A LeastSquaresBased Weak Galerkin Finite Element Method for Second Order Elliptic Equations
Here, in this article, we introduce a leastsquaresbased 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 leastsquares 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 leastsquaresbased weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reactiondiffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.
 Authors:

^{[1]}
;
^{[2]};
^{[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:
 Grant/Contract Number:
 AC0500OR22725
 Type:
 Accepted Manuscript
 Journal Name:
 SIAM Journal on Scientific Computing
 Additional Journal Information:
 Journal Volume: 39; Journal Issue: 4; Journal ID: ISSN 10648275
 Publisher:
 SIAM
 Research Org:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; weak Galerkin; nite element methods; leastsquares nite element methods; second order elliptic problems
 OSTI Identifier:
 1394348
Mu, Lin, Wang, Junping, and Ye, Xiu. A LeastSquaresBased Weak Galerkin Finite Element Method for Second Order Elliptic Equations. United States: N. p.,
Web. doi:10.1137/16M1083244.
Mu, Lin, Wang, Junping, & Ye, Xiu. A LeastSquaresBased Weak Galerkin Finite Element Method for Second Order Elliptic Equations. United States. doi:10.1137/16M1083244.
Mu, Lin, Wang, Junping, and Ye, Xiu. 2017.
"A LeastSquaresBased Weak Galerkin Finite Element Method for Second Order Elliptic Equations". United States.
doi:10.1137/16M1083244. https://www.osti.gov/servlets/purl/1394348.
@article{osti_1394348,
title = {A LeastSquaresBased 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 leastsquaresbased 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 leastsquares 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 leastsquaresbased weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reactiondiffusion 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 = {2017},
month = {8}
}