Locally conservative, stabilized finite element methods for variably saturated flow

Computer Methods in Applied Mechanics and Engineering

Volumn 197, Issue 51-52, 2008, Pages 4610-4625

  Kees, C E ^a   Farthing, M W ^a   Dawson, C N ^b  

^a US ARMY ENGINEER RESEARCH AND DEVELOPMENT CENTER   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Finite element method; Local conservation; Multiscale stabilization; Richards' equation

Indexed keywords

ADVECTION; FLOW OF WATER; FLUID MECHANICS; GROUNDWATER; GROUNDWATER FLOW; HYDRAULICS; HYDROGEOLOGY; INCOMPRESSIBLE FLOW; LINEAR EQUATIONS; MOTION ESTIMATION; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS; UNDERGROUND RESERVOIRS;

FINITE ELEMENT; FINITE ELEMENT DISCRETIZATIONS; FLOW MODELLING; FLOW PROBLEMS; GALERKIN FINITE ELEMENT METHODS; HETEROGENEOUS DOMAINS; LOCAL CONSERVATION; LOCAL OSCILLATIONS; MASS-LUMPING; MULTI SCALING; MULTISCALE STABILIZATION; NONLINEAR ADVECTION-DIFFUSION EQUATION; NONPHYSICAL OSCILLATIONS; PRESSURE FIELDS; RICHARDS' EQUATION; STABILIZED FINITE-ELEMENT METHODS; TO MANY; VARIABLY SATURATED; VARIABLY SATURATED FLOW; VELOCITY FIELDS;

FINITE ELEMENT METHOD;

EID: 53749088165     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2008.06.005     Document Type: Article

References (79)

