`Mathematical Problems in EngineeringVolume 2012, Article ID 187634, 23 pageshttp://dx.doi.org/10.1155/2012/187634`
Research Article

## The Finite Volume Formulation for 2D Second-Order Elliptic Problems with Discontinuous Diffusion/Dispersion Coefficients

Dipartimento di Economia e Ingegneria Agraria, Forestale e Ambientale (DEIAFA) Sez. Idraulica, Via Leonardo da Vinci 44, 10095 Grugliasco, Italy

Received 18 July 2011; Revised 2 November 2011; Accepted 17 November 2011

Copyright © 2012 Stefano Ferraris et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. R. Eymard, T. Gallouët, and R. Herbin, “Finite volume methods,” in Handbook of Numerical Analysis, vol. 7, pp. 713–1020, North-Holland, Amsterdam, The Netherlands, 2000.
2. Y. Coudière, Analyse de schémas volumes finis sur maillages non structurés pour des problèmes linéaires hyperboliques et elliptiques, Ph.D. thesis, Universitè “P. Sabatier” de Toulouse, Toulouse, France, 1999.
3. Y. Coudière, J.-P. Vila, and P. Villedieu, “Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem,” Mathematical Modelling and Numerical Analysis, vol. 33, no. 3, pp. 493–516, 1999.
4. Y. Coudière and P. Villedieu, “Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes,” M2AN. Mathematical Modelling and Numerical Analysis, vol. 34, no. 6, pp. 1123–1149, 2000.
5. E. Bertolazzi and G. Manzini, “A cell-centered second-order accurate finite volume method for convection-diffusion problems on unstructured meshes,” Mathematical Models & Methods in Applied Sciences, vol. 14, no. 8, pp. 1235–1260, 2004.
6. G. Manzini and A. Russo, “A finite volume method for advection-diffusion problems in convection-dominated regimes,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 13–16, pp. 1242–1261, 2008.
7. G. Manzini and S. Ferraris, “Mass-conservative finite volume methods on 2-D unstructured grids for the Richards' equation,” Advances in Water Resources, vol. 27, no. 12, pp. 1199–1215, 2004.
8. R. Haverkamp and M. Vauclin, “A note on estimating finite difference interblock hydraulic conductivity values for transient unsaturated flow problems,” Water Resources Research, vol. 15, no. 1, pp. 181–188, 1979.
9. N. Varado, I. Braud, P. J. Ross, and R. Haverkamp, “Assessment of an efficient numerical solution of the 1D Richards' equation on bare soil,” Journal of Hydrology, vol. 323, no. 1–4, pp. 244–257, 2006.
10. N. Romano, B. Brunone, and A. Santini, “Numerical analysis of one-dimensional unsaturated flow in layered soils,” Advances in Water Resources, vol. 21, no. 4, pp. 315–324, 1998.
11. J. M. Gastó, J. Grifoll, and Y. Cohen, “Estimation of internodal permeabilities for numerical simulation of unsaturated flows,” Water Resources Research, vol. 38, no. 12, pp. 621–6210, 2002.
12. A. Szymkiewicz and R. Helmig, “Comparison of conductivity averaging methods for one-dimensional unsaturated flow in layered soils,” Advances in Water Resources, vol. 34, no. 8, pp. 1012–1025, 2011.
13. P. Berg, “Long-term simulation of water movement in soils using mass-conserving procedures,” Advances in Water Resources, vol. 22, no. 5, pp. 419–430, 1999.
14. F. Chen and L. Ren, “Application of the finite difference heterogeneous multiscale method to the Richards' equation,” Water Resources Research, vol. 44, no. 7, article W07413, 2008.
15. L. Li, H. Zhou, and J. J. Gómez-Hernández J. Jaime, “A comparative study of three-dimensional hydraulic conductivity upscaling at the macro-dispersion experiment (MADE) site, Columbus Air Force Base, Mississippi (USA),” Journal of Hydrology, vol. 404, no. 3-4, pp. 278–293, 2011.
16. W. E and B. Engquist, “The heterogeneous multiscale methods,” Communications in Mathematical Sciences, vol. 1, no. 1, pp. 87–132, 2003.
17. Y. Efendiev and T. Y. Hou, Multiscale Finite Element Methods. Theory and Applications, vol. 4, Springer, New York, NY, USA, 2009.
18. I. Babuška, “Solution of interface problems by homogenization—I, II,” SIAM Journal on Mathematical Analysis, vol. 7, no. 5, pp. 603–634, 1976.
19. I. Babuška, “Homogenization and its application. Mathematical and computational problems,” in Numerical Solution of Partial Differential Equations III, B. Hubbard, Ed., pp. 89–116, Academic Press, New York, NY, USA, 1976.
20. I. Babuška, “Solution of interface problems by homogenization—III,” SIAM Journal on Mathematical Analysis, vol. 8, no. 6, pp. 923–937, 1977.
21. L. J. Durlofsky, Y. Efendiev, and V. Ginting, “An adaptive localglobal multiscale finite volume element method for two-phase flow simulations,” Advances in Water Resources, vol. 30, pp. 576–588, 2007.
22. P. Dostert, Y. Efendiev, and B. Mohanty, “Efficient uncertainty quantification techniques for Richards’ equation,” Advances in Water Resources, vol. 32, pp. 329–339, 2009.
23. L. Bergamaschi, S. Mantica, and G. Manzini, “A mixed finite element—finite volume formulation of the black-oil model,” SIAM Journal on Scientific Computing, vol. 20, no. 3, pp. 970–997, 1999.
