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A Modified Interior Penalty Virtual Element Method for Fourth-Order Singular Perturbation Problems

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Abstract

This paper is dedicated to the numerical solution of a fourth-order singular perturbation problem using the interior penalty virtual element method (IPVEM). Compared with the original IPVEM proposed in Zhao et al. (Math Comp 92(342):1543–1574, 2023), the study introduces modifications to the jumps and averages in the penalty term, as well as presents a mesh-dependent selection of the penalty parameter. Drawing inspiration from the modified Morley finite element methods, we leverage the conforming interpolation technique to handle the lower part of the bilinear form in the error analysis. We establish the optimal convergence in the energy norm and provide a rigorous proof of uniform convergence concerning the perturbation parameter in the lowest-order case.

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Acknowledgements

The authors deeply appreciate the anonymous referees for their meticulous line-by-line examination and for providing valuable comments and suggestions, which greatly improved an earlier version of the paper.

Funding

Fang Feng was supported by the National Science Foundation for Young Scientists of China (No. 12401528) and also partially supported by Central Government Special Fund for Basic Scientific Research Business Expenses of Colleges and Universities, No. 30924010837. Yue Yu was partially supported by the National Science Foundation for Young Scientists of China (No. 12301561).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, People’s Republic of China

    Fang Feng

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, National Center for Applied Mathematics in Hunan, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Yue Yu

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  1. Fang Feng
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Contributions

Fang Feng and Yue Yu collaborated closely to shape the conceptualization, methodology, and writing of this research. Additionally, Yue Yu took charge of implementing the discrete method in the study. Fang Feng also checked the implementation details.

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Correspondence to Yue Yu.

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Feng, F., Yu, Y. A Modified Interior Penalty Virtual Element Method for Fourth-Order Singular Perturbation Problems. J Sci Comput 101, 21 (2024). https://doi.org/10.1007/s10915-024-02665-4

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