数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (4): 1572-1593.doi: 10.1007/s10473-024-0421-7

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THE SUPERCLOSENESS OF THE FINITE ELEMENT METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM ON A BAKHVALOV-TYPE MESH IN 2D

Chunxiao Zhang, Jin Zhang*   

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
  • 收稿日期:2023-03-09 修回日期:2023-07-10 出版日期:2024-08-25 发布日期:2024-08-30

THE SUPERCLOSENESS OF THE FINITE ELEMENT METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM ON A BAKHVALOV-TYPE MESH IN 2D

Chunxiao Zhang, Jin Zhang*   

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
  • Received:2023-03-09 Revised:2023-07-10 Online:2024-08-25 Published:2024-08-30
  • Contact: *E-mail: jinzhangalex@hotmail.com
  • About author:E-mail: chunxiaozhangang@outlook.com
  • Supported by:
    J. Zhang's work was supported by National Natural Science Foundation of China (11771257) and the Shandong Provincial Natural Science Foundation of China (ZR2023YQ002, ZR2023MA007, ZR2021MA004).

摘要: For singularly perturbed convection-diffusion problems, supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes, especially in the case of 2D. The difficulties arise from the width of the mesh in the layer adjacent to the transition point, resulting in a suboptimal estimate for convergence. Existing analysis techniques cannot handle these difficulties well. To fill this gap, here a novel interpolation is designed delicately for the smooth part of the solution, bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method. Our theoretical result is uniform in the singular perturbation parameter $\varepsilon$ and is supported by the numerical experiments.

关键词: singularly perturbed, convection-diffusion, finite element method, supercloseness, Bakhvalov-type mesh

Abstract: For singularly perturbed convection-diffusion problems, supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes, especially in the case of 2D. The difficulties arise from the width of the mesh in the layer adjacent to the transition point, resulting in a suboptimal estimate for convergence. Existing analysis techniques cannot handle these difficulties well. To fill this gap, here a novel interpolation is designed delicately for the smooth part of the solution, bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method. Our theoretical result is uniform in the singular perturbation parameter $\varepsilon$ and is supported by the numerical experiments.

Key words: singularly perturbed, convection-diffusion, finite element method, supercloseness, Bakhvalov-type mesh

中图分类号: 

  • 65N12