A hybridized formulation for the weak Galerkin mixed finite element method

Mu, Lin; Wang, Junping; Ye, Xiu

doi:10.1016/j.cam.2016.01.004

Title: A hybridized formulation for the weak Galerkin mixed finite element method

Abstract

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising from the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.

Authors:

Mu, Lin; Wang, Junping;

Publication Date:: Thu Dec 01 00:00:00 EST 2016

Research Org.:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1769965

Alternate Identifier(s):: OSTI ID: 1323958; OSTI ID: 1338538

Grant/Contract Number:: ERKJE45; AC05-00OR22725

Resource Type:: Published Article

Journal Name:: Journal of Computational and Applied Mathematics

Additional Journal Information:: Journal Name: Journal of Computational and Applied Mathematics Journal Volume: 307 Journal Issue: C; Journal ID: ISSN 0377-0427

Publisher:: Elsevier

Country of Publication:: Belgium

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Weak Galerkin; Finite element methods; Discrete weak divergence; Second-order elliptic problems; Hybridized mixed finite element methods

Citation Formats


                    Mu, Lin, Wang, Junping, and Ye, Xiu. A hybridized formulation for the weak Galerkin mixed finite element method.  Belgium: N. p., 2016. 
Web.  doi:10.1016/j.cam.2016.01.004.

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                    Mu, Lin, Wang, Junping, & Ye, Xiu. A hybridized formulation for the weak Galerkin mixed finite element method.  Belgium.  https://doi.org/10.1016/j.cam.2016.01.004

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                    Mu, Lin, Wang, Junping, and Ye, Xiu. Thu .  
"A hybridized formulation for the weak Galerkin mixed finite element method".  Belgium.  https://doi.org/10.1016/j.cam.2016.01.004.

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@article{osti_1769965,

  title        = {A hybridized formulation for the weak Galerkin mixed finite element method},

  author       = {Mu, Lin and Wang, Junping and Ye, Xiu},

  abstractNote = {This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising from the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.},

  doi          = {10.1016/j.cam.2016.01.004},

  journal      = {Journal of Computational and Applied Mathematics},

  number       = C,

  volume       = 307,

  place        = {Belgium},

  year         = {Thu Dec 01 00:00:00 EST 2016},

  month        = {Thu Dec 01 00:00:00 EST 2016}

}

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Works referenced in this record:

Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates
journal, January 1985

Arnold, D. N.; Brezzi, F.
ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 19, Issue 1
DOI: 10.1051/m2an/1985190100071

Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes
journal, January 2011

da Veiga, L. Beirão; Lipnikov, K.; Manzini, G.
SIAM Journal on Numerical Analysis, Vol. 49, Issue 5
DOI: 10.1137/100807764

Convergence analysis of the high-order mimetic finite difference method
journal, May 2009

Beirão da Veiga, L.; Lipnikov, K.; Manzini, G.
Numerische Mathematik, Vol. 113, Issue 3
DOI: 10.1007/s00211-009-0234-6

Weak Galerkin finite element methods for elliptic PDEs
journal, July 2015

Wang, ChunMei; Wang, JunPing
SCIENTIA SINICA Mathematica, Vol. 45, Issue 7
DOI: 10.1360/N012014-00233

Mixed finite elements for second order elliptic problems in three variables
journal, March 1987

Brezzi, Franco; Douglas, Jim; Durán, Ricardo
Numerische Mathematik, Vol. 51, Issue 2
DOI: 10.1007/BF01396752

A weak Galerkin finite element method for second-order elliptic problems
journal, March 2013

Wang, Junping; Ye, Xiu
Journal of Computational and Applied Mathematics, Vol. 241
DOI: 10.1016/j.cam.2012.10.003

A weak Galerkin mixed finite element method for second order elliptic problems
journal, May 2014

Wang, Junping; Ye, Xiu
Mathematics of Computation, Vol. 83, Issue 289
DOI: 10.1090/S0025-5718-2014-02852-4

Basic Principles of Virtual Element Methods
journal, November 2012

BeirÃO Da Veiga, L.; Brezzi, F.; Cangiani, A.
Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 01
DOI: 10.1142/S0218202512500492

The finite element method with Lagrangian multipliers
journal, June 1973

Babuška, Ivo
Numerische Mathematik, Vol. 20, Issue 3
DOI: 10.1007/BF01436561

An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes
journal, December 2014

Wang, Chunmei; Wang, Junping
Computers & Mathematics with Applications, Vol. 68, Issue 12
DOI: 10.1016/j.camwa.2014.03.021

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
journal, January 2009

Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho
SIAM Journal on Numerical Analysis, Vol. 47, Issue 2
DOI: 10.1137/070706616

Virtual Elements for Linear Elasticity Problems
journal, January 2013

da Veiga, L. Beira͂o; Brezzi, F.; Marini, L. D.
SIAM Journal on Numerical Analysis, Vol. 51, Issue 2
DOI: 10.1137/120874746

Two families of mixed finite elements for second order elliptic problems
journal, June 1985

Brezzi, Franco; Douglas, Jim; Marini, L. D.
Numerische Mathematik, Vol. 47, Issue 2
DOI: 10.1007/BF01389710

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Similar Records

Title: A hybridized formulation for the weak Galerkin mixed finite element method

Abstract

Citation Formats

Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates journal, January 1985

Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes journal, January 2011

Convergence analysis of the high-order mimetic finite difference method journal, May 2009

Weak Galerkin finite element methods for elliptic PDEs journal, July 2015

Mixed finite elements for second order elliptic problems in three variables journal, March 1987

A weak Galerkin finite element method for second-order elliptic problems journal, March 2013

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Basic Principles of Virtual Element Methods journal, November 2012

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An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes journal, December 2014

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems journal, January 2009

Virtual Elements for Linear Elasticity Problems journal, January 2013

Two families of mixed finite elements for second order elliptic problems journal, June 1985

Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates
journal, January 1985

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journal, January 2011

Convergence analysis of the high-order mimetic finite difference method
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Weak Galerkin finite element methods for elliptic PDEs
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Mixed finite elements for second order elliptic problems in three variables
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A weak Galerkin finite element method for second-order elliptic problems
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Basic Principles of Virtual Element Methods
journal, November 2012

The finite element method with Lagrangian multipliers
journal, June 1973

An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes
journal, December 2014

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
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Two families of mixed finite elements for second order elliptic problems
journal, June 1985