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Volume 26, Issue 1
The Weak Galerkin Method for Elliptic Eigenvalue Problems

Qilong Zhai, Hehu Xie, Ran Zhang & Zhimin Zhang

DOI: 10.4208/cicp.OA-2018-0201

Commun. Comput. Phys., 26 (2019), pp. 160-191.

Published online: 2019-02

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  • Abstract

This article is devoted to studying the application of the weak Galerkin (WG) finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds. The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions. The non-conforming finite element space of the WG method is the key of the lower bound property. It also makes the WG method more robust and flexible in solving eigenvalue problems. We demonstrate that the WG method can achieve arbitrary high convergence order. This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements. Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.

  • Keywords

Weak Galerkin finite element method, elliptic eigenvalue problem, lower bounds, error estimate.

  • AMS Subject Headings

65N30, 65N15, 65N12, 74N20, 35B45, 35J50, 35J35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-26-160, author = {Qilong Zhai, Hehu Xie, Ran Zhang and Zhimin Zhang}, title = {The Weak Galerkin Method for Elliptic Eigenvalue Problems}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {1}, pages = {160--191}, abstract = {

This article is devoted to studying the application of the weak Galerkin (WG) finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds. The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions. The non-conforming finite element space of the WG method is the key of the lower bound property. It also makes the WG method more robust and flexible in solving eigenvalue problems. We demonstrate that the WG method can achieve arbitrary high convergence order. This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements. Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0201}, url = {http://global-sci.org/intro/article_detail/cicp/13030.html} }
TY - JOUR T1 - The Weak Galerkin Method for Elliptic Eigenvalue Problems AU - Qilong Zhai, Hehu Xie, Ran Zhang & Zhimin Zhang JO - Communications in Computational Physics VL - 1 SP - 160 EP - 191 PY - 2019 DA - 2019/02 SN - 26 DO - http://doi.org/10.4208/cicp.OA-2018-0201 UR - https://global-sci.org/intro/article_detail/cicp/13030.html KW - Weak Galerkin finite element method, elliptic eigenvalue problem, lower bounds, error estimate. AB -

This article is devoted to studying the application of the weak Galerkin (WG) finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds. The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions. The non-conforming finite element space of the WG method is the key of the lower bound property. It also makes the WG method more robust and flexible in solving eigenvalue problems. We demonstrate that the WG method can achieve arbitrary high convergence order. This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements. Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.

Qilong Zhai, Hehu Xie, Ran Zhang and Zhimin Zhang. (2019). The Weak Galerkin Method for Elliptic Eigenvalue Problems. Communications in Computational Physics. 26 (1). 160-191. doi:10.4208/cicp.OA-2018-0201
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