ON THE CONTROL VOLUME FINITE ELEMENT METHODS AND THEIR APPLICATIONS TO MULTIPHASE FLOW

Networks and Heterogeneous Media

Volumn 1, Issue 4, 2006, Pages 689-706

  Chen, Zhangxin ^a  

^a SOUTHERN METHODIST UNIVERSITY   (United States)

Author keywords

Control volume finite element; error analysis; Galerkin finite element; multiphase flow; numerical experiments; upstream weighting

Indexed keywords

FINITE ELEMENT METHOD; GALERKIN METHODS; NUMERICAL METHODS; PIECEWISE LINEAR TECHNIQUES; POROUS MATERIALS;

CONTROL VOLUME - FINITE ELEMENT; CONTROL-VOLUME FINITE ELEMENT METHODS; GALERKIN FINITE ELEMENT METHODS; GALERKIN FINITE ELEMENTS; NUMERICAL EXPERIMENTS; NUMERICAL SOLUTION; SECOND-ORDER PARTIAL DIFFERENTIAL EQUATION; STANDARD FINITE ELEMENT; UPSTREAM WEIGHTING; WEIGHTED CONTROLS;

MULTIPHASE FLOW;

EID: 52149101668     PISSN: 15561801     EISSN: 1556181X     Source Type: Journal    
DOI: 10.3934/nhm.2006.1.689     Document Type: Article

References (41)

