A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

Mu, Lin; Wang, Junping; Ye, Xiu

doi:10.1137/16M1083244

Title: A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

Abstract

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for both the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.

Authors:

^[1]; Wang, Junping ^[2]; Ye, Xiu ^[3]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
National Science Foundation, Arlington, VA (United States). Division of Mathematical Sciences
University of Arkansas at Little Rock, Little Rock, AR (United States). Department of Mathematics

Publication Date:: Thu Aug 17 00:00:00 EDT 2017

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:: 1394348

Grant/Contract Number:: AC05-00OR22725

Resource Type:: Accepted Manuscript

Journal Name:: SIAM Journal on Scientific Computing

Additional Journal Information:: Journal Volume: 39; Journal Issue: 4; Journal ID: ISSN 1064-8275

Publisher:: SIAM

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; weak Galerkin; nite element methods; least-squares nite element methods; second order elliptic problems

Citation Formats


                    Mu, Lin, Wang, Junping, and Ye, Xiu. A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations.  United States: N. p., 2017. 
Web.  doi:10.1137/16M1083244.

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                    Mu, Lin, Wang, Junping, & Ye, Xiu. A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations.  United States.  https://doi.org/10.1137/16M1083244

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                    Mu, Lin, Wang, Junping, and Ye, Xiu. Thu .  
"A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations".  United States.  https://doi.org/10.1137/16M1083244.  https://www.osti.gov/servlets/purl/1394348.

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@article{osti_1394348,

  title        = {A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations},

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

  abstractNote = {Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for both the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.},

  doi          = {10.1137/16M1083244},

  journal      = {SIAM Journal on Scientific Computing},

  number       = 4,

  volume       = 39,

  place        = {United States},

  year         = {Thu Aug 17 00:00:00 EDT 2017},

  month        = {Thu Aug 17 00:00:00 EDT 2017}

}

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Works referenced in this record:

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
journal, January 2002

Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo
SIAM Journal on Numerical Analysis, Vol. 39, Issue 5
DOI: 10.1137/S0036142901384162

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

Accuracy of least-squares methods for the Navier-Stokes equations
journal, July 1993

Bochev, Pavel B.; Gunzburger, Max D.
Computers & Fluids, Vol. 22, Issue 4-5
DOI: 10.1016/0045-7930(93)90025-5

Analysis of least squares finite element methods for the Stokes equations
journal, January 1994

Bochev, Pavel B.; Gunzburger, Max D.
Mathematics of Computation, Vol. 63, Issue 208
DOI: 10.1090/S0025-5718-1994-1257573-4

A locally conservative, discontinuous least-squares finite element method for the Stokes equations
journal, February 2011

Bochev, Pavel; Lai, James; Olson, Luke
International Journal for Numerical Methods in Fluids, Vol. 68, Issue 6
DOI: 10.1002/fld.2536

A least-squares approach based on a discrete minus one inner product for first order systems
journal, July 1997

Bramble, James H.; Lazarov, Raytcho D.; Pasciak, Joseph E.
Mathematics of Computation, Vol. 66, Issue 219
DOI: 10.1090/S0025-5718-97-00848-X

A least-squares finite element approximation for the compressible Stokes equations
journal, January 2000

Cai, Zhiqiang; Ye, Xiu
Numerical Methods for Partial Differential Equations, Vol. 16, Issue 1
DOI: 10.1002/(SICI)1098-2426(200001)16:1<62::AID-NUM5>3.0.CO;2-G

Least-squares finite element approximations for the Reissner-Mindlin plate
journal, September 1999

Cai, Zhiqiang; Ye, Xiu; Zhang, Huilong
Numerical Linear Algebra with Applications, Vol. 6, Issue 6
DOI: 10.1002/(SICI)1099-1506(199909)6:6<479::AID-NLA172>3.0.CO;2-K

First-Order System Least Squares for Second-Order Partial Differential Equations: Part I
journal, December 1994

Cai, Z.; Lazarov, R.; Manteuffel, T. A.
SIAM Journal on Numerical Analysis, Vol. 31, Issue 6
DOI: 10.1137/0731091

