## Journal of Applied Mathematics

### A Completely Discrete Heterogeneous Multiscale Finite Element Method for Multiscale Richards’ Equation of van Genuchten-Mualem Model

#### Abstract

We propose a fully discrete method for the multiscale Richards’ equation of van Genuchten-Mualem model which describes the flow transport in unsaturated heterogenous porous media. Under the framework of heterogeneous multiscale method (HMM), a fully discrete scheme combined with a regularized procedure is proposed. Including the numerical integration, the discretization is given by ${C}^{0}$ piecewise finite element in space and an implicit scheme in time. Error estimates between the numerical solution and the solution of homogenized problem are derived under the assumption that the permeability is periodic. Numerical experiments with periodic and random permeability are carried out for the van Genuchten-Mualem model of Richards’ equation to show the efficiency and accuracy of the proposed method.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 657816, 10 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305622

Digital Object Identifier
doi:10.1155/2014/657816

Mathematical Reviews number (MathSciNet)
MR3187048

Zentralblatt MATH identifier
07010711

#### Citation

Cao, Haitao; Yu, Tao; Yue, Xingye. A Completely Discrete Heterogeneous Multiscale Finite Element Method for Multiscale Richards’ Equation of van Genuchten-Mualem Model. J. Appl. Math. 2014 (2014), Article ID 657816, 10 pages. doi:10.1155/2014/657816. https://projecteuclid.org/euclid.jam/1425305622

#### References

• L. A. Richards, “Capillary conduction of liquids through porous mediums,” Physics, vol. 1, pp. 318–333, 1931.
• M. Th. van Genuchten, “A closed-form equation for predicting the hydraulic conductivity of unsaturated soils,” Soil Science Society of America Journal, vol. 44, no. 5, pp. 892–898, 1980.
• Y. Mualem, “A new model for predicting the hydraulic conductivity of unsaturated porous media,” Water Resources Research, vol. 12, no. 3, pp. 513–522, 1976.
• H. W. Alt and S. Luckhaus, “Quasilinear elliptic-parabolic differential equations,” Mathematische Zeitschrift, vol. 183, no. 3, pp. 311–341, 1983.
• R. H. Nochetto, “Error estimates for multidimensional singular parabolic problems,” Japan Journal of Applied Mathematics, vol. 4, no. 1, pp. 111–138, 1987.
• R. H. Nochetto and C. Verdi, “Approximation of degenerate parabolic problems using numerical integration,” SIAM Journal on Numerical Analysis, vol. 25, no. 4, pp. 784–814, 1988.
• I. S. Pop, “Error estimates for a time discretization method for the Richards' equation,” Computational Geosciences, vol. 6, no. 2, pp. 141–160, 2002.
• I. S. Pop and W. A. Yong, “A maximum principle based numerical approach to porous medium equation,” in Proceedings of the 14th Conference on Scientific Computing, pp. 207–218, 1997.
• F. Radu, I. S. Pop, and P. Knabner, “Order of convergence estimate for an euler implicit mixed finite element discretization of Richards' equation,” SIAM Journal on Numerical Analysis, vol. 4, pp. 1452–1478, 2004.
• W. Jäger and J. Kačur, “Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes,” RAIRO Modélisation Mathématique et Analyse Numérique, vol. 29, no. 5, pp. 605–627, 1995.
• D. Kavetski, P. Binning, and S. W. Sloan, “Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards equation,” Advances in Water Resources, vol. 24, no. 6, pp. 595–605, 2001.
• I. S. Pop, F. A. Radu, and P. Knabner, “Mixed finite elements for the Richards' equation: linearization procedure,” Journal of Computational and Applied Mathematics, vol. 168, pp. 1365–2373, 2004.
• M. Slodicka, “A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic paroblems arising in flow in porous medium,” SIAM Journal on Scientific Computing, vol. 23, no. 5, pp. 1593–1614, 2002.
• 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.
• E. Weinan and B. Engquist, “The Heterogeneous multi-scale methods,” Communications in Mathematical Sciences, vol. 1, pp. 87–132, 2003.
• Z. Chen, W. Deng, and H. Ye, “Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media,” Communications in Mathematical Sciences, vol. 3, no. 4, pp. 493–515, 2005.
• A. Abdulle, “The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs,” in Multiple Scales Problems in Biomathematics, Mechanics, Physics and Numerics, vol. 31 of GAKUTO International Series Mathematical Sciences & Applications, pp. 133–181, 2009.
• P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, The Netherlands, 1978.
• H. T. Cao and X. Y. Yue, “Homogenization of Richards' equation of van Genuchten–-Mualem model,” Journal of Mathematical Analysis and Applications, vol. 412, no. 1, pp. 391–400, 2014. \endinput