A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media

SIAM Journal on Numerical Analysis

Volumn 33, Issue 4, 1996, Pages 1669-1687

  Arbogast, Todd ^a,b   Wheeler, Mary F ^a,b   Zhang, Nai Ying ^a,c  

^a RICE UNIVERSITY   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^c CleverSys Inc   (United States)

Author keywords

Degenerate parabolic equation; Error estimates; Mixed finite element; Nonlinear; Porous media

Indexed keywords

EID: 0000325197     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142994266728     Document Type: Article

References (35)

1. H. W. ALT AND E. DIBENEDETTO, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), pp. 335-392.
2. H. W. ALT AND S. LUCKHAUS, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), pp. 311-341.
3. T. ARBOGAST, An Error Analysis for Galerkin Approximations to an Equation of Mixed Elliptic-Parabolic Type, Tech. Report TR 90-33, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1990.
4. _, The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19 (1992), pp. 1009-1031.
5. T. ARBOGAST, M. OBEYESEKERE, AND M. F. WHEELER, Numerical methods for the simulation of flow in root-soil systems, SIAM J. Numer. Anal., 30 (1993), pp. 1677-1702.
6. T. ARBOGAST, M. F. WHEELER, AND I. YOTOV, Mixed finite elements for elliptic problems with tensor coefficients as finite differences, SIAM J. Numer. Anal., 34 (1997), to appear.
7. J. W. BARRETT AND P. KNABNER, Finite element approximation of transport of reactive solutes in porous media, Part 2, Error estimates for equilibrium adsorption processes, SIAM J. Numer. Anal., 34 (1997), to appear.
8. J. BEAR, Dynamics of Fluids in Porous Media, Dover, New York, 1972.
9. F. BREZZI, J. DOUGLAS, JR., R. DURÁN, AND M. FORTIN, Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51 (1987), pp. 237-250.
10. 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. Numér., 21 (1987), pp. 581-604.
11. F. BREZZI, J. DOUGLAS, JR., AND L. D. MARINI, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217-235.
12. Z. CHEN AND J. DOUGLAS, JR., Prismatic mixed finite elements for second order elliptic problems, Calcolo, 26 (1989), pp. 135-148.
13. R. E. COLLINS, Flow of Fluids Through Porous Materials, Reinhold, New York, 1961.
14. J. DOUGLAS, JR. AND J. ROBERTS, Global estimates for mixed methods for second order elliptic problems, Math. Comp., 45 (1985), pp. 39-52.
15. C. J. VAN DUIJN AND P. KNABNER, Solute Transport Through Porous Media with Slow Adsorption, Report No. 54, Institut für Mathematik, Universitat Augsburg, Augsburg, Germany, 1988.
16. R. DURÁN, Superconvergence for rectangular mixed finite elements, Numer. Math., 58 (1990), pp. 287-298.
17. 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.
18. C. W. FETTER, Contaminant Hydrogeology, Macmillan, New York, 1993.
19. J. L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications 1, Springer-Verlag, New York, 1971.
20. E. MAGENES, R. H. NOCHETTO, AND C. VERDI, Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, RAIRO. Model. Math. Anal. Numér., 21 (1987), pp. 655-678.
21. F. A. MILNER, Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp., 44 (1985), pp. 303-320.
22. 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.
23. 3, Numer. Math., 35 (1980), pp. 315-341.
24. 3, Numer. Math., 50 (1986), pp. 57-81.
25. R. H. NOCHETTO, Error estimates for two-phase Stefan problems in several space variables, I: Linear boundary conditions, Calcolo, 22 (1985), pp. 457-499.
26. _, Finite element methods for parabolic free boundary problems, 1990 Lancaster Summer School Proceedings, in Advances in Numerical Analysis, Vol. I: Nonlinear Partial Differential Equations and Dynamical Systems, Oxford University Press, 1991, pp. 34-88.
27. R. H. NOCHETTO AND C. VERDI, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25 (1988), pp. 784-814.
28. D. W. PEACEMAN, Fundamentals of numerical reservoir simulation, Elsevier. Amsterdam, 1977.
29. P. 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.
30. M. E. ROSE, Numerical methods for flows through porous media. I, Math. Comp., 40 (1983), pp. 435-467.
31. _, Numerical methods for flows through porous media. II, Comput. Math. Appl., 20 (1980), pp. 99-122.
32. 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., Society for Industrial and Applied Mathematics, Philadelphia, 1983, pp. 35-106.
33. C. SRINIVAS, B. RAMASWAMY, M. F. WHEELER, Mixed finite element methods for flow through unsaturated porous media, in Computational Methods in Water Resources IX, Vol. 1: Numerical Methods 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. 239-246.
34. 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.
35. 2 error estimates for Calerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), pp. 723-759.