First-Order System Least Squares for Second-Order Partial Differential Equations: Part II
journal, April 1997

Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F.
SIAM Journal on Numerical Analysis, Vol. 34, Issue 2
DOI: 10.1137/S0036142994266066

An error estimate of the least squares finite element method for the Stokes problem in three dimensions
journal, September 1994

Chang, Ching Lung
Mathematics of Computation, Vol. 63, Issue 207
DOI: 10.1090/S0025-5718-1994-1234425-7

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

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
journal, December 1998

Cockburn, Bernardo; Shu, Chi-Wang
SIAM Journal on Numerical Analysis, Vol. 35, Issue 6
DOI: 10.1137/S0036142997316712

Analysis of the DPG Method for the Poisson Equation
journal, January 2011

Demkowicz, L.; Gopalakrishnan, J.
SIAM Journal on Numerical Analysis, Vol. 49, Issue 5
DOI: 10.1137/100809799

A primal DPG method without a first-order reformulation
journal, October 2013

Demkowicz, L.; Gopalakrishnan, J.
Computers & Mathematics with Applications, Vol. 66, Issue 6
DOI: 10.1016/j.camwa.2013.06.029

A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support
journal, November 2008

Edwards, Michael G.; Zheng, Hongwen
Journal of Computational Physics, Vol. 227, Issue 22
DOI: 10.1016/j.jcp.2008.05.028

Optimal least-squares finite element method for elliptic problems
journal, January 1993

Jiang, Bo-Nan; Povinelli, Louis A.
Computer Methods in Applied Mechanics and Engineering, Vol. 102, Issue 2
DOI: 10.1016/0045-7825(93)90108-A

Least-squares finite element method for fluid dynamics
journal, July 1990

Jiang, Bo-Nan; Povinelli, Louis A.
Computer Methods in Applied Mechanics and Engineering, Vol. 81, Issue 1
DOI: 10.1016/0045-7825(90)90139-D

Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity
journal, November 2014

Lin, Guang; Liu, Jiangguo; Mu, Lin
Journal of Computational Physics, Vol. 276
DOI: 10.1016/j.jcp.2014.07.001

Discontinuous Discretization for Least-Squares Formulation of Singularly Perturbed Reaction-Diffusion Problems in One and Two Dimensions
journal, January 2009

Lin, Runchang
SIAM Journal on Numerical Analysis, Vol. 47, Issue 1
DOI: 10.1137/070700267

Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities
journal, January 2009

Lin, Runchang
Numerische Mathematik, Vol. 112, Issue 2
DOI: 10.1007/s00211-008-0208-0

First‐Order System Least Squares for Geometrically Nonlinear Elasticity
journal, January 2006

Manteuffel, T. A.; McCormick, S. F.; Schmidt, J. G.
SIAM Journal on Numerical Analysis, Vol. 44, Issue 5
DOI: 10.1137/050628027

A weak Galerkin finite element method with polynomial reduction
journal, September 2015

Mu, Lin; Wang, Junping; Ye, Xiu
Journal of Computational and Applied Mathematics, Vol. 285
DOI: 10.1016/j.cam.2015.02.001

Least-Squares Mixed Finite Elements for Second-Order Elliptic Problems
journal, October 1994

Pehlivanov, A. I.; Carey, G. F.; Lazarov, R. D.
SIAM Journal on Numerical Analysis, Vol. 31, Issue 5
DOI: 10.1137/0731071

A hybrid high-order locking-free method for linear elasticity on general meshes
journal, January 2015

Di Pietro, Daniele A.; Ern, Alexandre
Computer Methods in Applied Mechanics and Engineering, Vol. 283
DOI: 10.1016/j.cma.2014.09.009

Hybrid high-order methods for variable-diffusion problems on general meshes
journal, January 2015

Di Pietro, Daniele A.; Ern, Alexandre
Comptes Rendus Mathematique, Vol. 353, Issue 1
DOI: 10.1016/j.crma.2014.10.013

