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A New Weak Galerkin Finite Element Scheme for the Brinkman Model

Part of:Partial differential equations, boundary value problemsPhase 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)

Abstract

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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 methods65N15: Error bounds65N12: Stability and convergence of numerical methods74N20: Dynamics of phase boundaries

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016, pp. 1409 - 1434
Copyright
Copyright © Global-Science Press 2016 

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