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Klausen, R. A.; Russell, T. F.

Relationships among some locally conservative discretization methods which handle discontinuous coefficients. (English) Zbl 1124.76030

Comput. Geosci. 8, No. 4, 341-377 (2005).
Summary: This paper presents the relationships between some numerical methods suitable for a heterogeneous elliptic equation with application to reservoir simulation. The methods discussed are the classical mixed finite element method (MFEM), the control-volume mixed finite element method (CVMFEM), the support operators method (SOM), the enhanced cell-centered finite difference method (ECCFDM), and the multi-point flux-approximation (MPFA) control-volume method. These methods are all locally mass conservative, and handle general irregular grids with anisotropic and heterogeneous discontinuous permeability. In addition to this, the methods have a common weak continuity in the pressure across the edges, which in some cases corresponds to Lagrange multipliers. It seems that this last property is an essential common quality for these methods.

Cited in 39 Documents

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs

Keywords:

relationships, mixed finite element method (MFEM); expanded mixed finite element method (EMFEM); enhanced cell-centered finite difference method (ECCFDM); control-volume mixed finite element method (CVMFEM); support operator method (SOM); multi-point flux-approximation (MPFA) control-volume method
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\textit{R. A. Klausen} and \textit{T. F. Russell}, Comput. Geosci. 8, No. 4, 341--377 (2005; Zbl 1124.76030)
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References:

[1] I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci. 6 (2002) 404–432. · Zbl 1094.76550
[2] 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
[3] 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
[4] I. Aavatsmark, T. Barkve and T. Mannseth, Control-volume discretization methods for 3D quadrilateral grids in inhomogeneous, anisotropic reservoirs, SPE Journal 3 (1998) 146–154. · Zbl 0951.65080
[5] 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
[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] K. Aziz and A. Settari, Petroleum Reservoir Simulation (Applied Science Publishers, London, 1979).
[8] M. Berndt, K. Lipnukov, J.D. Moulton and M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation, East–West J. Numer. Math. 9 (2001) 253–316. · Zbl 1014.65114
[9] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Vol. 15 (Springer, New York, 1991). · Zbl 0788.73002
[10] Z. Cai, J.E. Jones, S.F. McCormick and T.F. Russell, Control-volume mixed finite element methods, Comput. Geosci. 1 (1997) 289–315. · Zbl 0941.76050
[11] S. Chou, D.Y. Kwak and K.Y. Kim, A general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids: The overlapping covolume case, SIAM J. Numer. Anal. 39 (2001) 1170–1196. · Zbl 1007.65091
[12] L.J. Durlofsky, Accuracy of mixed and control-volume finite element approximations to Darcy velocity and related quantities, Water Resourc. Res. 30 (1994) 965–973.
[13] M.G. Edwards, Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids, Comput. Geosci. 6 (2002) 433–452. · Zbl 1036.76034
[14] M.G. Edwards, R.D. Lazarov and I. Yotov, ed., Special Issue on Locally Conservative Numerical Methods for Flow in Porous Media, Comput. Geosci. 6(3/4) (2002).
[15] M.G. Edwards and C.F. Rogers, A flux continous scheme for the full tensor pressure equation, in: Proc. of the 4th European Conf. on the Mathematics of Oil Recovery, Vol. D (Røros, 1994).
[16] M.G. Edwards and C.F. Rogers, Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Comput. Geosci. 2 (1998) 259–290. · Zbl 0945.76049
[17] G.T. Eigestad, I. Aavatsmark and M. Espedal, Symmetry and M-matrix issues for the O-method on an unstructured grid, Comput. Geosci. 6 (2002) 381–404. · Zbl 1094.76551
[18] G.T. Eigestad and R.A. Klausen: On the convergence of the MPFA O-method; Numerical experiments for discontinuous media, Numer. Methods Partial Differential Equations (2004) accepted. · Zbl 1089.76037
[19] R.E. Ewing, M. Liu and J. Wang, Superconvergence of mixed finite element approximation over quadrilaterals, SIAM J. Numer. Anal. 36 (1999) 772–787. · Zbl 0926.65107
[20] J. Hyman, J. Morel, M. Shashkov and S. Steinberg, Mimetic finite difference methods for diffusion equations, Comput. Geosci. 6 (2002) 333–352. · Zbl 1023.76033
[21] J. Hyman, M. Shashkov and S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, J. Comput. Phys. 132 (1997) 130–148. · Zbl 0881.65093
[22] R.A. Klausen and G.T. Eigestad, Multi-point flux approximations and finite element methods; practical aspects of discontinuous media, in: Proc. of ECMOR IX, Cannes, France (30 August–2 September 2004). · Zbl 1089.76037
[23] S.H. Lee, P. Jenny and H.A. Tchelepi, A finite-volume method with hexahedral multi-block grids for modeling low in porous media, Comput. Geosci. 6 (2002) 353–379. · Zbl 1094.76541
[24] 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(3) (1985). · Zbl 0573.65082
[25] I. Mishev, Nonconforming finite volume methods, Comput. Geosci. 6 (2002) 253–268. · Zbl 1036.76038
[26] R. Mosé, P. Siegel, P. Ackerer and G. Chavent, Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity? Water Resourc. Res. 30 (1994) 3001–3012.
[27] R.L. Naff, T.F. Russel and J.D. Wilson, Shape functions for velocity interpolation on general hexahedral cells, Comput. Geosci. 6 (2002) 285–314. · Zbl 1094.76542
[28] J.M. Nordbotten and I. Aavatsmark, Monotonicity conditions for control-volume methods on uniform parallelogram grids in homogeneous media, submitted.
[29] P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in: Mathematical Aspects of Finite Element Methods, eds. I. Galligani and E. Magenes, Lecture Notes in Mathematics, Vol. 606 (Springer, New York, 1977) pp. 292–315. · Zbl 0362.65089
[30] T.F. Russell, Relationships among some conservative discretization methods, in: Numerical Treatment of Multiphase Flows in Porous Media, eds. Z. Chen et al., Lecture Notes in Physics, Vol. 552 (Springer, Heidelberg, 2000) pp. 267–282. · Zbl 1010.76064
[31] 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, ed. R.E. Ewing, Frontiers in Applied Mathematics, Vol. 1 (SIAM, Philadelphia, PA, 1983) pp. 35–106.
[32] A.A. Samarskij, Theorie der Differenzenverfahren (Akademische Verlagsgesellschaft Geest & Poetig, Leipzig, 1984). · Zbl 0543.65067
[33] M. Shashkov and S. Steinberg, Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys. 129 (1996) 383–405. · Zbl 0874.65062
[34] G. Strang and G.J. Fix, An Analysis of the Finite Element Method (Wiley, New York, 1973). · Zbl 0356.65096
[35] J. Wang and T. Mathew, Mixed finite element methods over quadrilaterals, in: Proc. of the 3rd Internat. Conf. on Advances in Numerical Methods and Applications, eds. I.T. Dimov et al. (World Scientific, Singapore, 1994) pp. 203–214. · Zbl 0813.65120
[36] J.A. Wheeler, M.F. Wheeler and I. Yotov, Enhanced velocity mixed methods for flow in multiblock domains, Comput. Geosci. 6 (2002) 315–332. · Zbl 1023.76023
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