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Finite volume schemes for the biharmonic problem on general meshes


Authors: R. Eymard, T. Gallouët, R. Herbin and A. Linke
Journal: Math. Comp. 81 (2012), 2019-2048
MSC (2010): Primary 65N08
DOI: https://doi.org/10.1090/S0025-5718-2012-02608-1
Published electronically: April 16, 2012
MathSciNet review: 2945146
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Abstract: During the development of a convergence theory for Nicolaides' extension of the classical MAC scheme for the incompressible Navier-Stokes equations to unstructured triangle meshes, it became clear that a convergence theory for a new kind of finite volume discretizations for the biharmonic problem would be a very useful tool in the convergence analysis of the generalized MAC scheme. Therefore, we present and analyze new finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions on grids which satisfy an orthogonality condition. We prove that a piecewise constant approximate solution of the biharmonic problem converges in $ L^2(\Omega )$ to the exact solution. Similar results are shown for the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. Error estimates are also derived. This part of the paper is a first, significant step towards a convergence theory of Nicolaides' extension of the classical MAC scheme. Further, we show that finite volume discretizations for the biharmonic problem can also be defined on very general, nonconforming meshes, such that the same convergence results hold. The possibility to construct a converging lowest order finite volume method for the $ H^2$-regular biharmonic problem on general meshes seems to be an interesting result for itself and clarifies the necessary ingredients for converging discretizations of the biharmonic problem. All these results are confirmed by numerical results.


