Convergence of multipoint flux approximations on quadrilateral grids

Numerical Methods for Partial Differential Equations

Volumn 22, Issue 6, 2006, Pages 1438-1454

  Klausen, Runhild A ^a   Winther, Ragnar ^a  

^a UNIVERSITY OF OSLO   (Norway)

Author keywords

Convergence; Mixed finite element; Multipoint flux approximation

Indexed keywords

EID: 33750584064     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20158     Document Type: Article

References (25)

1. 1 I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids, Comput Geosci 6 ( 2002), 404–432.
2. 2 I. Aavatsmark, T. Barkve, T. Mannseth, Control‐volume discretization methods for 3D quadrilateral grids in inhomogeneous, anisotropic reservoirs, SPE J 3 ( 1998), 146–154.
3. 3 M. G. Edwards, C. F. Rogers, Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Comput Geosci 2 ( 1998), 259–290.
4. 4 R. A. Klausen, On Locally Conservative Numerical Methods for Elliptic Problems; Application to Reservoir Simulation, Ph.D. Thesis, Dissertation 297, Unipub AS, University of Oslo, 2003.
5. 5 R. A. Klausen, T. F. Russell, Relationships among some locally conservative discretization methods which handle discontinuous coefficients, Comput Geosci 8 ( 2004), 341–377.
6. 6 Schlumberger, Eclipse, industry standard in reservoir simulation, Schlumberger information solutions, 2003.
7. 7 A. Agouzal, J. Baranger, J.‐F. Maitre, F. Oudin, Connection between finite volume and mixed finite element methods for a diffusion problem with nonconstants coefficients, application to a convection diffusion problem, East‐West J Numer Math 3 ( 1995), 237–254.
8. 8 I. Aavatsmark, T. Barkve, Ø. Bøe, T. Mannseth, Discretization on non‐orthogonal, curvilinear grids for multi‐phase flow, Proc. 4th European Conference on the Mathematics of Oil Recovery, Røros, D, 1994.
9. 9 I. Aavatsmark, G. T. Eigestad, R. A. Klausen, Numerical convergence of MPFA for general quadrilateral grids in two and three dimensions, IMA Volumes in Mathematics and its Applications, vol. 142, Compatible spatial discretizations for Partial Differential quations, 2006, pp. 1–22.
10. 10 G. T. Eigestad, T. Aadland, I. Aavatsmark, R. A. Klausen, J. M. Nordbotten, Recent advances for mpfa methods, Proc. 9th European Conference on the Mathematics of Oil Recovery, Cannes, France, 2004.
11. 11 G. T. Eigestad, R. A. Klausen, On the convergence of the multi‐point flux approximation o‐method; numerical experiments for discontinous permeability, Numer Methods Partial Diff Eqns 2 (6) ( 2005), 1079–10981.
12. 12 M. F. Wheeler, I. Yotov, A cell‐centered finite difference method on quadrilaterals, IMA Volumes in Mathematics and its Applications, vol. 142, Compatible spatial discretizations for partial differential quations, 2006, pp. 189–208.
13. 13 M. F. Wheeler, I. Yotov, A multipoint flux mixed finite element method, Technical Report TR Math 05‐06 ( 2005), Department of Mathematics, University of Pittsburgh.
14. 14 D. N. Arnold, D. Boffi, R. S. Falk, Quadrilateral H(div) finite elements, SIAM J Numer Anal 42 ( 2005), 2429–2451.
15. 15 M. G. Edwards, C. F. Rogers, A flux continuous scheme for the full tensor pressure equation, Proc. 4th European Conference on the Mathematics of Oil Recovery, Røros, D, l994.
16. 16 I. Aavatsmark, T. Barkve, Ø. Bøe, T. Mannseth, Discretization on unstructured grids for inhomogeneous anisotropic media. Part i: Derivation of the methods. Part ii: Discussion and numerical results, SIAM J Sci Comput 19 ( 1998), 1700–1736.
17. 17 T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, I. Yotov, Enhanced cell‐centered finite differences for elliptic equations on general geometry, SIAM J Sci Comput 19 ( 1998), 404–425.
18. 18 S. Chou, D. Y. Kwak, 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.
19. 19 J. Hyman, M. Shashkov, S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non‐isotropic materials, J Comput Phys 132 ( 1997), 254–316.
20. 20 M. Berndt, K. Lipnikov, J. D. Moulton, M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation, East‐west J Numer Math 9 ( 2001), 130–148.
21. 21 P. G. Ciarlet, The finite element method for elliptic problems, North‐Holland Publishing Company, 1997.
22. 22 R. E. Ewing, M. L. Liu, J. Wang, Superconvergence of mixed finite element approximations over quadrilaterals, SIAM J Numer Anal 36 ( 1999), 772–787.
23. 23 F. Brezzi, M. Fortin, Mixed and hybrid finite element methods, Springer, New York, 1991.
24. 24 D. Braess, Finite elements, Cambridge University Press, 1997.
25. 25 J. Wang, T. Mathew, Mixed finite element methods over quadrilaterals, Proceedings of the 3th International Conference on Advances in Numerical Methods and Applications I. T. Dimov, Bl. Sendov, P. Vassilevskith, editors, 1994, pp. 203–214.