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Two-grid Raviart-Thomas mixed finite element methods combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations

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Abstract

In this paper, we discuss a priori error estimates of two-grid mixed finite element methods for a class of nonlinear parabolic equations. The lowest order Raviart-Thomas mixed finite element and Crank-Nicolson scheme are used for the spatial and temporal discretization. First, we derive the optimal a priori error estimates for all variables. Second, we present a two-grid scheme and analyze its convergence. It is shown that if the two mesh sizes satisfy h = H2, then the two-grid method achieves the same convergence property as the Raviart-Thomas mixed finite element method. Finally, we give a numerical example to verify the theoretical results.

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Funding

Luoping Chen is financially supported by the Natural Science Foundation of China (11501473) and the Fundamental Research Funds of the Central Universities of China (2682016CX108). Tianliang Hou and Yueting Yang are financially supported by the Innovation Talent Training Program of Science and Technology of Jilin Province of China (20180519011JH), Science and Technology Research Project of Jilin Provincial Department of Education (JJKH20190634KJ), and Beihua University Youth Research and Innovation Team Development Project. Yin Yang is financially supported by the National Natural Science Foundation of China (11671342, 11771369, and 11931003), Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2018JJ2374, 2018WK4006, and 2019YZ3003), and Key Project of Hunan Provincial Department of Education (17A210).

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, Sichuan, China

    Luoping Chen

  2. School of Mathematics and Statistics, Beihua University, Jilin, 132013, Jilin, China

    Tianliang Hou & Yueting Yang

  3. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China

    Yin Yang

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  1. Tianliang Hou
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  2. Luoping Chen
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  3. Yueting Yang
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  4. Yin Yang
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Correspondence to Luoping Chen.

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Communicated by: Carlos Garcia-Cervera

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Hou, T., Chen, L., Yang, Y. et al. Two-grid Raviart-Thomas mixed finite element methods combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations. Adv Comput Math 46, 24 (2020). https://doi.org/10.1007/s10444-020-09777-z

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  • DOI: https://doi.org/10.1007/s10444-020-09777-z

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