Finite Volume Box Schemes and Mixed Methods

Jean-Pierre Croisille

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 5, page 1087-1106
  • ISSN: 0764-583X

Abstract

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We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form div ϕ ( u , ∇ u ) = f . The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for u and the flux φ. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown u and the div-conforming space of Raviart-Thomas for the flux φ. The two test spaces are the functions constant per cell both for the conservative and for the flux equations. We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method.

How to cite

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Croisille, Jean-Pierre. "Finite Volume Box Schemes and Mixed Methods." ESAIM: Mathematical Modelling and Numerical Analysis 34.5 (2010): 1087-1106. <http://eudml.org/doc/197549>.

@article{Croisille2010,
abstract = { We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form $\mathop\{\rm div\}\nolimits\varphi(u,\nabla u)=f$. The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for u and the flux φ. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown u and the div-conforming space of Raviart-Thomas for the flux φ. The two test spaces are the functions constant per cell both for the conservative and for the flux equations. We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method. },
author = {Croisille, Jean-Pierre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Box method; box scheme; mixed finite element method; Petrov-Galerkin method; finite volume method.; Poisson equation; mixed Petrov-Galerkin finite volume schemes; error estimate},
language = {eng},
month = {3},
number = {5},
pages = {1087-1106},
publisher = {EDP Sciences},
title = {Finite Volume Box Schemes and Mixed Methods},
url = {http://eudml.org/doc/197549},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Croisille, Jean-Pierre
TI - Finite Volume Box Schemes and Mixed Methods
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 5
SP - 1087
EP - 1106
AB - We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form $\mathop{\rm div}\nolimits\varphi(u,\nabla u)=f$. The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for u and the flux φ. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown u and the div-conforming space of Raviart-Thomas for the flux φ. The two test spaces are the functions constant per cell both for the conservative and for the flux equations. We prove an optimal second order error estimate for the box scheme and we emphasize the link between this scheme and the post-processing of Arnold and Brezzi of the classical mixed method.
LA - eng
KW - Box method; box scheme; mixed finite element method; Petrov-Galerkin method; finite volume method.; Poisson equation; mixed Petrov-Galerkin finite volume schemes; error estimate
UR - http://eudml.org/doc/197549
ER -