1. I. Aavatsmark, E. Reiso, and R.Teigland, Control-volume discretization method for quadrilateral grids with faults and local refinements, Computational Geosciences, 5 (2001), 1–23.
2. R. A. Adams, “Sobolev Spaces,” Academic Press, Inc., New York, 1975.
3. S. N. Antontsev, On the solvability of boundary value problems for degenerate two-phase porous flow equations, Dinamika Splo˘snoĭ Sredy Vyp., 10 (1972), 28–53, in Russian.
4. K. Aziz and A. Settari, “Petroleum Reservoir Simulation,” Applied Science, Publishers Ltd, London, 1979.
5. R. E. Bank and D. J. Rose, Some error estimates for the box method, SIAM J. Numer. Anal., 24 (1987), 777–787.
6. Z. Cai, J. Mandel, and S. McCormick, The finite volume element method for diffusion equations on general triangulations, SIAM J Numer Anal., 28 (1991), 392–403.
7. G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1978.
8. Z. Chen, Analysis of mixed methods using conforming and nonconfirming finite element methods, RAIRO Model. Math. Anal. Numer., 27 (1993), 9–34.
9. Z. Chen, Degenerate two-phase incompressible flow I: Existence, uniqueness and regularity of a weak solution, Journal of Differential Equations, 171 (2001), 203–232.
10. Z. Chen, Degenerate two-phase incompressible flow II: Regularity, stability and stabilization, Journal of Differential Equations, 186 (2002), 345–376.
11. Z. Chen and R. E. Ewing, Comparison of various formulations of three-phase flow in porous media, J. Comp. Physics, 132 (1997), 362–373.
12. Z. Chen, G. Huan, and B. Li, An improved IMPES method for two-phase flow in porous media, Transport in Porous Media, 54 (2004), 361–376.
13. Z. Chen, G. Huan, and Y. Ma, “Computational Methods for Multiphase Flows in Porous Media,” in the Computational Science and Engineering Series, Vol. 2, SIAM, Philadelphia, PA, 2006.
14. Z. Chen and J. Li, A new stabilized finite volume method for the stationary Stokes equations, submitted for publication.
15. 2-estimate of finite volume element method, Adv. Comput. Math., 16 (2002), 291–303.
16. S. H. Chou and D. Y. Kwak, Analysis and convergence of a MAC scheme for the generalized Stokes problem, Numer. Meth. Partial Diff. Equ., 13 (1997), 147–162.
17. S. H. Chou and D. Y. Kwak, A covolume method based on rotated bilinears for the generalized Stokes problem, SIAM J. Numer. Anal., 35 (1998), 494–507.
18. ∞ in control volume methods for elliptic and parabolic problems: a unified approach, Math Comp., 69 (2000), 103–120.
19. S.-H. Chou and P. S. Vassilevski, A general mixed covolume framework for constructing conservative schemes for elliptic problems, Math. Comp., 68 (1999), 991–1011.
20. P. G. Ciarlet, “The Finite Element Method for Elliptic Problems,” North-Holland, Amster-dam, 1978.
21. E. F. D’Azevedo and R. B. Simpson, On optimal interpolation triangle incidences, SIAM J. Sci. Stat. Comput., 10 (1989), 1063–1075.
22. R. E. Ewing (ed.), “The Mathematics of Reservoir Simulation,” SIAM, Philadelphia, 1983.
23. R. E. Ewing, T. Lin, and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal., 39 (2002), 1865–1888.
24. P. A. Forsyth, A control volume finite element approach to NAPL groundwater contamination, SIAM J. Sci. Stat. Comput., 12 (1991), 1029–1057.
25. L. S. Fung, A. D. Hiebert, and L. Nghiem, “Reservoir Simulation with a Control Volume Finite Element Method,” SPE 21224, the 11th SPE Symp. Reser. Simul., Anaheim, 1991.
26. B. Heinrich, ”Finite Difference Methods on Irregular Networks,” Birkhauser, Basel, Boston, Stuttgart, 1987.
27. B. Joe, Delaunay triangular meshes in convex polygons, SIAM J. Sci. Stat. Comput., 7 (1986), 514–539.
28. R. D. Lazarov, I. D. Mishev, and P. S. Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal., 33 (1996), 31–55.
29. P. A. Lemonnier, Improvement of reservoir simulation by a triangular discontinuous finite element method, SPE paper 8249 presented at the 1979 Annual Fall Technical Conference and Exhibition of SPE of AIME, Las Vegas, Sept. 23–26.
30. B. Li, Z. Chen, and G. Huan, Control volume function approximation methods and their applications to modeling porous media flow I: The two-phase flow, Advances in Water Resources, 26 (2003), 435–444.
31. B. Li, Z. Chen, and G. Huan, Control volume function approximation methods and their applications to modeling porous media flow II: The black oil model, Advances in Water Resources, 27 (2004), 99–120.
32. R. Li, Generalized difference methods for a nonlinear Dirichlet problem, SIAM J Numer Anal., 24 (1987), 77–88.
33. R. Li, Z. Chen, and W. Wu, “The Generalized Difference Method for Differential Equations-Numerical Analysis of Finite Volume Methods,” Marcel Dekker, New York, 2000.
34. R. Li and P. Zhu, Generalized difference methods for second order elliptic partial differential equations (I) (in Chinese), Numer. Math., 4 (1982), 140–152.
35. C. Liu and S. McCormick, The finite volume element method (FVE) for planar cavity flow, Proc. 11th Internat. Conf. on CFD, Williamsburg, VA, June 28–July 2, 1988, 365–387.
36. D. W. Peaceman, “Fundamentals of Numerical Reservoir Simulation,” Elsevier, New York, 1977.
37. D. W. Peaceman, “Interpretation of Well-block Pressures in Numerical Reservoir Simulation,” SPE 6893, 52nd Annual Fall Technical Conference and Exhibition, Denver, 1977.
38. H. Rui, Analysis on a finite volume element method for the Stokes problems, Acta Mathematicae Applicatae Sinica, English Series, 3 (2005), 359–372.
39. R. S. Varga, “Matrix Iterative Analysis,” Prentice-Hall, Englewood Cliffs, NJ, 1962.
40. S. Verma and K. A. Aziz, Control volume scheme for flexible grids in reservoir simulation, Paper SPE37999, the 1997 SPE Symposium on Reservoir Simulation, Dallas, June 8-11, 1997.
41. X. Ye, On the relationship between finite volume and finite element methods applied to the Stokes equations, Numer. Methods Partial Diff. Equ., 5 (2001), 440–453.