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

Mu, Lin; Wang, Junping; Ye, Xiu

doi:10.1137/16M1083244

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

Journal Article · Thu Aug 17 00:00:00 EDT 2017 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/16M1083244· OSTI ID:1394348

^[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

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.

View Accepted Manuscript (DOE)

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

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1394348

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 4 Vol. 39; ISSN 1064-8275

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

References (31)

A least-squares finite element approximation for the compressible Stokes equations Cai, Zhiqiang; Ye, Xiu Numerical Methods for Partial Differential Equations, Vol. 16, Issue 1 https://doi.org/10.1002/(SICI)1098-2426(200001)16:1<62::AID-NUM5>3.0.CO;2-G	journal	January 2000
Least-squares finite element approximations for the Reissner-Mindlin plate Cai, Zhiqiang; Ye, Xiu; Zhang, Huilong Numerical Linear Algebra with Applications, Vol. 6, Issue 6 https://doi.org/10.1002/(SICI)1099-1506(199909)6:6<479::AID-NLA172>3.0.CO;2-K	journal	September 1999
Layer-Adapted Grids for Singular Perturbation Problems Roos, Hans-Görg ZAMM, Vol. 78, Issue 5 https://doi.org/10.1002/(SICI)1521-4001(199805)78:5<291::AID-ZAMM291>3.0.CO;2-R	journal	May 1998
A locally conservative, discontinuous least-squares finite element method for the Stokes equations Bochev, Pavel; Lai, James; Olson, Luke International Journal for Numerical Methods in Fluids, Vol. 68, Issue 6 https://doi.org/10.1002/fld.2536	journal	February 2011
Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities Lin, Runchang Numerische Mathematik, Vol. 112, Issue 2 https://doi.org/10.1007/s00211-008-0208-0	journal	January 2009
Convergence analysis of the high-order mimetic finite difference method Beirão da Veiga, L.; Lipnikov, K.; Manzini, G. Numerische Mathematik, Vol. 113, Issue 3 https://doi.org/10.1007/s00211-009-0234-6	journal	May 2009
Least-squares finite element method for fluid dynamics Jiang, Bo-Nan; Povinelli, Louis A. Computer Methods in Applied Mechanics and Engineering, Vol. 81, Issue 1 https://doi.org/10.1016/0045-7825(90)90139-D	journal	July 1990
Optimal least-squares finite element method for elliptic problems Jiang, Bo-Nan; Povinelli, Louis A. Computer Methods in Applied Mechanics and Engineering, Vol. 102, Issue 2 https://doi.org/10.1016/0045-7825(93)90108-A	journal	January 1993
Accuracy of least-squares methods for the Navier-Stokes equations Bochev, Pavel B.; Gunzburger, Max D. Computers & Fluids, Vol. 22, Issue 4-5 https://doi.org/10.1016/0045-7930(93)90025-5	journal	July 1993
A weak Galerkin finite element method for second-order elliptic problems Wang, Junping; Ye, Xiu Journal of Computational and Applied Mathematics, Vol. 241 https://doi.org/10.1016/j.cam.2012.10.003	journal	March 2013
A weak Galerkin finite element method with polynomial reduction Mu, Lin; Wang, Junping; Ye, Xiu Journal of Computational and Applied Mathematics, Vol. 285 https://doi.org/10.1016/j.cam.2015.02.001	journal	September 2015
A primal DPG method without a first-order reformulation Demkowicz, L.; Gopalakrishnan, J. Computers & Mathematics with Applications, Vol. 66, Issue 6 https://doi.org/10.1016/j.camwa.2013.06.029	journal	October 2013
A hybrid high-order locking-free method for linear elasticity on general meshes Di Pietro, Daniele A.; Ern, Alexandre Computer Methods in Applied Mechanics and Engineering, Vol. 283 https://doi.org/10.1016/j.cma.2014.09.009	journal	January 2015
Hybrid high-order methods for variable-diffusion problems on general meshes Di Pietro, Daniele A.; Ern, Alexandre Comptes Rendus Mathematique, Vol. 353, Issue 1 https://doi.org/10.1016/j.crma.2014.10.013	journal	January 2015
A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support Edwards, Michael G.; Zheng, Hongwen Journal of Computational Physics, Vol. 227, Issue 22 https://doi.org/10.1016/j.jcp.2008.05.028	journal	November 2008
Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity Lin, Guang; Liu, Jiangguo; Mu, Lin Journal of Computational Physics, Vol. 276 https://doi.org/10.1016/j.jcp.2014.07.001	journal	November 2014
