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Adaptive discontinuous Galerkin finite element methods for the Allen-Cahn equation on polygonal meshes

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Abstract

In this paper, we develop a polygonal mesh adaptation algorithm for a fully implicit scheme based on discontinuous Galerkin (DG) finite element methods in space and backward Euler method in time to solve the Allen-Cahn equation. The mathematical framework and a procedure for solving this nonlinear equation are given. We extend the DG discretization with a polygonal mesh adaptation method to save computation time and capture the thin interfaces more accurately. A criterion based on the local value of the phase field function gradient is used to select the target element for refinement and coarsening, and then a 4-node polygonal mesh refinement strategy is adopted by connecting the midpoint of each edge to the barycenter of the target element. Using numerical tests, including motion by the mean curvature, curvature-driven flow, the Allen-Cahn equation with a logarithmic free energy, the Allen-Cahn equation with advection, and the application for image segmentation, we verify the accuracy, efficiency, and capabilities of the adaptive DG on polygonal meshes and confirm the decreasing property of the discrete energy.

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The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We would like to thank the reviewers for carefully critiquing the manuscript. With their suggestions, we were able to better convey our arguments and present our results in a clearer manner.

Funding

Rui Li is partially supported by the National Natural Science Foundation of China (Nos. 11901372, 11931013, 12171296, 12071270), the Young Talent fund of University Association for Science and Technology in Shaanxi (No. 20200504), the Natural Science Foundation of Shaanxi Province (No. 2020JQ-403), the Fundamental Research Fund for the Central Universities of China (Nos. GK202103004, GK201901008), and the Sinopec Key Laboratory of Geophysics; Yali Gao is partially supported by the National Natural Science Foundation of China (No. 11901461), China Postdoctoral Science Foundation (No. 2020M673464), and Guangdong Basic and Applied Basic Research Foundation (2023A1515010697); Zhangxin Chen is financially supported by Foundation CMG.

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  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China

    Rui Li

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, People’s Republic of China

    Yali Gao

  3. Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen, 518063, People’s Republic of China

    Yali Gao

  4. Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta T2N 1N4, Canada

    Zhangxin Chen

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Rui Li, Yali Gao, and Zhangxin Chen contributed equally to this work.

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Li, R., Gao, Y. & Chen, Z. Adaptive discontinuous Galerkin finite element methods for the Allen-Cahn equation on polygonal meshes. Numer Algor 95, 1981–2014 (2024). https://doi.org/10.1007/s11075-023-01635-5

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