Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media

SIAM Journal on Numerical Analysis

Volumn 37, Issue 3, 2000, Pages 701-724

  Woodward, Carol S ^a   Dawson, Clint N ^b  

^a LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Degenerate parabolic equation; Error estimates; Mixed finite elements; Nonlinear; Porous media; Richards' equation

Indexed keywords

EID: 0001077416     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142996311040     Document Type: Article

References (17)

1. H. W. ALT AND S. LUCKHAUS, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), pp. 311-341.
2. T. ARBOGAST, An Error Analysis for Galerkin Approximations to an Equation of Mixed Elliptic-Parabolic Type, Technical report TR90-33, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1990.
3. 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.
4. T. ARBOGAST, M. F. WHEELER, AND NAI-YING ZHANG, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal., 33 (1996), pp. 1669-1687.
5. T. ARBOGAST, M. F. WHEELER, AND I. YOTOV, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal., 34 (1997), pp. 828-852.
6. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
7. 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.
8. M. A. CELIA, E. T. BOULOUTAS, AND R. L. ZARBA, A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res., 26 (1990), pp. 1483-1496.
9. C. N. DAWSON, M. F. WHEELER, AND C. S. WOODWARD, A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal., 35 (1998), pp. 435-452.
10. R. A. FREEZE AND J. A. CHERRY, Groundwater, Prentice Hall, Englewood Cliffs, NJ, 1979.
11. R. HAVERKAMP AND M. VAUCLIN, A comparative study of three forms of the Richard equation used for predicting one-dimensional infiltration in unsaturated soil, Soil Sci. Soc. Amer. J., 45 (1981), pp. 13-20.
12. M. T. VAN GENUCHTEN, A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Amer. J., 44 (1980), pp. 892-898.
13. R. H. NOCHETTO AND C. VERDI, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal., 25 (1988), pp. 784-814.
14. P. A. RAVIART AND J. M. THOMAS, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. 606, I. Galligani and E. Magenes, eds., Springer-Verlag, Berlin, 1977, pp. 292-315.
15. M. E. ROSE, Numerical methods for flows through porous media I, Math. Comp., 40 (1983), pp. 435-467.
16. P. J. ROSS AND K. L. BRISTOW, Simulating water movement in layered and gradational soils using the Kirchhoff transform, Soil Sci. Soc. Amer. J., 54 (1990), pp. 1519-1524.
17. A. WEISER AND M. WHEELER, On convergence of block-centered finite differences for elliptic problems, SIAM J. Numer. Anal., 25 (1988), pp. 351-375.