1. de Marsily G. Quantitative Hydrogeology: Groundwater Hydrology for Engineers (1986), Academic Press, San Diego, CA
2. Witelski T.P. Horizontal infiltration into wet soil. Water Resour. Res. 34 7 (1998) 1859-1863
3. Celia M.A., Bouloutas E.T., and Zarba R.L. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res. 26 7 (1990) 1483-1496
4. Kees C.E., and Miller C.T. Higher order time integration methods for two-phase flow. Adv. Water Resour. 25 2 (2002) 159-177
5. Manzini G., and Ferraris S. Mass-conservative finite volume methods on 2-D unstructured grids for the Richards' equation. Adv. Water Resour. 12 (2004) 1199-1215
6. Ern A., and Guermond J.-L. Theory and Practice of Finite Elements. Applied Mathematical Sciences vol. 159 (2004), Springer-Verlag, New York
7. Forsyth P., and Kropinksi M. Monotonicity considerations for saturated-unsaturated subsurface flow. SIAM J. Sci. Comput. 18 5 (1997) 1328-1354
8. Farthing M.W., Kees C.E., and Miller C.T. Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv. Water Resour. 26 4 (2003) 373-394
9. Bause M., and Knabner P. Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Adv. Water Resour. 27 (2004) 565-581
10. Srivastava R., and Yeh T.C.J. A 3-Dimensional numerical model for water flow and transport of chemically reactive solute through porous media under variably saturated conditions. Adv. Water Resour. 15 5 (1992) 275-287
11. Woodward C.S., and Dawson C.N. Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J. Numer. Anal. 37 3 (2000) 701-724
12. Li H., Farthing M.W., Dawson C.N., and Miller C.T. Local discontinuous Galerkin approximations to Richards' equation. Adv. Water Resour. 30 (2007) 555-575
13. Miller C.T., Abhishek C., Sallerson A.B., Prins J.F., and Farthing M.W. A comparison of computational and algorithmic advances for solving Richards' equation. In: Miller C.T., Farthing M.W., Gray W.G., and Pinder G.F. (Eds). Proceedings of the XVth International Conference on Computational Methods in Water Resources (2004), Elsevier, Chapel Hill, North Carolina 1131-1145
14. Hughes T., Feijóo G., Mazzei L., and Quincy J. The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24
15. Larson M., and Niklasson A. A conservative flux for the continuous Galerkin method based on discontinuous enrichment. CALCOLO 41 (2004) 65-76
16. Sun S.Y., and Wheeler M.F. A posteriori error estimation and dynamic adaptivity for symmetric discontinuous Galerkin approximations of reactive transport problems. Comput. Methods Appl. Mech. Engrg. 195 7-8 (2006) 632-652
17. Hughes T., and Wells G. Conservation properties for the Galerkin and stabilised forms of the advection-diffusion and incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1141-1159
18. Hughes T. Multiscale phenomena: Greens's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127 (1995) 387-401
19. Abriola L.M., and Rathfelder K. Mass balance errors in modeling two-phase immiscible flows: causes and remedies. Adv. Water Resour. 16 (1993) 223-239
20. Dawson C., Sun S., and Wheeler M.F. Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565-2580
21. Hughes T., Engel G., Mazzei L., and Larson M. The continuous Galerkin method is locally conservative. J. Comput. Phys. 163 (2000) 467-488
22. Chippada S., Dawson C.N., Martinez M.L., and Wheeler M.F. A projection method for constructing a mass conservative velocity field. Comput. Methods Appl. Mech. Engrg. 157 1-2 (1998) 1-10
23. Chavent G., and Roberts J.E. A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Adv. Water Resour. 14 6 (1991) 329-348
24. Dawson C. Coupling local discontinuous and continuous Galerkin methods for flow problems. Adv. Water Resour. 28 7 (2005) 729-744
25. Masud A., and Hughes T. A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4341-4370
26. Hughes T., Masud A., and Wan J. A stabilized mixed discontinuous Galerkin method for Darcy flow. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3347-3381
27. Bergamaschi L., and Putti M. Mixed finite elements and Newton-type linearizations for the solution of Richards' equation. Int. J. Numer. Methods Engrg. 45 (1999) 1025-1046
28. Farthing M.W., Kees C.E., Coffey T.S., Kelley C.T., and Miller C.T. Efficient steady-state solution techniques for variably saturated groundwater flow. Adv. Water Resour. 26 8 (2003) 833-849
29. Cordes C., and Kinzelbach K. Continuous groundwater velocity field and path lines in linear, bilinear, and trilinear finite elements. Water Resour. Res. 28 11 (1992) 2903-2911
30. Ackerer P., Mos R., Siegel P., and Chavent G. Reply application of the mixed hybrid finite element approximation in a groundwater flow model: luxury or necessity? by R. Mosé, P. Siegel, P. Ackerer, and G. Chavent. Water Resour. Res. 32 6 (1996) 1911-1913
31. Cordes C., and Kinzelbach K. Comment Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity? by R. Mosé, P. Siegel, P. Ackerer, and G. Chavent. Water Resources Research 32 6 (1996) 1905-1909
32. Durlofsky L.J. Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities. Water Resour. Res. 30 4 (1994) 965-973
33. Turek S. Efficient Solvers for Incompressible Flow Problems: An algorithmic and Computational Approach. Lecture Notes in Computational Science and Engineering vol. 6 (1999), Springer-Verlag
34. Codina R. Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comput. Methods Appl. Mech. Engrg. 156 (1998) 185-210
35. Franca L.P., Frey S.L., and Hughes T.J.R. Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg. 95 2 (1992) 253-276
36. Nagrath S., Jansen K.E., and Lahey R.T. Computation of incompressible bubble dynamics with a stabilized finite element level set method. Comput. Methods Appl. Mech. Engrg. 194 (2005) 4565-4587
37. Tezduyar T.E., Mittal S., Ray S.E., and Shih R. Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput. Methods Appl. Mech. Engrg. 95 (1992) 221-242
38. Arbogast T. Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal. 42 2 (2004) 576-598
