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Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations

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Abstract

In this paper, we present a second-order accurate Crank-Nicolson scheme for the two-grid finite element methods of the nonlinear Sobolev equations. This method involves solving a small nonlinear system on a coarse mesh with mesh size H and a linear system on a fine mesh with mesh size h, which can still maintain the asymptotically optimal accuracy compared with the standard finite element method. However, the two-grid scheme can reduce workload and save a lot of CPU time. The optimal error estimates in H1-norm show that the two-grid methods can achieve optimal convergence order when the mesh sizes satisfy h = O(H2). These estimates are shown to be uniform in time. Numerical results are provided to verify the theoretical estimates.

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Acknowledgments

The authors thank the referees for valuable constructive comments and suggestions, which led to a significant improvement of this paper.

Funding

The work is supported by the National Natural Science Foundation of China (Grant No. 11771375, 11671157, 11571297, 91430213), Shandong Province Natural Science Foundation(Grant No. ZR2018MAQ008), and China Postdoctoral Science Foundation funded project (Grant No. 2017M610501).

Author information

Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Chuanjun Chen & Kang Li

  2. School of Data and Computer Science, Sun Yat-sen University, Guangzhou, 510006, China

    Kang Li

  3. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Yanping Chen

  4. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    Yunqing Huang

Authors
  1. Chuanjun Chen
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  2. Kang Li
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  3. Yanping Chen
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Corresponding author

Correspondence to Yanping Chen.

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Communicated by: Carlos Garcia-Cervera

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Chen, C., Li, K., Chen, Y. et al. Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations. Adv Comput Math 45, 611–630 (2019). https://doi.org/10.1007/s10444-018-9628-2

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  • DOI: https://doi.org/10.1007/s10444-018-9628-2

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