- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 575298, 19 pages
New Scheme of Finite Difference Heterogeneous Multiscale Method to Solve Saturated Flow in Porous Media
1Department of Mathematics, Xiangnan University, Chenzhou 423000, China
2Department of Soil and Water Sciences, China Agricultural University and Key Laboratory of Plant-Soil Interactions, MOE, Beijing 100094, China
Received 4 October 2013; Accepted 16 January 2014; Published 2 March 2014
Academic Editor: Shuyu Sun
Copyright © 2014 Fulai Chen and Li Ren. 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.
- B. B. Dykaar and P. K. Kitanidis, “Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach 2: results,” Water Resources Research, vol. 28, no. 4, pp. 1167–1178, 1992.
- T. Y. Hou and X.-H. Wu, “A multiscale finite element method for elliptic problems in composite materials and porous media,” Journal of Computational Physics, vol. 134, no. 1, pp. 169–189, 1997.
- T. Y. Hou, X.-H. Wu, and Z. Cai, “Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,” Mathematics of Computation, vol. 68, no. 227, pp. 913–943, 1999.
- W. E and B. Engquist, “The heterogeneous multiscale methods,” Communications in Mathematical Sciences, vol. 1, no. 1, pp. 87–132, 2003.
- Y. Efendiev and A. Pankov, “Numerical homogenization of nonlinear random parabolic operators,” Multiscale Modeling & Simulation, vol. 2, no. 2, pp. 237–268, 2004.
- I. Babuška, “Solution of interface problems by homogenization. I,” SIAM Journal on Mathematical Analysis, vol. 7, no. 5, pp. 603–634, 1976.
- 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.
- I. Babuška, “Solution of interface problems by homogenization. III,” SIAM Journal on Mathematical Analysis, vol. 8, no. 6, pp. 923–937, 1977.
- L. J. Durlofsky, Y. Efendiev, and V. Ginting, “An adaptive local-global multiscale finite volume element method for two-phase flow simulations,” Advances in Water Resources, vol. 30, no. 3, pp. 576–588, 2007.
- S. Ye, Y. Xue, and C. Xie, “Application of the multiscale finite element method to flow in heterogeneous porous media,” Water Resources Research, vol. 40, Article ID W09202, 2004.
- Z. Chen and T. Y. Hou, “A mixed multiscale finite element method for elliptic problems with oscillating coefficients,” Mathematics of Computation, vol. 72, no. 242, pp. 541–576, 2003.
- X. He and L. Ren, “Finite volume multiscale finite element method for solving the groundwater flow problems in heterogeneous porous media,” Water Resources Research, vol. 41, no. 10, Article ID W10417, 2005.
- W. E, P. Ming, and P. Zhang, “Analysis of the heterogeneous multiscale method for elliptic homogenization problems,” Journal of the American Mathematical Society, vol. 18, no. 1, pp. 121–156, 2005.
- P. Ming and P. Zhang, “Analysis of the heterogeneous multiscale method for parabolic homogenization problems,” Mathematics of Computation, vol. 76, no. 257, pp. 153–177, 2007.
- P. Ming and X. Yue, “Numerical methods for multiscale elliptic problems,” Journal of Computational Physics, vol. 214, no. 1, pp. 421–445, 2006.
- X. Yue and W. E, “Numerical methods for multiscale transport equations and application to two-phase porous media flow,” Journal of Computational Physics, vol. 210, no. 2, pp. 656–675, 2005.
- F. Rotersa, P. Eisenlohra, L. Hantcherlia, et al., “Overview of constitutive laws, kinematics, homogenization and multiscalemethods in crystal plasticity finite-element modeling: theory, experiments, applications,” Acta Materialia, vol. 58, pp. 1152–1211, 2010.
- J. Principe, R. Codina, and F. Henke, “The dissipative structure of variational multiscale methods for incompressible flows,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 13–16, pp. 791–801, 2010.
- V. John and A. Kindl, “Numerical studies of finite element variational multiscale methods for turbulent flow simulations,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 13–16, pp. 841–852, 2010.
- A. Abdulle and W. E, “Finite difference heterogeneous multi-scale method for homogenization problems,” Journal of Computational Physics, vol. 191, no. 1, pp. 18–39, 2003.
- W. E, B. Engquist, and Z. Huang, “Heterogeneous multiscale method: a general methodology for multiscale modeling,” Physical Review B, vol. 67, Article ID 092101, 2003.
- F. Chen and L. Ren, “Application of the finite difference heterogeneous multiscale method to the Richards' equation,” Water Resources Research, vol. 44, Article ID W07413, 2008.
- X. Wen and J. J. Gomez-Hernandez, “Upscaling hydraulic conductivities in heterogeneous media: an overview,” Journal of Hydrology, vol. 183, no. 1-2.
- P. Renard and G. de Marsily, “Calculating equivalent permeability: a review,” Advances in Water Resources, vol. 20, no. 5-6, pp. 253–278, 1997.
- R. Du and P. Ming, “Heterogeneous multiscale finite element method with novel numerical integration schemes,” Communications in Mathematical Sciences, vol. 8, pp. 797–1091, 2010.
- P. M. de Zeeuw, “Matrix-dependent prolongations and restrictions in a blackbox multigrid solver,” Journal of Computational and Applied Mathematics, vol. 33, no. 1, pp. 1–27, 1990.
- L. W. Gelhar and C. L. Axness, “Three-dimensional stochastic analysis of macrodispersion in aquifers,” Water Resources Research, vol. 19, no. 1, pp. 161–180, 1983.
- H. Cao and X. Yue, “The discrete finite volume method on quadrilateral mesh,” Journal of Suzhou University, vol. 10, pp. 6–10, 2005.
- A. Mantoglou and J. L. Wilson, “The turning bands method for simulation of random fields using line generation by a spectral method,” Water Resources Research, vol. 18, no. 5, pp. 1379–1394, 1982.
- H. F. Wang and M. P. Anderson, Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods, W. H. Free-Man and Company, San Francisco, Calif, USA, 1982.
- X. He and L. Ren, “A modified multiscale finite element method for well-driven flow problems in heterogeneous porous media,” Journal of Hydrology, vol. 329, no. 3-4, pp. 674–684, 2006.