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Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations

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Abstract

We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection–diffusion equations. Since the discretization error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective field, and the geometric properties such as directed edges and the area of triangles. Based on this observation, the shape, size and equidistribution requirements are used to derive corresponding metric tensor and stabilization parameters. Numerical results are provided to validate the stability and efficiency of the proposed numerical strategy.

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Acknowledgements

Yana Di is supported in part by the National Natural Science Foundation of China (91630208, 11771437). Hehu Xie is supported in part by the National Natural Science Foundation of China (91730302, 11771434, 91330202, 11371026, 11001259, 11031006, 2011CB309703) and Science Challenge Project (TZ2016002). Xiaobo Yin is supported by National Natural Science Foundation of China (11671165, 91630201), Program for Changjiang Scholars and Innovative Research Team in University \(\sharp \) IRT13066, and self-determined research funds of Central China Normal University (CCNU16A02039). The authors would like to thank Professor Lutz Tobiska for his valuable suggestion to this work. We are also thankful to anonymous reviewers for their remarks and suggestions.

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Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Yana Di & Hehu Xie

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yana Di & Hehu Xie

  3. Hubei Key Laboratory of Mathematical Sciences, School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Xiaobo Yin

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  1. Yana Di
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Correspondence to Xiaobo Yin.

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Yana Di and Hehu Xie are supported in part by the National Center for Mathematics and Interdisciplinary Science, CAS.

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Di, Y., Xie, H. & Yin, X. Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations. J Sci Comput 76, 48–68 (2018). https://doi.org/10.1007/s10915-017-0610-9

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