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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2021, Vol. 38 ›› Issue (4): 553-563.doi: 10.3969/j.issn.1005-3085.2021.04.009

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Cahn-Hilliard方程的时间双层网格有限元方法

王旦霞,   贾宏恩,   李亚倩   

  1. 太原理工大学数学学院，太原  030024
  • 收稿日期:2019-03-25 接受日期:2019-09-19 出版日期:2021-08-15 发布日期:2021-10-15
  • 基金资助:
    国家自然科学基金 (11872264)；山西省自然科学基金 (201801D121016).

Two Time-mesh Finite Element Method for Cahn-Hilliard Equation

WANG Dan-xia,   JIA Hong-en,   LI Ya-qian   

  1. College of Mathematics, Taiyuan University of Technology, Taiyuan 030024
  • Received:2019-03-25 Accepted:2019-09-19 Online:2021-08-15 Published:2021-10-15
  • Supported by:
    The National Natural Science Foundation of China (11872264); the Natural Science Foundation of Shanxi Province (201801D121016).

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摘要： 针对用非线性数值格式求解Cahn-Hilliard方程时由非线性迭代引起的耗时问题，本文提出了一种时间双层网格（TT-M）有限元（FE）方法．该方法分为两步：第一步，在粗的时间步长上求解非线性Cahn-Hilliard系统，其中空间离散采用有限元方法，时间离散采用Crank-Nicolson格式；第二步，在细的时间步长上求解线性系统，然后证明了该方法的稳定性和误差估计，并通过数值算例对理论部分进行验证．结果表明，与传统的Galerkin有限元方法相比，该方法可以节省计算时间，说明了该方法的有效性和可行性．

关键词: Cahn-Hilliard方程, TT-M FE方法, 稳定性, 误差估计, CPU时间

Abstract: A time two-mesh (TT-M) finite element (FE) method is proposed for solving the Cahn-Hilliard equation in a nonlinear numerical scheme. The method is carried out in two steps. A nonlinear Cahn-Hilliard system is solved on time coarse mesh at the first step, where the finite element method is used for spatial discretisation, and the Crank-Nicolson scheme is used for time discretisation. The second step is that a linear problem is solved on time fine mesh. Finally, the stability analysis and error estimates of the proposed method is given. Numerical examples are given to confirm the theoretical analysis. The results show that the method of this paper can save computation time compared with the traditional Galerkin finite element method. The validity and feasibility of the proposed method are illustrated.

Key words: Cahn-Hilliard equation, TT-M FE method, stability, error estimate, CPU time

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