基于超收敛集群恢复的Cahn-Hilliard方程自适应有限元法
The SCR-Based Adaptive Finite ElementMethod for the Cahn-Hilliard Equation
DOI: 10.12677/AAM.2022.1111884, PDF, HTML, 下载: 161  浏览: 280  国家自然科学基金支持
作者: 田文艳, 贾宏恩*：太原理工大学数学学院，山西 太原；陈尧尧：安徽师范大学数学与统计学院，安徽 芜湖；孟朝霞：山西能源学院能源与动力工程系，山西 晋中
关键词: 误差估计；Cahn-Hilliard方程；自适应；SCR；有限元法；Error Estimate； The Cahn-Hilliard Equation； Adaptive； SCR； Finite Element Method
摘要: Cahn-Hilliard方程为四阶非线性的偏微分方程，在物理，生物，化学等各个领域都有广泛的应用，因此研究其数值方法具有实际的应用价值。本文通过分析Cahn-Hilliard方程的一种二阶数值 格式，证明了其误差估计和无条件能量稳定性，并且提出了一个基于后验误差估计的空间和时间 自适应策略，即超收敛集群恢复（superconvergent cluster recovery，简称为SCR）方法，用于数值求解Cahn-Hilliard方程，该策略的主要思想是基于误差估计的结果来控制网格大小，从而可以有效的降低计算成本，最后通过算例证明了SCR 算法的高效性和稳定性。
Abstract: The Cahn-Hilliard equation is a fourth-order nonlinear partial differential equation with a wide range of applications in various fields such as physics, biology, and chem- istry, so it is of practical application to study its numerical methods. In this study, we analyzed the Cahn-Hilliard equation in a second-order numerical format, demon- strated its error estimate and unconditional energy stability, and suggested a spatial and temporal adaptive strategy based on the posterior error estimate, namely the superconvergent cluster recovery (SCR) method, for numerical solutions.
文章引用：田文艳, 陈尧尧, 孟朝霞, 贾宏恩. 基于超收敛集群恢复的Cahn-Hilliard方程自适应有限元法[J]. 应用数学进展, 2022, 11(11): 8355-8367. https://doi.org/10.12677/AAM.2022.1111884

参考文献

[1] Cahn, J. and Hilliard, J. (1958) Free Energy of a Nonuniform System. I. Interfacial Free Energy. The Journal of Chemical Physics, 28, 258-267.
https://doi.org/10.1063/1.1744102
[2] Karma, A. and Rappel, W.J. (1998) Quantitative Phase-Field Modeling of Dendritic Growth in Two and Three Dimensions. Physical Review E, 57, 4323-4349.
https://doi.org/10.1103/PhysRevE.57.4323
[3] Allen, S. and Cahn, J. (1979) A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening. Acta Materialia, 27, 1085-1095.
https://doi.org/10.1016/0001-6160(79)90196-2
[4] Kobayashi, R. (1993) Modeling and Numerical Simulations of Dendritic Crystal Growth. Phys- ica D: Nonlinear Phenomena, 63, 410-423.
https://doi.org/10.1016/0167-2789(93)90120-P
[5] Gurtin, M., Polignone, D. and Vinals, J. (1996) Two-Phase Binary Fluids and Immiscible Flu- ids Described by an Order Parameter. Mathematical Models and Methods in Applied Sciences, 6, 815-831.
https://doi.org/10.1142/S0218202596000341
[6] Barret, J., Blowey, J. and Garcke, H. (1999) Finite Element Approximation of the Cahn- Hilliard Equation with Degenerate Mobility. SIAM Journal on Numerical Analysis, 37, 286- 318.
https://doi.org/10.1137/S0036142997331669
[7] Elliott, C. and French, D. (1989) A Nonconforming Finite-Element Method for the Two- Dimensional Cahn-Hilliard Equation. SIAM Journal on Numerical Analysis, 26, 884-903.
https://doi.org/10.1137/0726049
[8] Elliott, C. and French, D. (1987) Numerical Studies of the Cahn-Hilliard Equation for Phase Separation. IMA Journal of Applied Mathematics, 38, 97-128.
https://doi.org/10.1093/imamat/38.2.97
[9] Shen, J., Xu, J. and Yang, J. (2018) The Scalar Auxiliary Variable (SAV) Approach for Gradient Flows. Journal of Computational Physics, 353, 407-416.
https://doi.org/10.1016/j.jcp.2017.10.021
[10] Zhao, S., Xiao, X. and Feng, X. (2020) An Efficient Time Adaptivity Based on Chemical Potential for Surface Cahn-Hilliard Equation Using Finite Element Approximation. Applied Mathematics and Computation, 369, Article ID: 124901.
https://doi.org/10.1016/j.amc.2019.124901
[11] Mao, D., Shen, L. and Zhou, A. (2006) Adaptive Finite Element Algorithms for Eigenvalue Problems Based on Local Averaging Type a Posteriori Error Estimates. Advances in Compu- tational Mathematics, 25, 135-160.
https://doi.org/10.1007/s10444-004-7617-0
[12] Chen, Y., Huang, Y. and Yi, N. (2021) A Decoupled Energy Stable Adaptive Finite Ele- ment Method for Cahn-Hilliard-Navier-Stokes Equations. Communications in Computational Physics, 29, 1186-1212.
https://doi.org/10.4208/cicp.OA-2020-0032
[13] Zhang, Z. and Qiao, Z. (2012) An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation. Communications in Computational Physics, 11, 1261-1278.
https://doi.org/10.4208/cicp.300810.140411s
[14] Huang, Y. and Yi, N. (2010) The Superconvergent Cluster Recovery Method. Journal of Sci- entific Computing, 44, 301-322.
https://doi.org/10.1007/s10915-010-9379-9
[15] Chen, Y., Huang, Y. and Yi, N. (2019) A SCR-Based Error Estimation and Adaptive Finite Element Method for the Allen-Cahn Equation. Computers and Mathematics with Applications, 78, 204-223.
https://doi.org/10.1016/j.camwa.2019.02.022
[16] Guillen-Gonzalez, F. and Tierra, G. (2014) Second Order Schemes and Time-Step Adaptivity for Allen-Cahn and Cahn-Hilliard Models. Computers and Mathematics with Applications, 68, 821-846.
https://doi.org/10.1016/j.camwa.2014.07.014
[17] Feng, X. and Prohl, A. (2004) Error Analysis of a Mixed Finite Element Method for the Cahn- Hilliard Equation. Numerische Mathematik, 99, 47-84.
https://doi.org/10.1007/s00211-004-0546-5
[18] Shen, J. and Yang, X. (2010) Numerical Approximations of Allen-Cahn and Cahn-Hilliard Equations. Discrete and Continuous Dynamical Systems, 28, 1669-1691.
https://doi.org/10.3934/dcds.2010.28.1669
[19] Li, C., Huang, Y. and Yi, N. (2019) An Unconditionally Energy Stable Second Order Finite Element Method for Solving the Allen-Cahn Equation. Journal of Computational and Applied Mathematics, 353, 38-48.
https://doi.org/10.1016/j.cam.2018.12.024

为你推荐