Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, p. 367-391
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.
DOI : https://doi.org/10.1051/m2an:2006013
Classification:  76M10,  76M12,  76S05
Keywords: mixed finite element method, saddle-point problem, finite volume method, second-order elliptic equation, nonlinear parabolic convection-diffusion-reaction equation
@article{M2AN_2006__40_2_367_0,
     author = {Vohral\'\i k, Martin},
     title = {Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {2},
     year = {2006},
     pages = {367-391},
     doi = {10.1051/m2an:2006013},
     zbl = {1116.65121},
     mrnumber = {2241828},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_2_367_0}
}
Vohralík, Martin. Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, pp. 367-391. doi : 10.1051/m2an:2006013. http://www.numdam.org/item/M2AN_2006__40_2_367_0/

[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. | Zbl 0951.65080

[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. | Zbl 0951.65082

[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. | Zbl 0839.65116

[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. | Zbl 0856.76033

[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. | Zbl 0880.65084

[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. | Zbl 0947.65114

[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. | Numdam | Zbl 0567.65078

[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. | Numdam | Zbl 0857.65116

[10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[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. | Zbl 0599.65072

[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. | Zbl 0631.65107

[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. | Zbl 1091.76520

[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. | Zbl 0932.65126

[15] Y. Coudière, J.-P. Vila and Villedieu Ph., Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493-516. | Numdam | Zbl 0937.65116

[16] C. Dawson, Analysis of an upwind-mixed finite element method for nonlinear contaminnat transport equations. SIAM J. Numer. Anal. 35 (1998) 1709-1724. | Zbl 0954.76043

[17] C. Dawson and V. Aizinger, Upwind-mixed methods for transport equations. Comput. Geosci. 3 (1999) 93-110. | Zbl 0962.65084

[18] J. Douglas Jr. and J.E. Roberts, Global estimates for mixed methods for second order elliptic equations. Math. Comp. 44 (1985) 39-52. | Zbl 0624.65109

[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. | Zbl 0981.65095

[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. | Zbl 1093.65110

[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. | Zbl 0761.76068

[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. | Zbl 0752.65037

[23] M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49 (1952) 409-436. | Zbl 0048.09901

[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. | Zbl 1079.76581

[25] J. Jaffré, Éléments finis mixtes et décentrage pour les équations de diffusion-convection. Calcolo 23 (1984) 171-197. | Zbl 0562.65077

[26] L. Jeannin, I. Faille and T. Gallouët, Comment modéliser les écoulements diphasiques compressibles sur des grilles hybrides ? Oil & Gas Science and Technology - Rev. IFP 55 (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. | Zbl 1124.76030

[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. | Zbl 0573.65082

[30] J.C. Nédélec, Mixed finite elements in 3 . Numer. Math. 35 (1980) 315-341. | Zbl 0419.65069

[31] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994). | MR 1299729 | Zbl 0803.65088

[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. | Zbl 0362.65089

[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. | Zbl 0875.65090

[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. | Zbl 0572.76089

[35] Y. Saad, Iterative Methods for Sparse Linear Systems. PWS Publishing Company (1996). | Zbl 1031.65047

[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. | Zbl 0761.65023

[37] M. Vohralík, Equivalence between mixed finite element and multi-point finite volume methods. C. R. Acad. Sci. Paris., Ser. I 339 (2004) 525-528. | Zbl 1058.65132

[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. | Zbl 0923.65064

[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. | Zbl 1043.65131