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,    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

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