Stabilized finite element methods for miscible displacement in porous media

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1. [1] J. BEAR, 1988, Dynamics of Fluids in Porous Media, Dover Publication, Inc., New York. Zbl1191.76002
2. [2] R. B. BIRD, W. E. STEWART, E. N. LIGHFOOT, 1966, Transport Phenomena, John Wiley, New York. 
3. [3] F. BREZZI, M. FORTIN, 1991, Mixed and Hybrid Finite Element Methods, Springer-Verlag. Zbl0788.73002MR1115205
4. [4] A. N. BROOKS, T. J. R. HUGHES, 1982, Streamline upwind/Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Meths, 32, 199-259. Zbl0497.76041MR679322
5. [5] B. COCKBURN, C. W. SHU, 1991, The Runge-Kutta local projection P' discontinuous Galerkin finite element method for scalar conservation laws, RAIRO-Model. Math. Anal. Numer., 25, 337-361. Zbl0732.65094MR1103092
6. [6] C. T. DAWSON, 1991, Godunov-mixed methods for advective flow problems in one space dimension, SIAM J. Numer. Anal., 28, 1282-1309. Zbl0741.65068MR1119271
7. [7] J. Jr. DOUGLAS, 1982, Simulation of miscible displacement in porous media by a modified method of characteristics procedure, In Numerical Analysis, Dundee 1981, vol. 912 of Lecture Notes in Mathematics, Springer-Verlag, Berlin. Zbl0476.76100MR654343
8. [8] J. Jr. DOUGLAS, 1984, Numerical methods for the flow of miscible fluids in porous media, In Numerical Methods in Coupled Systems, pp. 405-439, John Wiley and Sons Ltd., London, R. W. Lewis, P. Bettess and E. Hinton, Eds. Zbl0585.76138
9. [9] J. Jr. DOUGLAS, R. E. EWING, M. F. WHEELER, 1983, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, R. A. I. R. O., Anal. Numér., 17, 249-265. Zbl0526.76094MR702137
10. [10] J. Jr. DOUGLAS, R. E. EWING, M. F. WHEELER, 1983, The approximation of the pressure by a mixed method in the simulation of miscible displacement, R.A.I.R.O., Anal. Numér., 17, 17-33. Zbl0516.76094MR695450
11. [11] J. Jr. DOUGLAS, J. L. HENSLEY, Y. WEI, L. YEH, J. JAFFRÉ, P. J. PAES LEME, T. S. RAMAKRISHNAM, D. J. WILKINSON, 1992, A derivation for Darcy's law for miscible fluids and a revised model for miscible displacement in porous media, In Mathematical Modeling in Water Resources, vol. 2, pp. 165-178, Computational Mechamcs Publications, Elsevier Applied Science, Southampton, Boston, T. F. RUSSELL, E. R. EWING, C. A. BREBBIA, W. G. GRAY, G. F. PINDER, Eds. 
12. [12] J. Jr. DOUGLAS, J. E. ROBERTS, 1983, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comp., 41, 441-459. Zbl0537.76062MR717695
13. [13] J. Jr. DOUGLAS, T. F. RUSSELL, 1982, Numerical methods for convectiondominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal.,19, 871-885. Zbl0492.65051MR672564
14. [14] R. DURÁN, 1988, On the approximation of miscible displacement in porous media by a method of characteristics combined with a mixed method, SIAM J. Numer. Anal., 25, 989-1001. Zbl0661.76096MR960861
15. [15] R. E. EWING, 1983, The mathematics of reservoir simulation, Frontiers in Applied Mathematics, SIAM, Philadelphia. Zbl0533.00031MR770577
16. [16] R. E EWING, T. F. RUSSELL, M. F. WHEELER, 1983, Simulation of miscible displacement using mixed methods and a modified method of characteristics, In Proceedings, Seventh SPE Symposium on Reservoir Simulation, pp. 71-81, Dallas, Texas, Society of Petroleum Engineers, Paper SPE 12241. 
17. [17] R. E. EWING, T. F. RUSSELL, M. F. WHEELER, 1984, Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comp. Meth. Appl. Mech. Eng., 47, 73-92. Zbl0545.76131MR777394
18. [18] L. P. FRANCA, S. L. FREY, 1992, Stabilized finite element methods: II. The incompresible Navier-Stokes Equations, Comput. Meths. Appl. Mech. Engrg., 99, 209-233. Zbl0765.76048MR1186727
19. [19] L. P. FRANCA, S. L. FREY, T. J. R. HUGHES, 1992, Stabilized finite element methods : I. Application to the advective-diffusive model, Comput. Meths. Appl. Mech. Engrg., 95, 253-276. Zbl0759.76040MR1155924
20. [20] V. GIRAULT, P.- A. RAVIART, 1986, Finite Element Methods for Navier-Stokes Equations, Theory and Algonthms, Springer-Verlag, Berlin, Heidelberg, New York. Zbl0585.65077MR851383
21. [21] A. HARTEN, S. OSHER, 1987, Uniformly high-order accurate non-oscilatory schemes I, SIAM J. Numer. Anal., 24 279-309 Zbl0627.65102MR881365
22. [22] T. J. R. HUGHES, A. N. BROOKS, 1979, A multidimensional upwind scheme with no cross-wind diffusion, In Finite Element Methods for Convection Dominated Flows, pp. 19-35. ASME, New York, 1979 T. J. R. HUGUES, ed. Zbl0423.76067MR571679
23. [23] T. J. R. HUGHES, A. N. BROOKS, 1982, A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions : application to the streamline upwind procedure, In Finite Element Methods in Fluids, Wiley, Chichester, R. H. GALLAGHER, ed. 
24. [24] T. J. R. HUGHES, L. P. FRANCA, G. M. HULBERT, 1989, A new finite element formulation for computational fluid dynamics : VIII The Galerkin/least-square method for convective-diffusive equations, Comput. Meths..Appl. Engrg., 73, 173-189. Zbl0697.76100MR1002621
25. [25] J. JAFFRE, J. E. ROBERTS, 1985, Upstream weighting and mixed finite elements in the simulation of miscible displacements, Modélisation Mathématique et Analyse Numérique, 19, 443-460. Zbl0568.76096MR807326
26. [26] C. JOHNSON, J. SARANEN, 1986, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp., 47, 1-18. Zbl0609.76020MR842120
27. [27] C. JOHNSON, V. THOMÉE, 1981, Error estimates for some mixed finite element methods for parabolic type problems, R. A. I. R. O., Anal. Numér., 14, 41-78. Zbl0476.65074MR610597
28. [28] S. OSHER, 1984, Convergence of generalized MUSCL schemes, SIAM J. Numer. Anal., 22, 947-961. Zbl0627.35061MR799122
29. [29] D. W. PEACEMAN, 1966, Improved treatment of dispersion in numerical calculation of multidimensional miscible displacement, Soc. Petroleum Engr. J., 6, 213-216. 
30. [30] D. W. PEACEMAN, 1977, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York. 
31. [31] T. S. RAMAKRISHNAM, D. J. WILKINSON, pivate communication. 
32. [32] T. F. RUSSELL, 1985, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal., 22, 970-1013. Zbl0594.76087MR799124
33. [33] C. W. SHU, 1987, TVB uniformly high-order schemes for conservation laws, Math. of. Comp., 49, 105-121. Zbl0628.65075MR890256
34. [34] A. SZEPESSY, 1991, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions, R. A. I. R. O. Modél. Math. Anal. Numér., 26, 749-782. Zbl0751.65061MR1135992
35. [35] Y. WEI, Discontinuous Galerkin - mixed finite element methods for convection - dominated diffusion problems, to appear.