Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T11:24:54.743Z Has data issue: false hasContentIssue false
  • English
  • Français

A Posteriori Error Estimates of a Combined Mixed Finite Element and Discontinuous Galerkin Method for a Kind of Compressible Miscible Displacement Problems

Published online by Cambridge University Press:  03 June 2015

Jiming Yang  and
Zhiguang Xiong
Jiming Yang*
Affiliation:
College of Science, Hunan Institute of Engineering, Xiangtan 411104, Hunan, China
Zhiguang Xiong*
Affiliation:
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, Hunan, China
*
Corresponding author. Email: [email protected]
Email: [email protected]
Get access

Abstract

A kind of compressible miscible displacement problems which include molecular diffusion and dispersion in porous media are investigated. The mixed finite element method is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin method. Based on a duality argument, employing projection estimates and approximation properties, a posteriori residual-type hp error estimates for the coupled system are presented, which is often used for guiding adaptivity. Comparing with the error analysis carried out by Yang (Int. J. Numer. Meth. Fluids, 65(7) (2011), pp. 781–797), the current work is more complicated and challenging.

Keywords

65M1265M60A posteriori errordiscontinuous Galerkin methodcompressible miscible displacementmixed finite elementduality argument
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 2 , April 2013 , pp. 163 - 179
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Babuška, I. and Suri, M., The optimal convergence rate of the p-version of the finite element method, SIAM J. Numer. Anal., 24 (1987), pp. 750–776.Google Scholar
[2]Babuška, I. and Suri, M., The h-p version of the finite element method with quasiuniform meshes, RAIRO Math. Model Numer. Anal., 21 (1987), pp. 199–238.CrossRefGoogle Scholar
[3]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.Google Scholar
[4]Chen, H. and Chen, Y., A combined mixed finite element and discontinuous galerkin method for compressible miscible displacement problem, Natur. Sci. J. Xiangtan Univ., 26(2) (2004), pp. 119–126.Google Scholar
[5]Chen, Y. and Tang, Y., Numerical methods for constrained elliptic optimal control problems with rapidly oscillating coefficients, East Asian J. Appl. Math., 1 (2011), pp. 235–247.CrossRefGoogle Scholar
[6]Cui, M. R., A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media, J. Comput. Appl. Math., 198(1) (2007), pp. 19–34.Google Scholar
[7]Cui, M. R., Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem, J. Comput. Appl. Math., 214(2) (2008), pp. 617–636.CrossRefGoogle Scholar
[8]Douglas, JR. J. and Roberts, J. E., Numerical methods for a model for compressible miscible displacement in porous media, Math. Comput., 41 (1983), pp. 441–459.Google Scholar
[9]Oden, J. T., Babuška, I. and Baumann, C. E., A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., 146 (1998), pp. 491–516.CrossRefGoogle Scholar
[10]Romkes, A., Prudhomme, S. and Oden, J. T., A priori error analysis of a stabilized discontinuous Galerkin method, Comput. Math. Appl., 46 (2003), pp. 1289–1311.CrossRefGoogle Scholar
[11]Raviart, R. A. and Thomas, J. M., A mixed finite element method for second order elliptic problems, Mathematics Aspects of the Finite Element Method, Lecture notes in Mathematics, Vol. 606, Springer, New York, (1977), pp. 292–315.Google Scholar
[12]Rivière, B., Wheeler, M. F. and Girault, V., A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal., 39 (2001), pp. 902–931.Google Scholar
[13]Rivière, B. and Wheeler, M. F., Discontinuous Galerkin methods for coupled flow and transport problems, Commun. Numer. Methods Eng., 18 (2002), pp. 63–68.Google Scholar
[14]Song, P. and Sun, S., Contaminant flow and transport simulation in cracked porous media using locally conservative schemes, Adv. Appl. Math. Mech., 4 (2012), pp. 389–421.Google Scholar
[15]Sun, S., Discontinuous Galerkin Methods for Reactive Transport in Porous Media, PhD. thesis, The university of Texas at Austin, 2003.Google Scholar
[16]Sun, S., Rivière, B. and Wheeler, M. F., A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media, Recent Progress in Computational and Applied PDEs, Kluwer Academic Publishers, Plenum Press, Dordrecht, New York, (2002), pp. 323–351.Google Scholar
[17]Sun, S. and Wheeler, M. F., L2(H 1) norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems, J. Sci. Comput., 22 (2005), pp. 511–540.Google Scholar
[18]Wheeler, M. F., Sun, S., Eslinger, O. and Rivière, B., Discontinuous Galerkin method for modeling flow and reactive transport in porous media, In Wendland, W., editor, Analysis and Simulation of Multified Problem, Springer Verlag, (2003), pp. 37–58.Google Scholar
[19]Yang, J. and Chen, Y., A unified a posteriori error analysis for discontinuous Galerkin approximations of reactive transport equations, J. Comput. Math., 24(3) (2006), pp. 425–434.Google Scholar
[20]Yang, J. and Chen, Y., A priori error analysis of a discontinuous Galerkin approximation for a kind of compressible miscible displacement problems, Sci. China Math., 53(10) (2010), pp. 2679–2696.Google Scholar
[21]Yang, J. and Chen, Y., A priori error estimates of a combined mixed finite element and discontinuous Galerkin method for compressible miscible displacement with molecular diffusion and dispersion, J. Comput. Math., 29(1) (2011), pp. 93–110.Google Scholar
[22]Yang, J., A posteriori error of a discontinuous Galerkin scheme for compressible miscible displacement problems with molecular diffusion and dispersion, Int. J. Numer. Meth. Fluids, 65(7) (2011), pp. 781–797.Google Scholar