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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 ›› 2016, Vol. 33 ›› Issue (3): 309-318.doi: 10.3969/j.issn.1005-3085.2016.03.009

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反应扩散方程的奇异摄动问题的最小二乘的有限元方法(英)

邱常新,  赵维加,  宋丽娜   

  1. 青岛大学数学与统计学院，青岛 266071
  • 收稿日期:2015-03-19 接受日期:2015-09-10 出版日期:2016-06-15 发布日期:2016-08-15
  • 基金资助:
    国家自然科学基金 (11401332; 11072120).

A Balanced Finite Element Method of Least-squares Formulation for Singularly Perturbed Reaction-diffusion Problems

QIU Chang-xin,  ZHAO Wei-jia,  SONG Li-na   

  1. School of Mathematics and Statistics, Qingdao University, Qingdao 266071
  • Received:2015-03-19 Accepted:2015-09-10 Online:2016-06-15 Published:2016-08-15
  • Supported by:
    The National Natural Science Foundation of China (11401332; 11072120).

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摘要： 对于一种类型的反应扩散方程的奇异摄动问题，当利用标准能量范数通过有限元方法进行误差计算时，该范数表现为一个弱范数．因为奇异摄动问题的参数使该范数的每个部分有不同的收敛阶，所以范数不稳定．这篇文章中我们引进一个新的强范数，并利用这个新范数在一维空间下构建奇异摄动反应扩散问题的最小二乘有限元方法(LSFEM)及误差估计，通过数值算例对理论结果进行验证．

关键词: 奇异摄动反应扩散问题, 最小二乘法, 误差估计, 有限元法

Abstract:

For a kind of the singularly perturbed reaction-diffusion problem, the standard energy norm is too weak to measure adequately the errors of solutions computed by finite element methods. The multiplier of this problem gives an unbalanced norm whose different components have different orders of convergence. In the paper, we introduce a new stronger norm, construct the least-squares finite element method (LSFEM) in this new norm and develop a robust and stable numerical approach for more general singularly perturbed reaction-diffusion problems in 1D spaces. At last, numerical examples are presented to illustrate the proposed method and theoretical results.

Key words: singularly perturbed reaction-diffusion problem, least-squares methods, error estimates, finite element method

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