24. C. Gallo and G. Manzini, “2-D numerical modeling of bioremediation in heterogeneous saturated soils,” Transport in Porous Media, vol. 31, no. 1, pp. 67–88, 1998.
25. C. Gallo and G. Manzini, “A mixed finite element finite volume approach for solving biodegradation transport in groundwater,” International Journal for Numerical Methods in Fluids, vol. 26, no. 5, pp. 533–556, 1998.
26. C. Gallo and G. Manzini, “A fully coupled numerical model for two-phase flow with contaminant transport biodegradation kinetics,” Communications in Numerical Methods in Engineering, vol. 17, no. 5, pp. 325–336, 2001.
27. L. Beirão da Veiga, V. Gyrya, K. Lipnikov, and G. Manzini, “Mimetic finite difference method for the Stokes problem on polygonal meshes,” Journal of Computational Physics, vol. 228, no. 19, pp. 7215–7232, 2009.
28. A. Bouziani and I. Mounir, “On the solvability of a class of reaction-diffusion systems,” Mathematical Problems in Engineering, vol. 2006, Article ID 47061, 15 pages, 2006.
29. A. Cangiani and G. Manzini, “Flux reconstruction and solution post-processing in mimetic finite difference methods,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 9–12, pp. 933–945, 2008.
30. W. Correa de Oliveira Casaca and M. Boaventura, “A decomposition and noise removal method combining diffusion equation and wave atoms for textured images,” Mathematical Problems in Engineering, vol. 2010, Article ID 764639, 20 pages, 2010.
31. M. Dehghan, “On the numerical solution of the diffusion equation with a nonlocal boundary condition,” Mathematical Problems in Engineering, vol. 2003, no. 2, pp. 81–92, 2003.
32. S. Hu, Z. Liao, D. Sun, and W. Chen, “A numerical method for preserving curve edges in nonlinear anisotropic smoothing,” Mathematical Problems in Engineering, vol. 2011, Article ID 186507, 14 pages, 2011.
33. Z. Liao, S. Hu, D. Sun, and W. Chen, “Enclosed Laplacian operator of nonlinear anisotropic diffusion to preserve singularities and delete isolated points in image smoothing,” Mathematical Problems in Engineering, vol. 2011, Article ID 749456, 15 pages, 2011.
34. R. J. Moitsheki, “Transient heat diffusion with temperature-dependent conductivity and time-dependent heat transfer coefficient,” Mathematical Problems in Engineering, vol. 2008, Article ID 347568, 9 pages, 2008.
35. N. P. Moshkin and D. Yambangwai, “On numerical solution of the incompressible Navier-Stokes equations with static or total pressure specified on boundaries,” Mathematical Problems in Engineering, vol. 2009, Article ID 372703, 26 pages, 2009.
36. W. Wenquan, Z. Lixiang, Y. Yan, and G. Yakun, “Finite element analysis of turbulent flows using les and dynamic subgridscale models in complex geometries,” Mathematical Problems in Engineering, vol. 2011, Article ID 712372, 20 pages, 2011.
37. J. Yu, Y. Yang, and A. Campo, “Approximate solution of the nonlinear heat conduction equation in a semi-infinite domain,” Mathematical Problems in Engineering, vol. 2010, Article ID 421657, 24 pages, 2010.
38. Y. Zheng and Z. Zhao, “A fully discrete Galerkin method for a nonlinear space-fractional diffusion equation,” Mathematical Problems in Engineering, vol. 2011, Article ID 171620, 20 pages, 2011.
39. P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24, Pitman, Boston, Mass, USA, 1985.
40. E. Bertolazzi and G. Manzini, “A second-order maximum principle preserving finite volume method for steady convection-diffusion problems,” SIAM Journal on Numerical Analysis, vol. 43, no. 5, pp. 2172–2199, 2005.
41. E. Bertolazzi and G. Manzini, “On vertex reconstructions for cell-centered finite volume approximations of 2D anisotropic diffusion problems,” Mathematical Models & Methods in Applied Sciences, vol. 17, no. 1, pp. 1–32, 2007.
42. G. Manzini and M. Putti, “Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations,” Journal of Computational Physics, vol. 220, no. 2, pp. 751–771, 2007.
43. G. P. Leonardi, F. Paronetto, and M. Putti, “Effective anisotropy tensor for the numerical solution of flow problems in heterogeneous porous media,” in Proceedings of the CMWR 16th International Conference, Copenhagen, Denmark, 2006.
44. M. Quintard and S. Whitaker, “Single-phase flow in porous media: effect of local heterogeneities,” Journal de Mecanique Theorique et Appliquee, vol. 6, no. 5, pp. 691–726, 1987.
45. L. Tartar, “Estimations fines des coefficients homogénéisés,” in Ennio De Giorgi Colloquium, vol. 125, pp. 168–187, Pitman, Boston, Mass, USA, 1985.
46. R. Eymard, G. Henri, R. Herbin, R. Klofkorn, and G. Manzini, “3D Benchmark on discretizations schemes for anisotropic diffusion problems on general grids,” in Finite Volumes for Complex Applications VI, Problems and Perspectives, J. Fort, J. Furst, J. Halama, R. Herbin, and F. Hubert, Eds., vol. 2, pp. 95–130, Springer, 2011.
47. L. Beirão da Veiga and G. Manzini, “A higher-order formulation of the mimetic finite difference method,” SIAM Journal on Scientific Computing, vol. 31, no. 1, pp. 732–760, 2008.