An error estimate of the least squares finite element method for the Stokes problem in three dimensions Chang, Ching Lung Mathematics of Computation, Vol. 63, Issue 207 https://doi.org/10.1090/S0025-5718-1994-1234425-7	journal	September 1994
Analysis of least squares finite element methods for the Stokes equations Bochev, Pavel B.; Gunzburger, Max D. Mathematics of Computation, Vol. 63, Issue 208 https://doi.org/10.1090/S0025-5718-1994-1257573-4	journal	January 1994
A weak Galerkin mixed finite element method for second order elliptic problems Wang, Junping; Ye, Xiu Mathematics of Computation, Vol. 83, Issue 289 https://doi.org/10.1090/S0025-5718-2014-02852-4	journal	May 2014
A least-squares approach based on a discrete minus one inner product for first order systems Bramble, James H.; Lazarov, Raytcho D.; Pasciak, Joseph E. Mathematics of Computation, Vol. 66, Issue 219 https://doi.org/10.1090/S0025-5718-97-00848-X	journal	July 1997
A note on the conditioning of upwind schemes on Shishkin meshes Roos, Hans-GäRg IMA Journal of Numerical Analysis, Vol. 16, Issue 4 https://doi.org/10.1093/imanum/16.4.529	journal	January 1996
First‐Order System Least Squares for Geometrically Nonlinear Elasticity Manteuffel, T. A.; McCormick, S. F.; Schmidt, J. G. SIAM Journal on Numerical Analysis, Vol. 44, Issue 5 https://doi.org/10.1137/050628027	journal	January 2006
Discontinuous Discretization for Least-Squares Formulation of Singularly Perturbed Reaction-Diffusion Problems in One and Two Dimensions Lin, Runchang SIAM Journal on Numerical Analysis, Vol. 47, Issue 1 https://doi.org/10.1137/070700267	journal	January 2009
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho SIAM Journal on Numerical Analysis, Vol. 47, Issue 2 https://doi.org/10.1137/070706616	journal	January 2009
Least-Squares Mixed Finite Elements for Second-Order Elliptic Problems Pehlivanov, A. I.; Carey, G. F.; Lazarov, R. D. SIAM Journal on Numerical Analysis, Vol. 31, Issue 5 https://doi.org/10.1137/0731071	journal	October 1994
First-Order System Least Squares for Second-Order Partial Differential Equations: Part I Cai, Z.; Lazarov, R.; Manteuffel, T. A. SIAM Journal on Numerical Analysis, Vol. 31, Issue 6 https://doi.org/10.1137/0731091	journal	December 1994
Analysis of the DPG Method for the Poisson Equation Demkowicz, L.; Gopalakrishnan, J. SIAM Journal on Numerical Analysis, Vol. 49, Issue 5 https://doi.org/10.1137/100809799	journal	January 2011
A Nonconforming High-Order Method for the Biot Problem on General Meshes Boffi, Daniele; Botti, Michele; Di Pietro, Daniele A. SIAM Journal on Scientific Computing, Vol. 38, Issue 3 https://doi.org/10.1137/15M1025505	journal	January 2016
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo SIAM Journal on Numerical Analysis, Vol. 39, Issue 5 https://doi.org/10.1137/S0036142901384162	journal	January 2002
First-Order System Least Squares for Second-Order Partial Differential Equations: Part II Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F. SIAM Journal on Numerical Analysis, Vol. 34, Issue 2 https://doi.org/10.1137/S0036142994266066	journal	April 1997
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems Cockburn, Bernardo; Shu, Chi-Wang SIAM Journal on Numerical Analysis, Vol. 35, Issue 6 https://doi.org/10.1137/S0036142997316712	journal	December 1998

Cited By (2)

A least-squares-based weak Galerkin finite element method for elliptic interface problems Deka, Bhupen; Roy, Papri Proceedings - Mathematical Sciences, Vol. 129, Issue 5 https://doi.org/10.1007/s12044-019-0518-4	journal	July 2019
A discontinuous least-squares finite-element method for second-order elliptic equations Ye, Xiu; Zhang, Shangyou International Journal of Computer Mathematics, Vol. 96, Issue 3 https://doi.org/10.1080/00207160.2018.1445230	journal	March 2018

Similar Records

Multilevel first-order system least squares for PDEs

Conference · Fri Dec 30 23:00:00 EST 1994 · OSTI ID:219586

A comparison of Galerkin and least squares finite element methods for nonlinear heat conduction in laminated composites

Book · Tue Oct 01 00:00:00 EDT 1996 · OSTI ID:376014

A discrete divergence free weak Galerkin finite element method for the Stokes equations

Journal Article · Thu Nov 30 23:00:00 EST 2017 · Applied Numerical Mathematics · OSTI ID:1415198

Related Subjects

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

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

Citation Formats

References (31)

Cited By (2)

Similar Records

Related Subjects