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Volume 41, Issue 6
A Modified Weak Galerkin Finite Element Method for Singularly Perturbed Parabolic Convection-Diffusion-Reaction Problems

Suayip Toprakseven & Fuzheng Gao

DOI: 10.4208/jcm.2203-m2021-0031

J. Comp. Math., 41 (2023), pp. 1246-1280.

Published online: 2023-11

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  • Abstract

In this work, a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations. The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators, respectively. We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh. The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods. Optimal order of convergences are obtained in suitable norms. We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method. Various numerical examples are presented to support the theoretical results. It is theoretically and numerically shown that the method is quite stable.

  • Keywords

The modified weak Galerkin finite element method, Backward Euler method, Parabolic convection-diffusion problems, Error estimates.

  • AMS Subject Headings

65N15, 65N30, 35J50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-41-1246, author = {Toprakseven , Suayip and Gao , Fuzheng}, title = {A Modified Weak Galerkin Finite Element Method for Singularly Perturbed Parabolic Convection-Diffusion-Reaction Problems}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {6}, pages = {1246--1280}, abstract = {

In this work, a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations. The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators, respectively. We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh. The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods. Optimal order of convergences are obtained in suitable norms. We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method. Various numerical examples are presented to support the theoretical results. It is theoretically and numerically shown that the method is quite stable.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2203-m2021-0031}, url = {http://global-sci.org/intro/article_detail/jcm/22111.html} }
TY - JOUR T1 - A Modified Weak Galerkin Finite Element Method for Singularly Perturbed Parabolic Convection-Diffusion-Reaction Problems AU - Toprakseven , Suayip AU - Gao , Fuzheng JO - Journal of Computational Mathematics VL - 6 SP - 1246 EP - 1280 PY - 2023 DA - 2023/11 SN - 41 DO - http://doi.org/10.4208/jcm.2203-m2021-0031 UR - https://global-sci.org/intro/article_detail/jcm/22111.html KW - The modified weak Galerkin finite element method, Backward Euler method, Parabolic convection-diffusion problems, Error estimates. AB -

In this work, a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations. The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators, respectively. We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh. The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods. Optimal order of convergences are obtained in suitable norms. We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method. Various numerical examples are presented to support the theoretical results. It is theoretically and numerically shown that the method is quite stable.

Suayip Toprakseven & Fuzheng Gao. (2023). A Modified Weak Galerkin Finite Element Method for Singularly Perturbed Parabolic Convection-Diffusion-Reaction Problems. Journal of Computational Mathematics. 41 (6). 1246-1280. doi:10.4208/jcm.2203-m2021-0031
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