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An adaptive discontinuous finite volume element method for the Allen-Cahn equation

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Abstract

In this paper, the discontinuous finite volume element method (DFVEM) is considered to solve the Allen-Cahn equation which contains strong nonlinearity. The method is based on the DFVEM in space and the backward Euler method in time. The energy stability and unique solvability of the proposed fully discrete scheme are derived. The error estimates for the semi-discrete and fully discrete scheme are also established. A series of numerical experiments verify the efficiency of the proposed numerical method. The results show that our method can not only accurately capture the dynamic information of the phase transition, but also ensure the stability of the system during long-term numerical simulations.

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Funding

Jian Li is financially supported in part by NSF of China (No. 11771259)Shaanxi Provincial Joint Laboratory of Artificial Intelligence (No. 2022JC-SYS-05), Innovative team project of Shaanxi Provincial Department of Education (No. 21JP013) and Shaanxi Province Natural Science basic research program key project (No. 2023-JC-ZD-02). Rui Li is financially supported by the National Natural Science Foundation of China (No. 11901372), the Young Talent fund of University Association for Science and Technology in Shaanxi (No. 20200504), the Fundamental Research Fund for the Central Universities of China (No. GK202103004), and Sinopec Key Laboratory of Geophysics.

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Authors and Affiliations

  1. School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi’an, 710021, People’s Republic of China

    Jian Li & Jiyao Zeng

  2. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, People’s Republic of China

    Rui Li

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  2. Jiyao Zeng
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  3. Rui Li
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Correspondence to Rui Li.

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Communicated by: Long Chen

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Li, J., Zeng, J. & Li, R. An adaptive discontinuous finite volume element method for the Allen-Cahn equation. Adv Comput Math 49, 55 (2023). https://doi.org/10.1007/s10444-023-10031-5

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