Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes

Martin Vohralík

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 2, page 367-391
  • ISSN: 0764-583X

Abstract

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We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive definite for a large class of problems, but in general nonsymmetric. We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection–diffusion–reaction problems. Besides the theoretical relationship between the two methods, the results allow for important computational savings in the mixed finite element method, which we finally illustrate on a set of numerical experiments.

How to cite

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Vohralík, Martin. "Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes." ESAIM: Mathematical Modelling and Numerical Analysis 40.2 (2006): 367-391. <http://eudml.org/doc/249720>.

@article{Vohralík2006,
abstract = { We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive definite for a large class of problems, but in general nonsymmetric. We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection–diffusion–reaction problems. Besides the theoretical relationship between the two methods, the results allow for important computational savings in the mixed finite element method, which we finally illustrate on a set of numerical experiments. },
author = {Vohralík, Martin},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Mixed finite element method; saddle-point problem; finite volume method; second-order elliptic equation; nonlinear parabolic convection–diffusion–reaction equation.; mixed methods; finite volume method; Raviart-Thomas element; elliptic problems of second order; nonlinear parabolic convection-diffusion equations; numerical examples},
language = {eng},
month = {6},
number = {2},
pages = {367-391},
publisher = {EDP Sciences},
title = {Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes},
url = {http://eudml.org/doc/249720},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Vohralík, Martin
TI - Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/6//
PB - EDP Sciences
VL - 40
IS - 2
SP - 367
EP - 391
AB - We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive definite for a large class of problems, but in general nonsymmetric. We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection–diffusion–reaction problems. Besides the theoretical relationship between the two methods, the results allow for important computational savings in the mixed finite element method, which we finally illustrate on a set of numerical experiments.
LA - eng
KW - Mixed finite element method; saddle-point problem; finite volume method; second-order elliptic equation; nonlinear parabolic convection–diffusion–reaction equation.; mixed methods; finite volume method; Raviart-Thomas element; elliptic problems of second order; nonlinear parabolic convection-diffusion equations; numerical examples
UR - http://eudml.org/doc/249720
ER -

References

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  1. I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM J. Sci. Comput.19 (1998) 1700–1716.  
  2. I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: Discussion and numerical results. SIAM J. Sci. Comput.19 (1998) 1717–1736.  
  3. M. Aftosmis, D. Gaitonde and T. Sean Tavares, On the accuracy, stability and monotonicity of various reconstruction algorithms for unstructured meshes. AIAA (1994), paper No. 94-0415.  
  4. A. Agouzal, J. Baranger, J.-F. Maître and F. Oudin, Connection between finite volume and mixed finite element methods for a diffusion problem with nonconstant coefficients. Application to a convection diffusion problem. East-West J. Numer. Math.3 (1995) 237–254.  
  5. T. Arbogast, M.F. Wheeler and N. Zhang, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal.33 (1996) 1669–1687.  
  6. T. Arbogast, M.F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal.34 (1997) 828–852.  
  7. T. Arbogast, C.N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput.19 (1998) 404–425.  
  8. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér.19 (1985) 7–32.  
  9. 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.  
  10. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).  
  11. F. Brezzi, J. Douglas Jr. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217–235.  
  12. F. Brezzi, J. Douglas Jr., R. Duran and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math.51 (1987) 237–250.  
  13. G. Chavent, A. Younès and Ph. Ackerer, On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles. Comput. Methods Appl. Mech. Engrg.192 (2003) 655–682.  
  14. Z. Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems. East-West J. Numer. Math.4 (1996) 1–33.  
  15. Y. Coudière, J.-P. Vila and Villedieu Ph., Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN33 (1999) 493–516.  
  16. C. Dawson, Analysis of an upwind-mixed finite element method for nonlinear contaminnat transport equations. SIAM J. Numer. Anal.35 (1998) 1709–1724.  
  17. C. Dawson and V. Aizinger, Upwind-mixed methods for transport equations. Comput. Geosci.3 (1999) 93–110.  
  18. J. Douglas Jr. and J.E. Roberts, Global estimates for mixed methods for second order elliptic equations. Math. Comp.44 (1985) 39–52.  
  19. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, Ph.G. Ciarlet and J.-L. Lions Eds. Elsevier Science B.V., Amsterdam 7 (2000) 713–1020.  
  20. R. Eymard, T. Gallouët and R. Herbin, A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal.26 (2006) 326–353.  
  21. I. Faille, A control volume method to solve an elliptic equation on a two-dimensional irregular mesh. Comput. Methods Appl. Mech. Engrg.100 (1992) 275–290.  
  22. J.R. Gilbert, C. Moler and R. Schreiber, Sparse matrices in MATLAB: Design and implementation. SIAM J. Matrix Anal. Appl.13 (1992) 333–356.  
  23. M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand.49 (1952) 409–436.  
  24. H. Hoteit, J. Erhel, R. Mosé, B. Philippe and Ph. Ackerer, Numerical reliability for mixed methods applied to flow problems in porous media. Comput. Geosci.6 (2002) 161–194.  
  25. J. Jaffré, Éléments finis mixtes et décentrage pour les équations de diffusion-convection. Calcolo23 (1984) 171–197.  
  26. L. Jeannin, I. Faille and T. Gallouët, Comment modéliser les écoulements diphasiques compressibles sur des grilles hybrides ? Oil & Gas Science and Technology – Rev. IFP55 (2000) 269–279.  
  27. R.A. Klausen and G.T. Eigestad, Multi point flux approximations and finite element methods; practical aspects of discontinuous media, Proc. 9th European Conference on the Mathematics of Oil Recovery, Cannes, France, B003 (2004).  
  28. R.A. Klausen and T.F. Russell, Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput. Geosci.8 (2004) 341–377.  
  29. 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.  
  30. J.C. Nédélec, Mixed finite elements in 3 . Numer. Math.35 (1980) 315–341.  
  31. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994).  
  32. P.-A. Raviart and J.-M. Thomas, A mixed finite element method for 2-nd order elliptic problems, in Mathematical Aspects of Finite Element Methods. Galligani I., Magenes E. Eds., Lect. Notes Math., Springer, Berlin 606 (1977) 292–315.  
  33. J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Ph.G. Ciarlet and J.-L. Lions Eds., Elsevier Science B.V., Amsterdam 2 (1991) 523–639.  
  34. T.F. Russell and M.F. Wheeler, Finite element and finite difference methods for continuous flows in porous media, in The Mathematics of Reservoir Simulation, R.E. Ewing Ed., SIAM, Philadelphia (1983) 35–106.  
  35. Y. Saad, Iterative Methods for Sparse Linear Systems. PWS Publishing Company (1996).  
  36. H.A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput.13 (1992) 631–644.  
  37. M. Vohralík, Equivalence between mixed finite element and multi-point finite volume methods. C. R. Acad. Sci. Paris., Ser. I339 (2004) 525–528.  
  38. M. Vohralík, Equivalence between mixed finite element and multi-point finite volume methods. Derivation, properties, and numerical experiments, in Proceedings of ALGORITMY 2005, Slovak University of Technology, Slovakia (2005) 103–112.  
  39. A. Younès, R. Mose, Ph. Ackerer and G. Chavent, A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. J. Comput. Phys.149 (1999) 148–167.  
  40. A. Younès, Ph. Ackerer and G. Chavent, From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions. Internat. J. Numer. Methods Engrg.59 (2004) 365–388.  

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