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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 ›› 2022, Vol. 39 ›› Issue (5): 775-796.doi: 10.3969/j.issn.1005-3085.2022.05.008

• • 上一篇    下一篇

H (curl)空间中椭圆最优控制问题的自适应有限元算法

贺之龙,   赵建平,    杨   欢,    李   兵,    席梦茹   

  1. 新疆大学数学与系统科学学院，乌鲁木齐  830017
  • 出版日期:2022-10-15 发布日期:2022-12-15
  • 通讯作者: 赵建平 E-mail: zhaojianping@126.com
  • 基金资助:
    新疆维吾尔自治区自然科学基金(2019D01C047)；国家自然科学基金(61962056).

The Adaptive Finite Element Method for Elliptic Optimal Control Problem in H (curl) Space

HE Zhilong,   ZHAO Jianping,    YANG Huan,   LI Bing,   XI Mengru   

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017
  • Online:2022-10-15 Published:2022-12-15
  • Contact: J. Zhao. E-mail address: zhaojianping@126.com
  • Supported by:
    The Natural Science Foundation of Xinjiang Uygur Autonomous Region (2019D01C 047); the National Natural Science Foundation of China (61962056).

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摘要：

针对H (curl)空间椭圆型最优控制问题提出了一种自适应有限元方法。首先将H (curl)空间Maxwell方程的最优控制模型转化为偏微分方程组，给出了偏微分方程组的解得正则性。其次利用自适应有限元方法求解此偏微分方程组，同时讨论了方法的后验误差及收敛性。最后通过数值算例给出了该方法的数值结果，验证了有限元方法的有效性和可靠性。这一方法可以应用于更复杂的最优控制问题的求解。

关键词: 椭圆最优控制问题, 后验误差估计, 收敛性

Abstract:

In this paper, we present and analyze the adaptive finite element method for elliptic optimal control problem in H (curl) space. Firstly, based on Maxwell equation, we propose the elliptical optimal control problem in H (curl) space and establish an optimal control system. The optimal control model is equivalently transformed into the PDE systems and present the regularity property of the OCM. Then we apply the AFEM to solve the system. For the finite element approximation, we introduce the limitations of the posterior error estimator of the residual type and prove the convergence. The numerical examples are presented to verify theoretical results and indicate that the AFEM is more reliable and effective under the same conditions. Finally, our theoretical analysis and numerical algorithms can be promoted and applied to more complex problems.

Key words: elliptic optimal control problem, posteriori error estimate, adaptive convergence

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