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Equivalence between lowest-order mixed finite elementand multi-pointfinite volume methods on simplicial meshes

Published online by Cambridge University Press:  21 June 2006

Martin Vohralík
Martin Vohralík*
Affiliation:
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic. [email protected] Laboratoire de Mathématiques, Analyse Numérique et EDP, Université de Paris-Sud, Bât. 425, 91405 Orsay, France. [email protected]
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Abstract

We consider the lowest-order Raviart–Thomas mixed finite elementmethod for second-order elliptic problems on simplicial meshes intwo and three space dimensions. This method produces saddle-pointproblems for scalar and flux unknowns. We show how to easily andlocally eliminate the flux unknowns, which implies the equivalencebetween this method and a particular multi-point finite volumescheme, without any approximate numerical integration. The matrixof the final linear system is sparse, positive definite for alarge class of problems, but in general nonsymmetric. We next showthat these ideas also apply to mixed and upwind-mixed finiteelement discretizations of nonlinear parabolicconvection–diffusion–reaction problems. Besides the theoreticalrelationship between the two methods, the results allow forimportant computational savings in the mixed finite elementmethod, which we finally illustrate on a set of numericalexperiments.

Keywords

Mixed finite element methodsaddle-point problemfinite volume methodsecond-order elliptic equationnonlinearparabolic convection–diffusion–reaction equation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 2 , March 2006 , pp. 367 - 391
Copyright
© EDP Sciences, SMAI, 2006

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References

Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700–1716. CrossRef
Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 1717–1736. CrossRef
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.
Agouzal, A., Baranger, J., Maître, J.-F. and Oudin, F., 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.
Arbogast, T., Wheeler, M.F. and Zhang, N., 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. CrossRef
Arbogast, T., Wheeler, M.F. and Yotov, I., Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828–852. CrossRef
Arbogast, T., Dawson, C.N., Keenan, P.T., Wheeler, M.F. and Yotov, I., Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19 (1998) 404–425.
Arnold, D.N. and Brezzi, F., Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7–32. CrossRef
Baranger, J., Maître, J.-F. and Oudin, F., Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 445–465. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
Brezzi, F., Douglas Jr, J.. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. CrossRef
Brezzi, F., Douglas Jr, J.., R. Duran and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. CrossRef
Chavent, G., Younès, A. and Ackerer, Ph., 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. CrossRef
Chen, Z., Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems. East-West J. Numer. Math. 4 (1996) 1–33.
Coudière, Y., Villedieu Ph, J.-P. Vila., Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493–516. CrossRef
Dawson, C., Analysis of an upwind-mixed finite element method for nonlinear contaminnat transport equations. SIAM J. Numer. Anal. 35 (1998) 1709–1724. CrossRef
Dawson, C. and Aizinger, V., Upwind-mixed methods for transport equations. Comput. Geosci. 3 (1999) 93–110. CrossRef
Douglas Jr, J.. and J.E. Roberts, Global estimates for mixed methods for second order elliptic equations. Math. Comp. 44 (1985) 39–52.
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.
Eymard, R., Gallouët, T. and Herbin, R., 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. CrossRef
Faille, I., A control volume method to solve an elliptic equation on a two-dimensional irregular mesh. Comput. Methods Appl. Mech. Engrg. 100 (1992) 275–290. CrossRef
Gilbert, J.R., Moler, C. and Schreiber, R., Sparse matrices in MATLAB: Design and implementation. SIAM J. Matrix Anal. Appl. 13 (1992) 333–356. CrossRef
Hestenes, M.R. and Stiefel, E., Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49 (1952) 409–436. CrossRef
Hoteit, H., Erhel, J., Mosé, R., Philippe, B. and Ackerer, Ph., Numerical reliability for mixed methods applied to flow problems in porous media. Comput. Geosci. 6 (2002) 161–194. CrossRef
Jaffré, J., Éléments finis mixtes et décentrage pour les équations de diffusion-convection. Calcolo 23 (1984) 171–197.
Jeannin, L., Faille, I. and Gallouët, T., Comment modéliser les écoulements diphasiques compressibles sur des grilles hybrides ? Oil & Gas Science and Technology – Rev. IFP 55 (2000) 269–279. CrossRef
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).
Klausen, R.A. and Russell, T.F., Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput. Geosci. 8 (2004) 341–377. CrossRef
Marini, L.D., 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. CrossRef
Nédélec, J.C., Mixed finite elements in $\xR^3$ . Numer. Math. 35 (1980) 315–341.
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994).
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.
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.
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.
Y. Saad, Iterative Methods for Sparse Linear Systems. PWS Publishing Company (1996).
van der Vorst, H.A., 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. CrossRef
M. Vohralík, Equivalence between mixed finite element and multi-point finite volume methods. C. R. Acad. Sci. Paris., Ser. I 339 (2004) 525–528.
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.
Younès, A., Mose, R., Ackerer, Ph. and Chavent, G., 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.
Younès, A., Ackerer, Ph. and Chavent, G., From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions. Internat. J. Numer. Methods Engrg. 59 (2004) 365–388.