 1574-1603. MR 2991838,
  • [2] Boris Andreianov, Franck Boyer, and Florence Hubert, Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes, Numer. Methods Partial Differential Equations 23 (2007), no. 1, 145-195. MR 2275464 (2008c:65283),
  • [3] D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984), no. 4, 337-344 (1985). MR 799997 (86m:65136),
  • [4] L. Beirão da Veiga, V. Gyrya, K. Lipnikov, and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes, J. Comput. Phys. 228 (2009), no. 19, 7215-7232. MR 2568590 (2010k:65229),
  • [5] L. Beirão da Veiga and K. Lipnikov, A mimetic discretization of the Stokes problem with selected edge bubbles, SIAM J. Sci. Comput. 32 (2010), no. 2, 875-893. MR 2609344 (2011e:65255),
  • [6] L. Beirão da Veiga, K. Lipnikov, and G. Manzini, Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes, SIAM J. Numer. Anal. 48 (2010), no. 4, 1419-1443. MR 2684341 (2011m:65246),
  • [7] Daniele Boffi, Franco Brezzi, Leszek F. Demkowicz, Ricardo G. Durán, Richard S. Falk, and Michel Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications, Lecture Notes in Mathematics, vol. 1939, Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008. Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26-July 1, 2006; Edited by Boffi and Lucia Gastaldi. MR 2459075 (2010h:65219)
  • [8] Franck Boyer and Pierre Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, vol. 183, Springer, New York, 2013. MR 2986590
  • [9] Franco Brezzi and Michel Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • [10] Bernardo Cockburn, Guido Kanschat, Dominik Schötzau, and Christoph Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal. 40 (2002), no. 1, 319-343 (electronic). MR 1921922 (2003g:65141),
  • [11] Yves Coudière and Florence Hubert, A 3D discrete duality finite volume method for nonlinear elliptic equations, SIAM J. Sci. Comput. 33 (2011), no. 4, 1739-1764. MR 2831032 (2012m:65378),
  • [12] Y. Coudière, C. Pierre, O. Rousseau, and R. Turpault, A 2D/3D discrete duality finite volume scheme. Application to ECG simulation, Int. J. Finite Vol. 6 (2009), no. 1, 24. MR 2500950 (2010e:65187)
  • [13] 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 0343661 (49 #8401)
  • [14] S. Delcourte, Développement de méthodes de volumes finis pour la mécanique des fluides, Ph.D. thesis,, Université Paul Sabatier, Toulouse, France, 2007.
  • [15] Daniele Antonio Di Pietro and Alexandre Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148
  • [16] Komla Domelevo and Pascal Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, M2AN Math. Model. Numer. Anal. 39 (2005), no. 6, 1203-1249. MR 2195910 (2006j:65312),
  • [17] Jérôme Droniou and Robert Eymard, Study of the mixed finite volume method for Stokes and Navier-Stokes equations, Numer. Methods Partial Differential Equations 25 (2009), no. 1, 137-171. MR 2473683 (2009k:65208),
  • [18] Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138 (2005d:65002)
  • [19] 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 (2002e:65138)
  • [20] 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 (2007d:65100),
  • [21] Michel Fortin, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér. 11 (1977), no. 4, 341-354, iii (English, with French summary). MR 0464543 (57 #4473)
  • [22] Vivette Girault and Pierre-Arnaud Raviart, Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383 (88b:65129)
  • [23] Vivette Girault, Béatrice Rivière, and Mary F. Wheeler, A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems, Math. Comp. 74 (2005), no. 249, 53-84 (electronic). MR 2085402 (2005f:65149),
  • [24] F. Harlow and J. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, The physics of fluids 8 (1965), no. 12, 2182-2189.
  • [25] F. Hermeline, Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 21-24, 2497-2526. MR 2319051 (2008d:65123),
  • [26] Stella Krell, Stabilized DDFV schemes for Stokes problem with variable viscosity on general 2D meshes, Numer. Methods Partial Differential Equations 27 (2011), no. 6, 1666-1706. MR 2838314 (2012j:65368),
  • [27] Stella Krell and Gianmarco Manzini, The discrete duality finite volume method for Stokes equations on three-dimensional polyhedral meshes, SIAM J. Numer. Anal. 50 (2012), no. 2, 808-837. MR 2914287,
  • [28] D. S. Malkus, Eigenproblems associated with the discrete LBB condition for incompressible finite elements, Internat. J. Engrg. Sci. 19 (1981), no. 10, 1299-1310. MR 660563 (83k:73053),
  • [29] R. A. Nicolaides, Analysis and convergence of the MAC scheme. I. The linear problem, SIAM J. Numer. Anal. 29 (1992), no. 6, 1579-1591. MR 1191137 (93j:65143),
  • [30] Yousef Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. MR 1990645 (2004h:65002)
  • [31] R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations, RAIRO Anal. Numér. 18 (1984), no. 2, 175-182. MR 743884 (85i:65156)

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Additional Information

Franck Boyer
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France

Stella Krell
Affiliation: Université de Nice Sophia-Antipolis, CNRS, LJAD UMR 7351, Nice, France

Flore Nabet
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France

Keywords: Finite-volume methods, Stokes problem, DDFV methods, Inf-Sup stability
Received by editor(s): February 27, 2013
Received by editor(s) in revised form: December 20, 2013, and March 12, 2014
Published electronically: April 29, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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