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

Boyer, Franck; Krell, Stella; Nabet, Flore

doi:10.1090/mcom/2956

Inf-Sup stability of the discrete duality finite volume method for the 2D Stokes problem
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by Franck Boyer, Stella Krell and Flore Nabet PDF

Math. Comp. 84 (2015), 2705-2742 Request permission

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.

References

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, DOI 10.1093/imanum/drr046
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, DOI 10.1002/num.20170
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, DOI 10.1007/BF02576171
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, DOI 10.1016/j.jcp.2009.06.034
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, DOI 10.1137/090767029
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, DOI 10.1137/090757411
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, DOI 10.1007/978-3-540-78319-0
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, DOI 10.1007/978-1-4614-5975-0
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, DOI 10.1007/978-1-4612-3172-1
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. MR 1921922, DOI 10.1137/S0036142900380121
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, DOI 10.1137/100786046
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
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
S. Delcourte, Développement de méthodes de volumes finis pour la mécanique des fluides, Ph.D. thesis, http://tel.archives-ouvertes.fr/tel-00200833/fr/, Université Paul Sabatier, Toulouse, France, 2007.
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, DOI 10.1007/978-3-642-22980-0
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, DOI 10.1051/m2an:2005047
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, DOI 10.1002/num.20333
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
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
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 464543, DOI 10.1051/m2an/1977110403411
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, DOI 10.1007/978-3-642-61623-5
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. MR 2085402, DOI 10.1090/S0025-5718-04-01652-7
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.
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, DOI 10.1016/j.cma.2007.01.005
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, DOI 10.1002/num.20603
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, DOI 10.1137/110831593
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, DOI 10.1016/0020-7225(81)90013-6
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, DOI 10.1137/0729091
Yousef Saad, Iterative methods for sparse linear systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. MR 1990645, DOI 10.1137/1.9780898718003
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, DOI 10.1051/m2an/1984180201751