Discontinuous Galerkin methods for coupled flow and reactive transport problems

Applied Numerical Mathematics

Volumn 52, Issue 2-3 SPEC. ISS., 2005, Pages 273-298

  Sun, Shuyu ^a   Wheeler, Mary F ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

A priori error estimates; Coupled system; Discontinuous Galerkin methods; Flow; IIPG; NIPG; Reaction; SIPG; Transport

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; MATHEMATICAL OPERATORS; POLYNOMIAL APPROXIMATION; PROBLEM SOLVING; TENSORS;

A PRIORI ERROR ESTIMATES; COUPLED SYSTEM; DISCONTINUOUS GALERKIN METHODS; FLOW; IIPG; NIPG; REACTION; SIPG; TRANSPORT;

GALERKIN METHODS;

EID: 10444249692     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2004.08.035     Document Type: Conference Paper

References (55)

1. R. A. Adams, Sobolev Spaces, Academic Press, San Diego, CA, 1975.
2. V. Aizinger, C.N. Dawson, B. Cockburn, P. Castillo, The local discontinuous Galerkin method for contaminant transport, Adv. Water Resourc. 24 (2000) 73-87.
3. T. Arbogast, S. Bryant, C. Dawson, F. Saaf, C. Wang, M. Wheeler, Computational methods for multiphase flow and reactive transport problems arising in subsurface contaminant remediation, J. Comput. Appl. Math. 74 (1-2) (1996) 19-32.
4. T. Arbogast, M.F. Wheeler, A characteristics-mixed finite element method for advection-dominated transport problems, SIAM J. Numer. Anal. 32 (1995) 404-424.
5. D.N. Arnold, An Interior Penalty Finite Element Method with Discontinuous Elements, Ph.D. Thesis, The University of Chicago, Chicago, IL, 1979.
6. D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982) 742-760.
7. D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (5) (2002) 1749-1779.
8. I. Babuska, M. Suri, The h-p version of the finite element method with quasi-uniform meshes, RAIRO Model. Math. Anal. Numer. 21 (1987) 199-238.
9. I. Babuska, M. Suri, The optimal convergence rates of the p version of the finite element methos, SIAM J. Numer. Anal. 24 (1987) 750-776.
10. G. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977) 45-59.
11. C.E. Baumann, J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (3-4) (1999) 311-341.
12. C.M. Bethke, J. van der Lee, J.M. Schmitt, The chemistry beneath our feet modeling reacting flow in the earth's crust, in: Scientific bridges for 2000 and beyond, Académie des Sciences, 1999.
13. R.C. Borden, P.B. Bedient, Transport of dissolved hydrocarbons influenced by oxygen-limited biodegradation 1. Theoretical development, Water Resourc. Res. 22 (1986) 1973-1982.
14. S.L. Bryant, K.E. Thompson, Theory, modeling and experiment in reactive transport in porous media, Current Opinion Colloid Interface Sci. 6 (2001) 217-222.
15. C.Y. Chiang, C.N. Dawson, M.F. Wheeler, Modeling of in-situ biorestoration of organic compounds in groundwater, Transport Porous Media 6 (1991) 667-702.
16. B. Cockburn, C. Dawson, Approximation of the velocity by coupling discontinuous and mixed element methods for flow problems, Comput. Geosci. 6 (2002) 502-522.
17. B. Cockburn, C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (6) (1998) 2440-2463 (electronic).
18. C. Dawson, S. Sun, M.F. Wheeler, Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565-2580.
19. C.N. Dawson, Godunov-mixed methods for advection-diffusion equations in multidimensions, SIAM J. Numer. Anal. 30 (1993) 1315-1332.
20. C.N. Dawson, T.F. Russell, M.F. Wheeler, Some improved error estimates for the modified method of characteristics, SIAM J. Numer. Anal. 26 (1989) 1487-1512.
21. C.N. Dawson, M.F. Wheeler, Time-splitting methods for advection-diffusion-reaction equations arising in contaminant transport, in: R.E. O'Malley Jr (Ed.), Proceedings of the Second International Conference on Industrial and Applied Mathematics, ICIAM '91, SIAM, Philadelphia, PA, 1991, pp. 71-82.
22. P. Engesgaard, K.L. Kipp, A geochemical transport model for redox-controlled movement of mineral fronts in groundwater flow systems: A case of nitrate removal by oxidation of pyrite, Water Resourc. Res. 28 (10) (1992) 2829-2843.
23. A.K. Gupta, P.R. Bishnoi, N. Kalogerakis, A method for the simultaneous phase equilibria and stability calculations for multiphase reacting and non-reacting systmes, Fluid Phase Equilibria 63 (8) (1991) 65-89.
24. J.F. Kanney, C.T. Miller, D.A. Barry, Comparison of fully coupled approaches for approximating nonlinear transport and reaction problems, Adv. Water Resour. 26 (2003) 353-372.
25. J.S. Kindred, M.A. Celia, Contaminant transport and biodegradation 2. Conceptual model and test simulations, Water Resourc. Res. 25 (1989) 1149-1159.
26. P.M. Morel, J.G. Hering, Principles and Applications of Aquatic Chemistry, Wiley, New York, 1993.
27. J.T. Oden, I. Babuška, C.E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys. 146 (1998) 491-516.
28. B. Rivière, Discontinuous Galerkin Finite Element Methods for Solving the Miscible Displacement Problem in Porous Media, Ph.D. Thesis, The University of Texas at Austin, 2000.
