Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences

SIAM Journal on Numerical Analysis

Volumn 34, Issue 2, 1997, Pages 828-852

  Arbogast, Todd ^a   Wheeler, Mary F ^a   Yotov, Ivan ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Error estimates; Finite difference; Mixed finite element; Superconvergence; Tensor coefficient

Indexed keywords

EID: 0001198473     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142994262585     Document Type: Article

References (33)

1. 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., 18 (1997), to appear.
2. T. ARBOGAST, P. T. KEENAN, M. F. WHEELER, AND I. YOTOV, Logically rectangular mixed methods for Darcy flow on general geometry, SPE 29099, in Proceedings of the 13th SPE Symposium on Reservoir Simulation, Society of Petroleum Engineers, Richardson, Texas, 1995, pp. 51-59.
3. T. ARBOGAST, M. F. WHEELER, AND I. YOTOV, Logically rectangular mixed methods for groundwater flow and transport on general geometry, in Computational Methods in Water Resources X, Vol. 1, A. Peters, G. Whittum, B. Herrling, U. Meissner, C. A. Brebbia, W. G. Gray, and G. F. Pindar, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994, pp. 149-156.
4. D. N. ARNOLD AND F. BREZZI, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, RAIRO Model. Math. Anal. Numér., 19 (1985), pp. 7-32.
5. J. BEAR, Dynamics of Fluids in Porous Media, Dover, New York, 1972.
6. A. BENSOUSSAN, J. L. LIONS, AND G. PAPANICOLAOU, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
7. F. BREZZI, J. DOUGLAS, JR., R. DURN, AND M. FORTIN, Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51 (1987), pp. 237-250.
8. F. BREZZI, J. DOUGLAS, JR., M. FORTIN, AND L. D. MARINI, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Model. Math. Anal. Numer., 21 (1987), pp. 581-604.
9. F. BREZZI, J. DOUGLAS, JR., AND L. D. MARINI, Two families of mixed elements for second order elliptic problems, Numer. Math., 88 (1985), pp. 217-235.
10. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
11. Z. CHEN, BDM mixed methods for a nonlinear elliptic problem, J. Comput. Appl. Math., 53 (1994), pp. 207-223.
12. Z. CHEN, Expanded mixed finite element methods for linear second order elliptic problems I, IMA preprint series # 1219, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN, 1994.
13. Z. CHEN AND J. DOUGLAS, JR., Prismatic mixed finite elements for second order elliptic problems, Calcolo, 26 (1989), pp. 135-148.
14. P. J. DAVIS, Interpolation and Approximation, Dover Publications, New York, 1975.
15. J. DOUGLAS, JR., AND J. E. ROBERTS, Global estimates for mixed methods for second order elliptic equations, Math. Comp., 44 (1985), pp. 39-52.
16. J. DOUGLAS, JR., AND J. WANG, Superconvergence for mixed finite element methods on rectangular domains, Calcolo, 26 (1989), pp. 121-134.
17. R. DURÁN, Superconvergence for rectangular mixed finite elements, Numer. Math., 58 (1990), pp. 287-298.
18. L. J. DURLOFSKY, Representation of grid block permeability in coarse scale models of randomly heterogeneous porous media, Water Resources Research, 28 (1992), pp. 1791-1800.
19. R. E. EWING, R. D. LAZAROV, AND J. WANG, Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal., 28 (1991), pp. 1015-1029.
20. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
21. E. KASAP AND L. W. LAKE, An analytical method to calculate the effective permeability tensor of a grid block and its application in an outcrop study, in Proceedings of the 10th SPE Symposium on Reservoir Simulation, Society of Petroleum Engineers, Richardson, Texas, 1989, pp. 355-366.
22. J. KOEBBE, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors, Numer. Methods Partial Differential Equations, 9 (1993), pp. 339-355.
23. J. L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications 1, Springer-Verlag, Berlin, 1970.
24. A. R. MITCHELL AND D. F. GRIFFITHS, The Finite Difference Method in Partial Differential Equations, John Wiley, New York, 1980.
25. M. NAKATA, A. WEISER, AND M. F. WHEELER, Some superconvergence results for mixed finite element methods for elliptic problems on rectangular domains, in The Mathematics of Finite Elements and Applications V, J. R. Whiteman, ed., Academic Press, London, 1985.
26. 3, Numer. Math., 35 (1980), pp. 315-341.
27. D. W. PEACEMAN, Fundamentals of Numerical Reservoir Simulation, Elsevier, Amsterdam, 1977.
28. R. A. RAVIART AND J. M. THOMAS, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics 606, Springer-Verlag, New York, 1977, pp. 292-315.
29. 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., Frontiers in Applied Mathematics 1, SIAM, Philadelphia, PA, 1983, pp. 35-106.
30. J. M. THOMAS, Sur l'analyze numerique des methodes d'elements finis hybrides et mixtes, These de Doctorat d'etat, Sciences Mathematiques, à l'Universite Pierre et Marie Curie, 1977.
31. J. F. THOMPSON, Z. U. A. WARSI, AND C. W. MASTIN, Numerical Grid Generation: Foundations and Applications, Elsevier, New York, 1985.
32. A. WEISER AND M. F. WHEELER, On convergence of block-centered finite-differences for elliptic problems, SIAM J. Numer. Anal., 25 (1988), pp. 351-375.
33. M. F. WHEELER, K. R. ROBERSON, AND A. CHILAKAPATI, Three-dimensional bioremediation modeling in heterogeneous porous media, in Computational Methods in Water Resources IX, Vol. 2, Mathematical, Modeling in Water Resources, T. F. Russell, R. E. Ewing, C. A. Brebbia, W. G. Gray, and G. F. Pindar, eds., Computational Mechanics Publications, Southampton, U. K., 1992, pp. 299-315.