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Volume 22, Issue 3
A Conforming Discontinuous Galerkin Finite Element Method for Second-Order Parabolic Equation

Fuchang Huo, Yuchen Sun, Jiageng Wu & Liyuan Zhang

DOI: 10.4208/ijnam2025-1019

Int. J. Numer. Anal. Mod., 22 (2025), pp. 432-458.

Published online: 2025-03

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  • Abstract

The conforming discontinuous Galerkin (CDG) finite element method is an innovative and effective numerical approach to solve partial differential equations. The CDG method is based on the weak Galerkin (WG) finite element method, and removes the stabilizer in the numerical scheme. And the CDG method uses the average of the interior function to replace the value of the boundary function in the standard WG method. The integration by parts is used to construct the discrete weak gradient operator in the CDG method. This paper uses the CDG method to solve the parabolic equation. Firstly, the semi-discrete and full-discrete numerical schemes of the parabolic equation and the well-posedness of the numerical methods are presented. Then, the corresponding error equations for both numerical schemes are established, and the optimal order error estimates of $H^1$ and $L^2$ are provided, respectively. Finally, the numerical results of the CDG method are verified.

  • Keywords

Conforming discontinuous Galerkin finite element method, parabolic equation, weak Galerkin finite element method, optimal order convergence.

  • AMS Subject Headings

65N30, 65N15, 65N12, 35K20, 35J50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-432, author = {Huo , FuchangSun , YuchenWu , Jiageng and Zhang , Liyuan}, title = {A Conforming Discontinuous Galerkin Finite Element Method for Second-Order Parabolic Equation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {3}, pages = {432--458}, abstract = {

The conforming discontinuous Galerkin (CDG) finite element method is an innovative and effective numerical approach to solve partial differential equations. The CDG method is based on the weak Galerkin (WG) finite element method, and removes the stabilizer in the numerical scheme. And the CDG method uses the average of the interior function to replace the value of the boundary function in the standard WG method. The integration by parts is used to construct the discrete weak gradient operator in the CDG method. This paper uses the CDG method to solve the parabolic equation. Firstly, the semi-discrete and full-discrete numerical schemes of the parabolic equation and the well-posedness of the numerical methods are presented. Then, the corresponding error equations for both numerical schemes are established, and the optimal order error estimates of $H^1$ and $L^2$ are provided, respectively. Finally, the numerical results of the CDG method are verified.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1019}, url = {http://global-sci.org/intro/article_detail/ijnam/23888.html} }
TY - JOUR T1 - A Conforming Discontinuous Galerkin Finite Element Method for Second-Order Parabolic Equation AU - Huo , Fuchang AU - Sun , Yuchen AU - Wu , Jiageng AU - Zhang , Liyuan JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 432 EP - 458 PY - 2025 DA - 2025/03 SN - 22 DO - http://doi.org/10.4208/ijnam2025-1019 UR - https://global-sci.org/intro/article_detail/ijnam/23888.html KW - Conforming discontinuous Galerkin finite element method, parabolic equation, weak Galerkin finite element method, optimal order convergence. AB -

The conforming discontinuous Galerkin (CDG) finite element method is an innovative and effective numerical approach to solve partial differential equations. The CDG method is based on the weak Galerkin (WG) finite element method, and removes the stabilizer in the numerical scheme. And the CDG method uses the average of the interior function to replace the value of the boundary function in the standard WG method. The integration by parts is used to construct the discrete weak gradient operator in the CDG method. This paper uses the CDG method to solve the parabolic equation. Firstly, the semi-discrete and full-discrete numerical schemes of the parabolic equation and the well-posedness of the numerical methods are presented. Then, the corresponding error equations for both numerical schemes are established, and the optimal order error estimates of $H^1$ and $L^2$ are provided, respectively. Finally, the numerical results of the CDG method are verified.

Huo , FuchangSun , YuchenWu , Jiageng and Zhang , Liyuan. (2025). A Conforming Discontinuous Galerkin Finite Element Method for Second-Order Parabolic Equation. International Journal of Numerical Analysis and Modeling. 22 (3). 432-458. doi:10.4208/ijnam2025-1019
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