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  • 1. L. Agelas, D. A. Di Pietro, and R. Masson.
    A symmetric and coercive finite volume scheme for multiphase porous media flow problems with applications in the oil industry.
    In Finite volumes for complex applications V, pages 35-51. ISTE, London, 2008.
  • 2. M. Ben-Artzi, I. Chorev, J.-P. Croisille, and D. Fishelov.
    A compact difference scheme for the biharmonic equation in planar irregular domains.
    SIAM J. Numer. Anal., 47(4):3087-3108, 2009. MR 2551159 (2010i:65235)
  • 3. M. Ben-Artzi, J.-P. Croisille, and D. Fishelov.
    A fast direct solver for the biharmonic problem in a rectangular grid.
    SIAM J. Sci. Comput., 31(1):303-333, 2008. MR 2460780 (2009j:35066)
  • 4. C. Bi and L. Li.
    Mortar finite volume method with Adini element for biharmonic problem.
    J. Comput. Math., 22(3):475-488, 2004. MR 2056302 (2005a:65126)
  • 5. S. C. Brenner and L.-Y. Sung.
    $ C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains.
    J. Sci. Comput., 22/23:83-118, 2005. MR 2142191 (2005m:65258)
  • 6. F. Brezzi and M. Fortin.
    Mixed and hybrid finite element methods.
    Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 7. G. Chen, Z. Li, and P. Lin.
    A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow.
    Adv. Comput. Math., 29(2):113-133, 2008. MR 2420868 (2009d:65145)
  • 8. P. Ciarlet.
    The finite element method.
    In P. G. Ciarlet and J.-L. Lions, editors, Part I, Handbook of Numerical Analysis, III. North-Holland, Amsterdam, 1991. MR 1115235 (92f:65001)
  • 9. P. Destuynder and M. Salaün.
    Mathematical analysis of thin plate models.
    In Mathématiques & Applications [Mathematics & Applications], volume 24. Springer-Verlag, Berlin, 1996. MR 1422248 (2001e:74056)
  • 10. R. Eymard, J. Fuhrmann, and A. Linke.
    MAC schemes on triangular meshes.
    WIAS in: Finite Volumes for Complex Applications VI, Problems and Perspectives, Springer Proccedings in Mathematics, 2011, pp. 399-407.
  • 11. R. Eymard and T. Gallouët.
    H-convergence and numerical schemes for elliptic equations.
    SIAM Journal on Numerical Analysis, 41(2):539-562, 2003. MR 2004187 (2004h:65077)
  • 12. R. Eymard, T. Gallouët, and R. Herbin.
    Finite volume methods.
    In P. G. Ciarlet and
    J.-L. Lions, editors, Techniques of Scientific Computing, Part III, Handbook of Numerical Analysis, VII, pages 713-1020. North-Holland, Amsterdam, 2000. MR 1804748 (2002e:65138)
  • 13. R. Eymard, T. Gallouët, and R. Herbin.
    A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension.
    IMA J. Numer. Anal., 26(2):326-353, 2006. MR 2218636 (2007a:65170)
  • 14. R. Eymard, T. Gallouët, and R. Herbin.
    Cell centred discretization of fully nonlinear elliptic problems on general multidimensional polyhedral grids.
    J. Numer. Math., 17(3):173-193, 2009. MR 2573566 (2010k:65230)
  • 15. R. Eymard, T. Gallouët, and R. Herbin.
    Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, sushi: a scheme using stabilisation and hybrid interfaces.
    IMA J. Numer. Anal., 2009.
    Advance Access published on September 7, 2009 IMAJNA, see also http://hal.archives-ouvertes.fr/. MR 2727814
  • 16. R. Eymard, T. Gallouët, R. Herbin, and J.-C. Latché.
    Analysis tools for finite volume schemes.
    Acta Math. Univ. Comenian. (N.S.), 76(1):111-136, 2007. MR 2331058 (2008e:65243)
  • 17. T. Gallouët, R. Herbin, and M.-H. Vignal.
    Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions.
    SIAM J. Numer. Anal., 37(6):1935-1972 (electronic), 2000. MR 1766855 (2001e:65168)
  • 18. M. Gander and G. Wanner.
    From Euler, Ritz and Galerkin to modern computing.
    to appear in SIAM Review, 2011.
  • 19. E. H. Georgoulis and P. Houston.
    Discontinuous Galerkin methods for the biharmonic problem.
    IMA J. Numer. Anal., 29(3):573-594, 2009. MR 2520159 (2010g:65209)
  • 20. T. Gudi, N. Nataraj, and A. K. Pani.
    Mixed discontinuous Galerkin finite element method for the biharmonic equation.
    J. Sci. Comput., 37(2):139-161, 2008. MR 2453216 (2009j:65319)
  • 21. C. A. Hall, J. C. Cavendish, and W. H. Frey.
    The dual variable method for solving fluid flow difference equations on Delaunay triangulations.
    Comput. & Fluids, 20(2):145-164, 1991. MR 1123813 (92g:76059)
  • 22. F. H. Harlow and J. E. Welch.
    Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface.
    Physics of fluids, 8(12):2182-2189, 1965.
  • 23. I. Mozolevski, E. Süli, and P. R. Bösing.
    $ hp$-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation.
    J. Sci. Comput., 30(3):465-491, 2007. MR 2295480 (2008c:65341)
  • 24. J. Nicolaides, T. A. Porsching, and C. A. Hall.
    Covolume methods in computational fluid dynamics.
    In M. Hafez and K. Oshma, editors, Computation Fluid Dynamics Review, pages 279-299. John Wiley and Sons, New York, 1995.
  • 25. R. A. Nicolaides.
    Analysis and convergence of the MAC scheme. I. The linear problem.
    SIAM J. Numer. Anal., 29(6):1579-1591, 1992. MR 1191137 (93j:65143)
  • 26. S. V. Patankar.
    Numerical heat transfer and fluid flow.
    Series in Computational Methods in Mechanics and Thermal Sciences, Minkowycz and Sparrow Eds., 1980.
  • 27. W. Ritz.
    Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik.
    Journal für die reine und angewandte Mathematik:111-114, 1857, Analysis and Applications, 135:1-61, 1908.
  • 28. E. Süli and I. Mozolevski.
    $ hp$-version interior penalty DGFEMs for the biharmonic equation.
    Comput. Methods Appl. Mech. Engrg., 196(13-16):1851-1863, 2007. MR 2298696 (2008c:65350)
  • 29. T. Wang.
    A mixed finite volume element method based on rectangular mesh for biharmonic equations.
    J. Comput. Appl. Math., 172(1):117-130, 2004. MR 2091134 (2005h:65190)

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Additional Information

R. Eymard
Affiliation: Université Paris-Est Marne-la-Valleé, Laboratoire d′Analyse et Mathématiques Appliquées, 5 boulevard Descartes, F-77454 Marne la Vallée Cedex 2
Email: robert.eymard@univ-mlv.fr

T. Gallouët
Affiliation: Université Aix-Marseille, Laboratoire d′Analyse, Topologie et Probabilités, 39 rue Joliot-Curie, F-13453 Marseille cedex France
Email: thierry.gallouet@latp.univ-mrs.fr

R. Herbin
Affiliation: Université Aix-Marseille, Laboratoire d′Analyse, Topologie et Probabilités, 39 rue Joliot-Curie, F-13453 Marseille cedex France
Email: raphaele.herbin@latp.univ-mrs.fr

A. Linke
Affiliation: Weierstrass Institute, Mohrenstrasse 39, D-10117 Berlin, Germany
Email: alexander.linke@wias-berlin.de

DOI: https://doi.org/10.1090/S0025-5718-2012-02608-1
Keywords: Biharmonic problem, finite volume scheme, convergence analysis, error estimate
Received by editor(s): April 16, 2010
Received by editor(s) in revised form: April 22, 2011
Published electronically: April 16, 2012
Article copyright: © Copyright 2012 American Mathematical Society

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