Analysis of a new stabilized finite element method for the reaction-convection-diffusion equations with a large reaction coefficient

Huo-Yuan Duan Wuhan University Po-Wen Hsieh National Central University Roger Choon Ee Tan National University of Singapore Suh-Yuh Yang National Central University

Numerical Analysis and Scientific Computing mathscidoc:1703.25035

Computer Methods in Applied Mechanics and Engineering, 247, 2012.11
In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction-convection-diffusion equations. The equation under consideration involves a small diffusivity $\varepsilon$ and a large reaction coefficient $\sigma$, leading to high P\'eclet number and high Damk\"{o}hler number. In addition to giving error estimates of the approximations in $L^2$ and $H^1$ norms, we explicitly establish the dependence of error bounds on the diffusivity, the $L^{\infty}$ norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh P\'eclet number and a large mesh Damk\"{o}hler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method. The results obtained are also compared with those of existing stabilization methods.
reaction-convection-diffusion equation; boundary layer; interior layer; finite element method; stabilization method
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@inproceedings{huo-yuan2012analysis,
  title={Analysis of a new stabilized finite element method for the reaction-convection-diffusion equations with a large reaction coefficient },
  author={Huo-Yuan Duan, Po-Wen Hsieh, Roger Choon Ee Tan, and Suh-Yuh Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314211101449222708},
  booktitle={Computer Methods in Applied Mechanics and Engineering},
  volume={247},
  year={2012},
}
Huo-Yuan Duan, Po-Wen Hsieh, Roger Choon Ee Tan, and Suh-Yuh Yang. Analysis of a new stabilized finite element method for the reaction-convection-diffusion equations with a large reaction coefficient . 2012. Vol. 247. In Computer Methods in Applied Mechanics and Engineering. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314211101449222708.
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