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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 ›› 2020, Vol. 37 ›› Issue (4): 478-486.doi: 10.3969/j.issn.1005-3085.2020.04.008

• • 上一篇    下一篇

四阶Cahn-Hilliard方程的间断有限元方法

邹乐强1,   刘丽杰2,   韦雷雷2   

  1. 1- 河南工业和信息化职业学院，焦作  454000
    2- 河南工业大学理学院，郑州  450000
  • 收稿日期:2019-04-09 接受日期:2019-09-26 出版日期:2020-08-15 发布日期:2020-10-15
  • 基金资助:
    国家自然科学基金(11426090; 11461072)；河南省高等学校重点科研项目(19A110005).

A Discontinuous Galerkin Finite Element Method for the Fourth-order Cahn-Hilliard Equation

ZOU Le-qiang1,   LIU Li-jie2,   WEI Lei-lei2   

  1. 1- Henan College of Industry and Information Technology, Jiaozuo 454000 
    2- College of Science, Henan University of Technology, Zhengzhou 450000
  • Received:2019-04-09 Accepted:2019-09-26 Online:2020-08-15 Published:2020-10-15
  • Supported by:
    The National Natural Science Foundation of China (11426090; 11461072); the Foundation of He'nan Educational Committee (19A110005).

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摘要： Cahn-Hilliard方程是一类非常重要的四阶扩散方程，具有深刻的物理背景和丰富的理论内涵，对其设计高精度的数值格式具有重要的工程实践价值和科学意义．在本文中我们对四阶Cahn-Hilliard方程设计一种高精度的间断有限元，该方法不同于传统的局部间断有限元方法，不需要引进另外的辅助变量或将方程转化为一阶方程组，能够显著降低计算量和存储量．通过选取合适的数值流通量，我们证明了方法的稳定性和收敛性．数值实验结果表明该方法求解Cahn-Hilliard方程是收敛的和有效的．

关键词: Cahn-Hilliard方程, 间断有限元方法, 稳定性, 误差估计

Abstract: The Cahn-Hilliard equation which is a very important fourth-order diffusion model has profound physical background and rich theoretical connotation, and its numerical methods are of important scientific significance and engineering application value. In this paper, we develop and analyze the Discontinuous Galerkin (DG) finite element method for the fourth-order Cahn-Hilliard equation in one dimension. The method, which is different from the traditional local discontinuous Galerkin (LDG) method, can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system and reduce the amount of computation and storage. We prove stability and convergence by choosing interface numerical fluxes carefully. Some numerical tests is provided to illustrate the accuracy and capability of the scheme.

Key words: Cahn-Hilliard equation, discontinuous Galerkin method, stability, error estimates

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