References

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  1. B. Achchab, A. Agouzal, J. Baranger and J-F. Maître, Estimateur d'erreur a posteriori hiérarchique. Application aux éléments finis mixtes. Numer. Math.80 (1998) 159-179.  
  2. D.N. Arnold and F. Brezzi, Mixed and non-conforming finite elements methods: implementation, postprocessing and error estimates. RAIRO - Modél. Math. Anal. Numér.19 (1985) 7-32.  
  3. I. Babuska, Error-Bounds for Finite Elements Method. Numer. Math.16 (1971) 322-333.  
  4. R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal.24 (1987) 777-787.  
  5. J. Baranger, J.F. Maître and F. Oudin, Connection between finite volume and mixed finite element methods. RAIRO - Modél. Math. Anal. Numér.30 (1996) 445-465.  
  6. C. Bernardi, C. Canuto and Y. Maday, Un problème variationnel abstrait. Application à une méthode de collocation pour les équations de Stokes. C. R. Acad. Sci. Paris, t.303, Série I19 (1986) 971-974.  
  7. C. Bernardi, C. Canuto and Y. Maday, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal.25 (1988) 1237-1271.  
  8. D. Braess, Finite Elements. Cambridge Univ. Press (1997).  
  9. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Texts Appl. Math.15 (1994) Springer, New-York.  
  10. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems, arising from lagrangian multipliers. RAIRO8 (1974) R-2, 129-151.  
  11. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series Comp. Math.15, Springer Verlag, New-York (1991).  
  12. F. Brezzi, J. Douglas and L.D. Marini, Two families of Mixed Finite Element for second order elliptic problems. Numer. Math.47 (1985) 217-235.  
  13. Z. Cai, J. Mandel and S. McCormick, The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal.28 (1991) 392-402.  
  14. F. Casier, H. Deconninck and C. Hirsch, A class of central bidiagonal schemes with implicit boundary conditions for the solution of Euler's equations. AIAA-83-0126 (1983).  
  15. J.J. Chattot, Box-schemes for First Order Partial Differential Equations. Adv. Comp. Fluid Dynamics, Gordon Breach Publ. (1995) 307-331.  
  16. J.J. Chattot, A Conservative Box-scheme for the Euler Equations. Int. J. Num. Meth. Fluids (to appear).  
  17. J.J. Chattot and S. Malet, A box-schemefor the Euler equations. Lect. Notes Math.1270, Springer-Verlag, Berlin (1987) 82-99.  
  18. Y. Coudière, J-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. Math. Model. Numer.33 (1999) 493-516.  
  19. B. Courbet, Schémas boîte en réseau triangulaire, Rapport technique 18/3446 EN (1992), ONERA, unpublished.  
  20. B. Courbet, Schémas à deux points pour la simulation numérique des écoulements, La Recherche Aérospatiale n°4 (1990) 21-46.  
  21. B. Courbet, Étude d'une famille de schémas boîtes à deux points et application à la dynamique des gaz monodimensionnelle, La Recherche Aérospatiale n°5 (1991) 31-44.  
  22. B. Courbet and J.P. Croisille, Finite Volume Box Schemes on triangular meshes. Math. Model. Numer.32 (1998) 631-649.  
  23. J-P. Croisille, Finite Volume Box Schemes, in Proc. of the 2nd Int. Symp. on Finite Volume for Complex Applications. Hermes, Paris (1999).  
  24. M. Crouzeix and P.A. Raviart, Conforming and non conforming finite element methods for solving the stationary Stokes equations I. RAIRO7 (1973) R-3, 33-76.  
  25. F. Dubois, Finite volumes and mixed Petrov-Galerkin finite elements; the unidimensional problem. Num. Meth. PDE (to appear).  
  26. R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, in Handbook of Numerical Analysis, Ciarlet-Lions Eds. 5 (1997).  
  27. G. Fairweather and R.D. Saylor, The reformulation and numerical solution of certain nonclassical initial-boundary value problems. SIAM J. Sci. Stat. Comput.12 (1991) 127-144.  
  28. L. Fezoui and B. Stoufflet, A class of implicit upwind schemes for Euler equations on unstructured grids. J. Comp. Phys.84 (1989) 174-206.  
  29. V. Girault and P.A. Raviart, Finite Element Approximation of the Navier-Stokes equations. Lect. Notes Math.749, Springer, Berlin (1979).  
  30. W. Hackbusch, On first and second order box schemes. Computing41 (1989) 277-296.  
  31. H.B. Keller, A new difference scheme for parabolic problems, Numerical solutions of partial differential equations, II, B. Hubbard Ed., Academic Press, New-York (1971) 327-350.  
  32. R.D. Lazarov, J.E. Pasciak and P.S. Vassilevski, Coupling mixed and finite volume discretizations of convection-diffusion-reaction equations on non-matching grids, in Proc. of the 2nd Int. Symp. on Finite Volume for Complex Applications, Hermes, Paris (1999).  
  33. L.D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal.22 (1985) 493-496.  
  34. P.C. Meek and J. Norbury, Nonlinear moving boundary problems and a Keller box scheme. SIAM J. Numer. Anal.21 (1984) 883-893.  
  35. R.A. Nicolaides, Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal.19 (1982) 349-357.  
  36. P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems. Lect. Notes Math.606, Springer-Verlag, Berlin (1977) 292-315.  
  37. E. Süli, Convergence of finite volume schemes for Poisson's equation on non-uniform meshes. SIAM J. Numer. Anal.28 (1991) 1419-1430.  
  38. E. Süli, The accuracy of cell vertex finite volume methods on quadrilateral meshes. Math. of Comp.59 (1992) 359-382.  
  39. T. Schmidt, Box Schemes on quadrilateral meshes. Computing51 (1993) 271-292.  
  40. J-M Thomas and D. Trujillo, Mixed Finite Volume methods. Int. J. Num. Meth. Eng.45 (1999) to appear.  
  41. S.F. Wornom, Application of compact difference schemes to the conservative Euler equations for one-dimensional flows. NASA Tech. Mem.83262 (1982).  
  42. S.F. Wornom and M.M. Hafez, Implicit conservative schemes for the Euler equations. AIAA J.24 (1986) 215-233.  
  43. A. Younes, R. Mose, P. Ackerer and G. Chavent, A new formulation of the Mixed Finite Element Method for solving elliptic and parabolic PDE. J. Comp. Phys.149 (1999) 148-167.  

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