Relationships among some locally conservative discretization methods which handle discontinuous coefficients

Computational Geosciences

Volumn 8, Issue 4, 2004, Pages 341-377

  Klausen, Runhild A ^a   Russell, Thomas F ^b  

^a UNIVERSITY OF OSLO   (Norway)

^b UNIVERSITY OF COLORADO   (United States)

Author keywords

Control volume mixed finite element method (CVMFEM); Enhanced cell centered finite difference method (ECCFDM); Expanded mixed finite element method (EMFEM); Mixed finite element method (MFEM); Multi point flux approximation (MPFA) control volume method; Relationships; Support operator method (SOM)

Indexed keywords

FINITE ELEMENT METHOD;

EID: 22144493785     PISSN: 14200597     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10596-005-1815-9     Document Type: Article

References (36)

1. I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci. 6 (2002) 404-432. 10.1023/ A:1021291114475
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. 10.1137/S1064827595293594
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. 10.1137/S1064827595293594
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.
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. 10.1137/ S1064827594264545
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. 10.1137/ S0036142994262585
7. K. Aziz A. Settari 1979 Petroleum Reservoir Simulation Applied Science Publishers London
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.
9. F. Brezzi M. Fortin 1991 Mixed and Hybrid Finite Element Methods Springer Series in Computational Mathematics 15 Springer New York
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. 10.1023/ A:1011577530905
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. 10.1137/S003614290037544X
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. 10.1029/94WR00061
13. M.G. Edwards, Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids, Comput. Geosci. 6 (2002) 433-452. 10.1023/A:1021243231313
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. 10.1023/A:1011510505406
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. 10.1023/A:1021239130404
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.
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. 10.1137/S0036142997322801
20. J. Hyman, J. Morel, M. Shashkov and S. Steinberg, Mimetic finite difference methods for diffusion equations, Comput. Geosci. 6 (2002) 333-352. 10.1023/A:1021282912658
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. 10.1006/jcph.1996.5633
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).
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. 10.1023/A:1021287013566
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).
25. I. Mishev, Nonconforming finite volume methods, Comput. Geosci. 6 (2002) 253-268. 10.1023/A:1021214424953
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. 10.1029/ 94WR01786
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. 10.1023/A:1021218525861
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 J.M. Thomas 1977 A mixed finite element method for 2nd order elliptic problems eds. I. Galligani E. Magenes Mathematical Aspects of Finite Element Methods Lecture Notes in Mathematics 606 Springer New York 292-315
30. T.F. Russell 2000 Relationships among some conservative discretization methods ed. Z. Chen Numerical Treatment of Multiphase Flows in Porous Media Lecture Notes in Physics 552 Springer Heidelberg 267-282
31. T.F. Russell M.F. Wheeler 1983 Finite element and finite difference methods for continuous flows in porous media ed. R.E. Ewing The Mathematics of Reservoir Simulation Frontiers in Applied Mathematics 1 SIAM Philadelphia, PA 35-106
32. A.A. Samarskij 1984 Theorie der Differenzenverfahren Akademische Verlagsgesellschaft Geest & Poetig Leipzig
33. M. Shashkov and S. Steinberg, Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys. 129 (1996) 383-405. 10.1006/jcph.1996.0257
34. G. Strang G.J. Fix 1973 An Analysis of the Finite Element Method Wiley New York
35. J. Wang T. Mathew 1994 Mixed finite element methods over quadrilaterals eds. I.T. Dimov et al. Proc. of the 3rd Internat. Conf. on Advances in Numerical Methods and Applications World Scientific Singapore 203-214
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. 10.1023/A:1021270509932