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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:
ORCiD logo [1];  [2];  [3]
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
  2. National Science Foundation, Arlington, VA (United States). Division of Mathematical Sciences
  3. University of Arkansas at Little Rock, Little Rock, AR (United States). Department of Mathematics
Publication Date:
Research Org.:
Oak Ridge National Lab. (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. https://doi.org/10.1137/16M1083244.
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
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.
@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 = {2017},
month = {8}
}

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    journal, July 2019


    A discontinuous least-squares finite-element method for second-order elliptic equations
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