Layer-Adapted Grids for Singular Perturbation Problems
journal, May 1998

Roos, Hans-Görg
ZAMM, Vol. 78, Issue 5
DOI: 10.1002/(SICI)1521-4001(199805)78:5<291::AID-ZAMM291>3.0.CO;2-R

A note on the conditioning of upwind schemes on Shishkin meshes
journal, January 1996

Roos, Hans-GäRg
IMA Journal of Numerical Analysis, Vol. 16, Issue 4
DOI: 10.1093/imanum/16.4.529

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

Works referencing / citing this record:

A least-squares-based weak Galerkin finite element method for elliptic interface problems
journal, July 2019

Deka, Bhupen; Roy, Papri
Proceedings - Mathematical Sciences, Vol. 129, Issue 5
DOI: 10.1007/s12044-019-0518-4

A discontinuous least-squares finite-element method for second-order elliptic equations
journal, March 2018

Ye, Xiu; Zhang, Shangyou
International Journal of Computer Mathematics, Vol. 96, Issue 3
DOI: 10.1080/00207160.2018.1445230

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Title: A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

Abstract

Citation Formats

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems journal, January 2002

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

Accuracy of least-squares methods for the Navier-Stokes equations journal, July 1993

Analysis of least squares finite element methods for the Stokes equations journal, January 1994

A locally conservative, discontinuous least-squares finite element method for the Stokes equations journal, February 2011

A least-squares approach based on a discrete minus one inner product for first order systems journal, July 1997

A least-squares finite element approximation for the compressible Stokes equations journal, January 2000

Least-squares finite element approximations for the Reissner-Mindlin plate journal, September 1999

First-Order System Least Squares for Second-Order Partial Differential Equations: Part I journal, December 1994

First-Order System Least Squares for Second-Order Partial Differential Equations: Part II journal, April 1997

An error estimate of the least squares finite element method for the Stokes problem in three dimensions journal, September 1994

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

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems journal, December 1998

Analysis of the DPG Method for the Poisson Equation journal, January 2011

A primal DPG method without a first-order reformulation journal, October 2013

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Optimal least-squares finite element method for elliptic problems journal, January 1993

Least-squares finite element method for fluid dynamics journal, July 1990

Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity journal, November 2014

Discontinuous Discretization for Least-Squares Formulation of Singularly Perturbed Reaction-Diffusion Problems in One and Two Dimensions journal, January 2009

Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities journal, January 2009

First‐Order System Least Squares for Geometrically Nonlinear Elasticity journal, January 2006

A weak Galerkin finite element method with polynomial reduction journal, September 2015

Least-Squares Mixed Finite Elements for Second-Order Elliptic Problems journal, October 1994

A hybrid high-order locking-free method for linear elasticity on general meshes journal, January 2015

Hybrid high-order methods for variable-diffusion problems on general meshes journal, January 2015

Layer-Adapted Grids for Singular Perturbation Problems journal, May 1998

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

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

A least-squares-based weak Galerkin finite element method for elliptic interface problems journal, July 2019

A discontinuous least-squares finite-element method for second-order elliptic equations journal, March 2018

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
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Convergence analysis of the high-order mimetic finite difference method
journal, May 2009

Accuracy of least-squares methods for the Navier-Stokes equations
journal, July 1993

Analysis of least squares finite element methods for the Stokes equations
journal, January 1994

A locally conservative, discontinuous least-squares finite element method for the Stokes equations
journal, February 2011

A least-squares approach based on a discrete minus one inner product for first order systems
journal, July 1997

A least-squares finite element approximation for the compressible Stokes equations
journal, January 2000

Least-squares finite element approximations for the Reissner-Mindlin plate
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First-Order System Least Squares for Second-Order Partial Differential Equations: Part I
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First-Order System Least Squares for Second-Order Partial Differential Equations: Part II
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An error estimate of the least squares finite element method for the Stokes problem in three dimensions
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Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
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The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
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Analysis of the DPG Method for the Poisson Equation
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A primal DPG method without a first-order reformulation
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A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support
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