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

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.

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

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

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1394348

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

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 15 works

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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

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