29. B. Rivière, M.F. Wheeler, Nonconforming methods for transport with nonlinear reaction, in: Contemporary Mathematics, vol. 295, 2002, pp. 421-432.
30. B. Rivière, M.F. Wheeler, V. Girault, Part I: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems, Comput. Geosci. 3 (1999) 337-360.
31. B. Rivière, M.F. Wheeler, V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal. 39 (3) (2001) 902-931.
32. J. Rubin, Transport of reacting solutes in porous media: Relation between mathematical nature of problem formulation and chemical nature of reactions, Water Resourc. Res. 19 (5) (1983) 1231-1252.
33. J. Rubin, R.V. James, Dispersion-affected transport of reacting solutes in saturated porous media: Galerkin method applied to equilibrium-controlled exchange in unidirectional steady water flow, Water Resourc. Res. 9 (5) (1973) 1332-1356.
34. F. Saaf, A Study of Reactive Transport Phenomena in Porous Media, Ph.D. Thesis, Rice University, 1996.
35. Ch. Schwab, p- and hp-Finite Element Methods, Theory and Applications in Solid and Fluid Mechanics, Oxford Science Publications, Oxford, 1998.
36. J.V. Smith, R.W. Missen, W.R. Smith, General optimality criteria for multiphase multireaction chemical equilibrium, AIChE J. 39 (4) (1993) 707-710.
37. C.I. Steefel, P. Van Cappellen, Special issue: Reactive transport modeling of natural systems, J. Hydrology 209 (1-4) (1998) 1-388.
38. S. Sun, Discontinuous Galerkin Methods for Reactive Transport in Porous Media, Ph.D. Thesis, The University of Texas at Austin, 2003.
39. S. Sun, B. Rivière, M.F. Wheeler, A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media, in: Recent Progress in Computational and Applied PDEs, Conference Proceedings for the International Conference, Zhangjiaje, July 2001, pp. 321-348.
40. S. Sun, S. Wang, K.T. Yu, Three-dimensional simulation of distillation column containing structured packing (I) physical model and equations, Chinese J. Chem. Indust. Engrg. 49 (5) (1998) 549-559.
41. S. Sun, S. Wang, K.T. Yu, Theoretical study on the effect of axial liquid back-mixing on distillation in a packed column, Chemical Engineering (China) 27 (4) (1999) 7-10.
42. 1) norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems, J. Sci. Comput. (2004), in press.
43. S. Sun, M.F. Wheeler, A posteriori error analyses for symmetric discontinuous Galerkin approximations of reactive transport problems, TICAM Report 03-19, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 2003.
44. S. Sun, M.F. Wheeler, Mesh adaptation strategies for discontinuous Galerkin methods applied to reactive transport problems, in: Proceedings of International Conference on Computing, Communications and Control Technologies, CCCT 2004, vol. 1, 2004, pp. 223-228.
45. S. Sun, A. Zeng, S. Wang, K.T. Yu, Three-dimensional simulation of distillation column containing structured packing (II) validation of model and the effect of liquid distribution and mixing behavior on distillation, Chinese J. Chem. Indust. Engrg. 49 (5) (1998) 560-565.
46. A.J. Valocchi, M. Malmstead, Accuracy of operator splitting for advection-dispersion-reaction problems, Water Resourc. Res. 28 (5) (1992) 1471-1476.
47. J. van der Lee, L. De Windt, Present state and future directions of modeling of geochemistry in hydrogeological systems, J. Contam. Hydrol. 47 (2(4)) (2000) 265-282.
48. M.Th. van Genuchten, Analytical soultions for chemical transport with simultaneous adsorption, zero-order production and first-order decay, J. Hydrology 49 (1981) 213-233.
49. M.F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1) (1978) 152-161.
50. M.F. Wheeler, B.L. Darlow, Interior penalty Galerkin procedures for miscible displacement problems in porous media, in: Computational Methods in Nonlinear Mechanics, Proc. Second Internat. Conf., Univ. Texas, Austin, TX, 1979, North-Holland, Amsterdam, 1980, pp. 485-506.
51. M.F. Wheeler, C.N. Dawson, An operator-splitting method for advection-diffusion-reaction problems, in: J.R. Whiteman (Ed.), The Mathematics of Finite Elements and Applications VI, Academic Press, London, 1987, pp. 463-482.
52. M.F. Wheeler, C.N. Dawson, P.B. Bedient, C.Y. Chiang, R.C. Bordern, H.S. Rifai, Numerical simulation of microbial biodegradation of hydrocarbons in groundwater, in: Proceedings of AGWSE/IGWMCH Conference on Solving Ground Water Problems with Models, National Water Wells Association, 1987, pp. 92-108.
53. M.F. Wheeler, S. Sun, O. Eslinger, B. Rivière, Discontinuous Galerkin method for modeling flow and reactive transport in porous media, in: W. Wendland (Ed.), Analysis and Simulation of Multifield Problem, Springer, Berlin, 2003, pp. 37-58.
54. G.T. Yeh, V.S. Tripathi, A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components, Water Resourc. Res. 25 (1) (1989) 93-108.
55. G.T. Yeh, V.S. Tripathi, A model for simulating transport of reactive multispecies components: model development and demonstration, Water Resourc. Res. 27 (12) (1991) 3075-3094.