Abstract
Isogeometric analysis (IGA) has been widely used as a spatial discretization method for phase field models since the seminal work of Gómez et al. (Comput. Methods Appl. Mech. Engrg. 197(49), pp. 4333–4352, 2008), and the first numerical convergence study of IGA for the Cahn-Hilliard equation was presented by Kästner et al. (J. Comput. Phys. 305(15), pp. 360–371, 2016). However, to the best of our knowledge, the theoretical convergence analysis of IGA for the Cahn-Hilliard equation is still missing in the literature. In this paper, we provide the convergence analysis of IGA for the multi-dimensional Cahn-Hilliard equation for the first time. The two important steps to carry out the convergence analysis are (1) we rigorously prove that the \(L^{\infty }\) norm of IGA solution is uniformly bounded for all mesh sizes, and (2) we construct an appropriate Ritz projection operator for the bi-Laplacian term in the Cahn-Hilliard equation. The first- and second-order stabilized semi-implicit schemes are used to obtain the fully discrete schemes. The energy stability analyses are rigorously proved for the resulting fully discrete schemes. Finally, several two- and three-dimensional numerical examples are presented to verify the theoretical results.
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Data sets generated during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
The authors would like to thank the referees for their careful reading and for their critical questions, which helped to improve the paper.
Funding
The research of X. Meng was partially supported by the National Natural Science Foundation of China (Grant No. 12101057), the Scientific Research Fund of Beijing Normal University (Grant No. 28704-111032105), and the Guangdong Higher Education Upgrading Plan (UIC-R0400024-21). The research of Y. Qin was partially supported by the National Natural Science Foundation of China (Grant No. 12201369), and the Natural Science Foundation of Shanxi Province (Grant No. 202303021211004). G. Hu would like to thank the support from the National Natural Science Foundation of China (Grant No. 11922120), the Science and Technology Development Fund of Macao SAR (File/Project Nos. 001/2024/SKL and 0068/2024/RIA1), and the MYRG of University of Macau (Grant No. MYRG-GRG2023-00157-FST-UMDF).
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Meng, X., Qin, Y. & Hu, G. The Convergence Analysis of a Class of Stabilized Semi-Implicit Isogeometric Methods for the Cahn-Hilliard Equation. J Sci Comput 102, 26 (2025). https://doi.org/10.1007/s10915-024-02753-5
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DOI: https://doi.org/10.1007/s10915-024-02753-5
Keywords
- Isogeometric analysis
- Cahn-Hilliard equation
- Convergence analysis
- Stabilized semi-implicit scheme
- Energy stability analysis