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Supercloseness and postprocessing for linear finite element method on Bakhvalov-type meshes

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Abstract

Supercloseness and postprocessing of the linear finite element method are studied on the Bakhvalov-type mesh for a singularly perturbed convection diffusion problem. Finite element analysis on this kind of mesh has always been an open problem. The difficulties arise from the width \(\mathcal {O}(\varepsilon \ln (1/\varepsilon ) )\) of subdomain for the layer and nonuniformity of meshes in the layer. A novel interpolation is introduced to address difficulties from the width of subdomain for the layer. As a result, supercloseness of order two is obtained for the linear finite element method. Based on this supercloseness result, we propose and analyze a new postprocessing operator according to the mesh’s structure. Its stability is proved by means of numerical quadrature. Then, it is proved that the numerical solution after postprocessing converges second order. Numerical experiments verify these theoretical results.

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Funding

This research is supported by the National Natural Science Foundation of China (11771257), Shandong Provincial Natural Science Foundation, China (ZR2021MA004).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Jin Zhang

  2. School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

    Xiaowei Liu

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  1. Jin Zhang
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Correspondence to Jin Zhang.

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Zhang, J., Liu, X. Supercloseness and postprocessing for linear finite element method on Bakhvalov-type meshes. Numer Algor 92, 1553–1570 (2023). https://doi.org/10.1007/s11075-022-01353-4

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Mathematics Subject Classification (2010)