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山东大学学报（理学版） ›› 2016, Vol. 51 ›› Issue (2): 21-28.doi: 10.6040/j.issn.1671-9352.0.2015.190

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椭圆方程约束的最优边界控制问题的非重叠型区域分解迭代方法

刘文月,孙同军*   

  1. 山东大学数学学院, 山东 济南 250100
  • 收稿日期:2015-04-23 出版日期:2016-02-16 发布日期:2016-03-11
  • 通讯作者: 孙同军(1970— ),男,博士,副教授,研究方向为偏微分方程数值解法. E-mail:tjsun@sdu.edu.cn E-mail:1412635020@qq.com
  • 作者简介:刘文月(1988— ),女,硕士研究生,研究方向为偏微分方程最优控制问题的数值解法. E-mail:1412635020@qq.com
  • 基金资助:
    国家自然科学基金资助项目(11301300,11271231)

Iterative non-overlapping domain decomposition method for optimal boundary control problems governed by elliptic equations

LIU Wen-yue, SUN Tong-jun*   

  1. School of Mathematics, Shandong University, Jinan 250100, Shandong, China
  • Received:2015-04-23 Online:2016-02-16 Published:2016-03-11

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摘要： 研究了一类椭圆方程约束的最优边界控制问题的数值求解方法。为了避免运用传统数值方法所产生庞大的计算量,我们采用非重叠型区域分解迭代方法。 即:将求解区域Ω分解成若干个非重叠子区域,把上述的最优边界控制问题分解成这些子区域上的局部问题,这些局部问题间的内边界条件采用Robin条件。建立了求解这些局部问题的迭代格式,推导证明了迭代格式的收敛性。最后,给出一个数值算例,验证了迭代格式的有效性。

关键词: 非重叠型区域分解, Robin条件, 最优边界控制问题, 椭圆方程, 迭代方法

Abstract: A numerical method for solving optimal boundary control problems governed by elliptic equations is considered. In order to avoid large amounts of calculation produced by traditional numerical methods. An iterative non-overlapping domain decomposition method is established. The whole domain is divided into many non-overlapping subdomains, and the optimal boundary control problem is decomposed into local problems in these subdomains. Robin conditions are used to communicate the local problems on the interfaces between subdomains. The iterative scheme for solving these local problems is studied, and prove the convergence of the scheme is proved. Finally, a numerical example to prove the validity of the scheme is presented.

Key words: optimal boundary control problem, elliptic equation, Robin conditions, iterative method, non-overlapping domain decomposition method

中图分类号: 

  • O241.82
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