Abstract and Applied Analysis

Discontinuous Galerkin Immersed Finite Volume Element Method for Anisotropic Flow Models in Porous Medium

Abstract

By choosing the trial function space to the immersed finite element space and the test function space to be piecewise constant function space, we develop a discontinuous Galerkin immersed finite volume element method to solve numerically a kind of anisotropic diffusion models governed by the elliptic interface problems with discontinuous tensor-conductivity. The existence and uniqueness of the discrete scheme are proved, and an optimal-order energy-norm estimate and ${L}^{2}$-norm estimate for the numerical solution are derived.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 520404, 10 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412279690

Digital Object Identifier
doi:10.1155/2014/520404

Mathematical Reviews number (MathSciNet)
MR3219374

Zentralblatt MATH identifier
07022544

Citation

Liu, Zhong-yan; Chen, Huan-zhen. Discontinuous Galerkin Immersed Finite Volume Element Method for Anisotropic Flow Models in Porous Medium. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 520404, 10 pages. doi:10.1155/2014/520404. https://projecteuclid.org/euclid.aaa/1412279690

References

• J. H. Bramble, “A finite element method for interface problems in domains with smooth boundaries and interfaces,” Advances in Computational Mathematics, vol. 6, no. 1, pp. 109–138, 1996.
• I. Babuska, “The finite element method for elliptic equations with discontinuous coefficients,” Computing, vol. 5, no. 3, pp. 207–213, 1970.
• Z. Chen and J. Zou, “Finite element methods and their convergence for elliptic and parabolic interface problems,” Numerische Mathematik, vol. 79, no. 2, pp. 175–202, 1998.
• R. J. Leveque and Z. Li, “The immersed interface method for elliptic equations with discontinuous coefficients and singular sources,” SIAM Journal on Numerical Analysis, vol. 31, no. 4, pp. 1019–1044, 1994.
• Z. Li, T. Lin, and X. Wu, “New Cartesian grid methods for interface problems using the finite element formulation,” Numerische Mathematik, vol. 96, no. 1, pp. 61–98, 2003.
• Y. Gong, B. Li, and Z. Li, “Immersed-interface finite-element methods for elliptic interface problems with nonhomogeneous jump conditions,” SIAM Journal on Numerical Analysis, vol. 46, no. 1, pp. 472–495, 2008.
• Z. Li, T. Lin, Y. Lin, and R. C. Rogers, “An immersed finite element space and its approximation capability,” Numerical Methods for Partial Differential Equations, vol. 20, no. 3, pp. 338–367, 2004.
• Z. Li, “The immersed interface method using a finite element formulation,” Applied Numerical Mathematics, vol. 27, no. 3, pp. 253–267, 1998.
• Z. Li and K. Ito, The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, vol. 33 of Frontiers in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2006.
• R. E. Ewing, Z. Li, T. Lin, and Y. Lin, “The immersed finite volume element methods for the elliptic interface problems,” Mathematics and Computers in Simulation, vol. 50, no. 1–4, pp. 63–76, 1999.
• Z. Cai, “On the finite volume element method,” Numerische Mathematik, vol. 58, no. 1, pp. 713–735, 1991.
• S. H. Chou, “Analysis and convergence of a covolume method for the generalized stokes problem,” Mathematics of Computation, vol. 66, no. 217, pp. 85–104, 1997.
• Z. Cai and S. Mccormick, “On the accuracy of the finite volume element method for diffusion equations on composite grids,” SIAM Journal on Numerical Analysis, vol. 27, no. 3, pp. 636–655, 1990.
• R. E. Ewing, R. D. Lazarov, and Y. Lin, “Finite volune element approximations of non-local in time one-dimensional reactive flows in porous media,” Tech. Rep. ISC-98-06-MATH, Institute for Scientific Computation.
• E. Süli, “Convergence of finite volume schemes for Poisson's equation on nonuniform meshes,” SIAM Journal on Numerical Analysis, vol. 28, no. 5, pp. 1419–1430, 1991.
• J. M. Thomas and D. Trujillo, “Analysis of finite volume methods,” Tech. Rep. 19, Université de Pau et des Pays de L'adour, Pau, France, 1995.
• N. An and H. Z. Chen, “A partially penalty čommentComment on ref. [1?]: Please update the information of this reference, if possible.immersed finite element method for anisotropic flow medels in porous medium,” Numerical Methods for PDE. In press.
• R. G. Zhang and H. Z. Chen, “An immersed finite element method for anisotropic flow models in porous medium,” in Proceedings of the International Conference on Information Science and Technology (ICIST '11), pp. 168–175, Nanjing, China, March 2011.
• D. N. Arnold, F. Brezzi, B. Cockburn, and L. Donatella Marini, “Unified analysis of discontinuous Galerkin methods for elliptic problems,” SIAM Journal on Numerical Analysis, vol. 39, no. 5, pp. 1749–1779, 2002.
• B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, vol. 35 of Frontiers in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2008.
• X. Ye, “A new discontinuous finite volume method for elliptic problems,” SIAM Journal on Numerical Analysis, vol. 42, no. 3, pp. 1062–1072, 2004. \endinput