Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T15:28:02.357Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Weak Galerkin Finite Element Scheme for the Brinkman Model

Part of: Partial differential equations, boundary value problems Phase transformations in solids

Published online by Cambridge University Press:  17 May 2016

Qilong Zhai ,
Ran Zhang  and
Lin Mu
Qilong Zhai*
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, P.R. China
Ran Zhang*
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, P.R. China
Lin Mu*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing MI48824, United States
*
*Corresponding author. Email addresses:diql15@mails.jlu.edu.cn (Q. Zhai), zhangran@jlu.edu.cn (R. Zhang), linmu@math.msu.edu (L. Mu)
*Corresponding author. Email addresses:diql15@mails.jlu.edu.cn (Q. Zhai), zhangran@jlu.edu.cn (R. Zhang), linmu@math.msu.edu (L. Mu)
*Corresponding author. Email addresses:diql15@mails.jlu.edu.cn (Q. Zhai), zhangran@jlu.edu.cn (R. Zhang), linmu@math.msu.edu (L. Mu)
Get access

Abstract

The Brinkman model describes flow of fluid in complex porous media with a high-contrast permeability coefficient such that the flow is dominated by Darcy in some regions and by Stokes in others. A weak Galerkin (WG) finite element method for solving the Brinkman equations in two or three dimensional spaces by using polynomials is developed and analyzed. The WG method is designed by using the generalized functions and their weak derivatives which are defined as generalized distributions. The variational form we considered in this paper is based on two gradient operators which is different from the usual gradient-divergence operators for Brinkman equations. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Some computational results are presented to demonstrate the robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman equations.

Keywords

Weak Galerkin finite element methodsweak gradientBrinkman equationpolyhedral meshes

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N15: Error bounds 65N12: Stability and convergence of numerical methods 74N20: Dynamics of phase boundaries
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1409 - 1434
Copyright
Copyright © Global-Science Press 2016 

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]Brezzi, F., On the existence, uniqueness, and approximation of saddle point problems arising from Lagrange multipliers, RAIRO, 8 (1974), 129–151.Google Scholar
[2]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Elements, Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
[3]Brinkman, H.C., A calculation of the viscous force exerted by a flowing fluid in a dense swarm of particles, Appl. Sci. Res. A1, (1949), 27–34.CrossRefGoogle Scholar
[4]Burman, E. and Hansbo, P., Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem, Numer. Methods Partial Differential Equations, 21 (2005), 986–997.CrossRefGoogle Scholar
[5]Burman, E. and Hansbo, P., A unified stabilized method for Stokes' and Darcy's equations, J. Comput. Appl. Math., 198 (2007), 35–51.CrossRefGoogle Scholar
[6]Correa, M. R. and Loula, A. F. D., A unified mixed formulation naturally coupling Stokes and Darcy flows, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 2710–2722.CrossRefGoogle Scholar
[7]Crouzeix, M. and Raviart, P. A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numer., 7 (1973), 33–76.Google Scholar
[8]Girault, V. and Raviart, P.A., Finite Element Methods for the Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, 1986.CrossRefGoogle Scholar
[9]Guzmán, J. and Neilan, M., A family of nonconforming elements for the Brinkman problem, IMA Journal of Numerical Analysis, 32 (2012), 1484–1508.Google Scholar
[10]Iliev, O., Lazarov, R. and Willems, J., Variational multiscale finite element method for flows in highly porous media, Multiscale Modeling & Simulation, 9 (2011), 1350–1372.CrossRefGoogle Scholar
[11]Ligaarden, I.S., Krotkiewski, M., Lie, K.-A., Pal, M. and Schmid, D.W., On the Stokes-Brinkman equations for modeling flow in carbonate reservoirs. in: Proceedings of the ECMOR XII-12th European Conference on the Mathematics of Oil Recovery, 6-9 September 2010, Oxford, UK.CrossRefGoogle Scholar
[12]Mardal, K. A., Tai, X. C. and Winther, R., A Robust Finite Element Method for Darcy - Stokes Flow, SIAM J. Numer. Anal., 40 (2002), 1605–1631.CrossRefGoogle Scholar
[13]Mu, L., Wang, J. and Ye, X.. Weak Galerkin finite element methods on polytopal meshes, Numer. Methods Partial Differential Equations, 30 (2014), 1003–1029.CrossRefGoogle Scholar
[14]Mu, L., Wang, J. and Ye, X., A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods, J. of Comput. Phys., 273 (2014), 327–342.CrossRefGoogle Scholar
[15]Mu, L., Wang, J., Ye, X. and Zhang, S., Weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput. 59 (2014), 473–495.CrossRefGoogle Scholar
[16]Popov, P., Qin, G., Bi, L., Efendiev, Y., Ewing, R., Z. Kang, and J. Li, , Multiscale methods for modeling fluid flow through naturally fractured carbonate karst reservoirs, in: Proceedings of the SPE Annual Technical Conference and Exhibition, 2007.CrossRefGoogle Scholar
[17]Wang, J. and Ye, X., A weak Galerkin finite element method for second-order elliptic problems, J. Comp. Appl. Math., 241 (2013), 103–115.CrossRefGoogle Scholar
[18]Wang, J. and Ye, X., A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., 83(2014), 2101–2126.CrossRefGoogle Scholar
[19]Wang, J.and Ye, X., A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155–174.CrossRefGoogle Scholar
[20]Willems, J., Numerical upscaling for multiscale flow problems, Ph.D Thesis, 2009.Google Scholar
[21]Vafai, K., Porous Media: Applications in Biological Systems and Biotechnology, CRC Press, USA, 2011.Google Scholar
[22]Wehrspohn, R.B., Ordered Porous Nanostructures and Applications. Springer Science + Business Media, New York, 2005.CrossRefGoogle Scholar
[23]Xie, X., Xu, J. and Xue, G., Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models, J. Comput. Math., 26 (2008), 437–455.Google Scholar
[24]Zhai, Q., Zhang, R. and Wang, X., A hybridized weak Galerkin finite element scheme for the Stokes equations, Sci. China. Math., 58 (2015), 2455–2472.CrossRefGoogle Scholar
[25]Zhang, R. and Zhai, Q.. A new weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J. Sci. Comput., 64 (2015), 559–585.CrossRefGoogle Scholar