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Volume 19, Issue 6
Analysis of Weak Galerkin Finite Element Methods with Supercloseness

Ahmed AL-Taweel, Saqib Hussain & Xiaoshen Wang

Int. J. Numer. Anal. Mod., 19 (2022), pp. 761-776.

Published online: 2022-09

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  • Abstract

In [15], the computational performance of various weak Galerkin finite element methods in terms of stability, convergence, and supercloseness is explored and numerical results are listed in 31 tables. Some of the phenomena can be explained by the existing theoretical results and the others are to be explained. The main purpose of this paper is to provide a unified theoretical foundation to a class of WG schemes, where $(P_k(T), P_{k+1}(e), [P_{k+1}(T)]^2)$ elements are used for solving the second order elliptic equations (1)-(2) on a triangle grid in 2D. With this unified treatment, all of the existing results become special cases. The theoretical conclusions are corroborated by a number of numerical examples.

  • Keywords

Weak Galerkin, finite element methods, weak gradient, second-order elliptic problems, supercloseness, superconvergence.

  • AMS Subject Headings

Primary: 65N15, 65N30, Secondary: 35J50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-19-761, author = {AL-Taweel , AhmedHussain , Saqib and Wang , Xiaoshen}, title = {Analysis of Weak Galerkin Finite Element Methods with Supercloseness }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2022}, volume = {19}, number = {6}, pages = {761--776}, abstract = {

In [15], the computational performance of various weak Galerkin finite element methods in terms of stability, convergence, and supercloseness is explored and numerical results are listed in 31 tables. Some of the phenomena can be explained by the existing theoretical results and the others are to be explained. The main purpose of this paper is to provide a unified theoretical foundation to a class of WG schemes, where $(P_k(T), P_{k+1}(e), [P_{k+1}(T)]^2)$ elements are used for solving the second order elliptic equations (1)-(2) on a triangle grid in 2D. With this unified treatment, all of the existing results become special cases. The theoretical conclusions are corroborated by a number of numerical examples.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/21032.html} }
TY - JOUR T1 - Analysis of Weak Galerkin Finite Element Methods with Supercloseness AU - AL-Taweel , Ahmed AU - Hussain , Saqib AU - Wang , Xiaoshen JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 761 EP - 776 PY - 2022 DA - 2022/09 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/21032.html KW - Weak Galerkin, finite element methods, weak gradient, second-order elliptic problems, supercloseness, superconvergence. AB -

In [15], the computational performance of various weak Galerkin finite element methods in terms of stability, convergence, and supercloseness is explored and numerical results are listed in 31 tables. Some of the phenomena can be explained by the existing theoretical results and the others are to be explained. The main purpose of this paper is to provide a unified theoretical foundation to a class of WG schemes, where $(P_k(T), P_{k+1}(e), [P_{k+1}(T)]^2)$ elements are used for solving the second order elliptic equations (1)-(2) on a triangle grid in 2D. With this unified treatment, all of the existing results become special cases. The theoretical conclusions are corroborated by a number of numerical examples.

AL-Taweel , AhmedHussain , Saqib and Wang , Xiaoshen. (2022). Analysis of Weak Galerkin Finite Element Methods with Supercloseness . International Journal of Numerical Analysis and Modeling. 19 (6). 761-776. doi:
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