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Virtual element method for nonlinear Sobolev equation on polygonal meshes

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Abstract

In this work, the virtual element method (VEM) on convex polygonal meshes for the nonlinear Sobolev equations is developed, where the semi-discrete and fully discrete formulations are presented and analyzed. To overcome the complexity of nonlinear terms, the nonlinear coefficient is approximated by employing the orthogonal \(\varvec{L^{2}} \) projection operator, which is directly computable from the degrees of freedom. Under some assumptions about the nonlinear coefficient, the existence and uniqueness of the semi-discrete solution are analyzed. Furthermore, a priori error estimate showing optimal order of convergence with respect to the \(\varvec{H^{1}}\) semi-norm was derived. Finally, some numerical experiments are conducted to illustrate the theoretical convergence rate.

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The authors confirm that the data supporting the findings of this study are available within the manuscript. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

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Acknowledgements

The authors wish to thank the anonymous referees for their remarks that contributed to improve the presentation.

Funding

This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (41974133), Hunan Provincial Innovation Foundation for Postgraduate (XDCX2022Y065, XDCX2021B098), and Postgraduate Scientific Research Innovation Project of Hunan Province (CX20220639, CX20210597).

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Author notes

  1. Yanping Chen, Qiling Gu, and Yunqing Huang contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Wanxiang Liu, Qiling Gu & Yunqing Huang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guandong, People’s Republic of China

    Yanping Chen

Authors
  1. Wanxiang Liu
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  2. Yanping Chen
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  3. Qiling Gu
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  4. Yunqing Huang
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Contributions

The authors confirm contribution to the manuscript as follows: Wanxiang Liu: designing research plans, conducting numerical experiments and wrote the main manuscript text; Yanping Chen and Yunqing Huang: feasibility analysis of research scheme and revision of paper; Qiling Gu: collated documents and participated in writing papers. All authors reviewed the results and approved the final version of the manuscript.

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Correspondence to Yanping Chen.

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Liu, W., Chen, Y., Gu, Q. et al. Virtual element method for nonlinear Sobolev equation on polygonal meshes. Numer Algor 94, 1731–1761 (2023). https://doi.org/10.1007/s11075-023-01553-6

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