39. Codina R., and Blasco J. Analysis of a stabilized finite element approximation of the transient convection-diffusion-reaction equation using orthogonal subscales. Comput. Visualization Sci. 4 (2002) 167-174
40. Arbogast T. Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci. 6 (2002) 453-481
41. Juanes R., and Patzek T. Multiscale-stabilized solutions to one-dimensional systems of conservation laws. Comput. Methods Appl. Mech. Engrg. 194 (2005) 2781-2805
42. Brezzi F., Franca L., Hughes T., and Russo A. b = ∫ g. Comput. Methods Appl. Mech. Engrg. 145 (1997) 329-339
43. Arbogast T. An overview of subgrid upscaling for elliptic problems in mixed form. In: Chen Z., Glowinksi R., and Li K. (Eds). Current Trends in Scientific Computing, Contemporary Mathematics (2003), AMS 21-32
44. Aarnes J. On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. SIAM Multiscale Model. Simulat. 2 3 (2004) 421-439
45. Chen Y., Durlofsky L., Gerritsen M., and Wen X. A coupled local-global upscaling approach for simulating highly heterogeneous formations. Adv. Water Resour. 26 10 (2003) 1041-1060
46. Hou T.Y., and Wu X.H. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189
47. Jenny P., Lee S.H., and Tchelepi H.A. Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47-67
48. Hughes T.J.R., and Franca L.P. A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Comput. Methods Appl. Mech. Engrg. 65 (1987) 85-96
49. Chen Z., and Ewing R.E. From single-phase to compositional flow: applicability of mixed finite elements. Transport Porous Med. 27 (1997) 225-242
50. Klausen R., and Russell T. Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput. Geosci. 8 4 (2004) 341-377
51. Arnold D., and Brezzi F. Mixed and nonconforming finite element methods: implementation, postprocessing, and error estimates. Math. Model. Numer. Anal. 19 1 (1985) 7-32
52. Marini L.D. An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22 3 (1985) 493-496
53. Arbogast T., and Chen Z. On the implementation of mixed methods as nonconforming methods for second order elliptic problems. Math. Comput. 64 211 (1995) 943-972
54. Chen Z. Equivalence between and multigrid algorithms for nonconforming and mixed methods for second order elliptic problems. East-West J. Numer. Math. 4 1 (1996) 1-82
55. 1 conforming and nonconforming Galerkin FEMs: effective flux evaluation via a nonmixed method approach. SIAM J. Numer. Anal. 38 2 (2000) 660-680
56. Miller C.T., Christakos G., Imhoff P.T., McBride J.F., Pedit J.A., and Trangenstein J.A. Multiphase flow and transport modeling in heterogeneous porous media: challenges and approaches. Adv. Water Resour. 21 2 (1998) 77-120
57. van Genuchten M.T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44 (1980) 892-898
58. Mualem Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12 3 (1976) 513-522
59. Lang J. Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems (2001), Springer-Verlag, Berlin, Germany
60. Hairer E., and Wanner G. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems (1996), Springer-Verlag, Berlin
61. Hauke G. A simple subgrid scale stabilized method for the advection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2925-2947
62. Burman E., and Ern A. Nonlinear diffusion and discrete maximum principle for stabilized Galerkin approximations of the convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 191 (2002) 3833-3855
63. Hughes T.J.R., Mallet M., and Mizukami A. A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Comput. Methods Appl. Mech. Engrg. 54 3 (1986) 341-355
64. Brezzi F., Douglas Jr. J., Durán R., and Fortin M. Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237-250
65. Bazilevs Y., and Hughes T. Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput. Fluids 36 (2007) 12-26
66. Paniconi C., and Putti M. A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems. Water Resour. Res. 30 12 (1994) 3357-3374
67. Lehmann F., and Ackerer P.H. Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media. Transport Porous Med. 31 3 (1998) 275-292
68. C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Society for Industrial and Applied Mathematics, Philadelphia, 1995.
69. W.L. Briggs, V.E. Henson, S.F. McCormick, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, 2000.
70. C.E. Kees, M.W. Farthing, E.W. Jenkins, C.T. Kelley, S.E. Howington, Multiscale models of air/water flow in porous media, in: P.J. Binning, P.K. Engesgaard, H.K. Dahle, G.F. Pinder, W.G. Gray (Eds.), XVI International Conference on Computational Methods in Water Resources, Copenhagen, Denmark, 2006.
71. Li X. An overview of SuperLU: Algorithms, implementation, and user interface. ACM Trans. Math. Softw. 31 3 (2005) 302-325
72. Hairer E., Norsett S.P., and Wanner G. Solving Ordinary Differential Equations I. Nonstiff Problems (1993), Springer-Verlag, Berlin
73. Tezduyar T., and Osawa Y. Finite element stabilization parameters computed from element matrices and vectors. Comput. Methods Appl. Mech. Engrg. 3-4 (2000) 411-430
74. Miller C.T., Williams G.A., Kelley C.T., and Tocci M.D. Robust solution of Richards' equation for non-uniform porous media. Water Resour. Res. 34 10 (1998) 2599-2610
75. Farthing M.W., Kees C.E., and Miller C.T. Mixed finite element methods and higher-order temporal approximations. Adv. Water Resour. 25 1 (2002) 85-101
76. 1 nonconforming finite element multigrid method for radiation transport. SIAM J. Sci. Comput. 25 2 (2003) 369-384
77. Shewchuk J. Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. Applied Computational Geometry: Towards Geometric Engineering. In: Lin M., and Manocha D. (Eds). Lecture Notes in Computer Science vol. 1148 (1996), Springer-Verlag 203-222. http://www.cs.cmu.edu/~quake/triangle.html
78. Codina R., Principe J., Guasch O., and Badia S. Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2413-2430
79. Kees C.E., and Miller C.T. C++ implementations of numerical methods for solving differential-algebraic equations: design and optimization considerations, association for computing machinery. Trans. Math. Softw. 25 4 (1999) 377-403