Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 1-40.doi: 10.1007/s42967-024-00397-8

• •    下一篇

Adaptive Finite Element Method for an Elliptic Optimal Control Problem with Integral State Constraints

Pratibha Shakya, Kamana Porwal   

  1. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, 110016, India
  • 收稿日期:2023-10-02 修回日期:2024-02-02 出版日期:2026-02-20 发布日期:2026-02-11
  • 通讯作者: Pratibha Shakya,E-mail:shakya.pratibha10@gmail.com E-mail:shakya.pratibha10@gmail.com
  • 作者简介:Kamana Porwal,E-mail:kamana@maths.iitd.ac.in
  • 基金资助:
    The first author would like to thank the Indian Institute of Technology Delhi and the Department of Science and Technology (DST) for providing financial assistance under (PDF/2021/000444), New Delhi, India. The second author wants to thank the Council of Scientific and Industrial Research under the CSIR Extramural Research Grant (Grant No. 25(0297)/19/EMRII).

Adaptive Finite Element Method for an Elliptic Optimal Control Problem with Integral State Constraints

Pratibha Shakya, Kamana Porwal   

  1. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, 110016, India
  • Received:2023-10-02 Revised:2024-02-02 Online:2026-02-20 Published:2026-02-11
  • Contact: Pratibha Shakya,E-mail:shakya.pratibha10@gmail.com E-mail:shakya.pratibha10@gmail.com
  • Supported by:
    The first author would like to thank the Indian Institute of Technology Delhi and the Department of Science and Technology (DST) for providing financial assistance under (PDF/2021/000444), New Delhi, India. The second author wants to thank the Council of Scientific and Industrial Research under the CSIR Extramural Research Grant (Grant No. 25(0297)/19/EMRII).

摘要: In this article, we develop a posteriori error analysis of a nonconforming finite element method for a linear quadratic elliptic distributed optimal control problem with two different sets of constraints, namely (i) integral state constraint and integral control constraint; (ii) integral state constraint and pointwise control constraints. In the analysis, we have taken the approach of reducing the state-control constrained minimization problem into a state minimization problem obtained by eliminating the control variable. The reliability and efficiency of a posteriori error estimator are discussed. Numerical results are reported to illustrate the behavior of the error estimator.

关键词: Elliptic optimal control problem, Fourth-order variational inequality, Integral state constraints, Adaptive finite element method

Abstract: In this article, we develop a posteriori error analysis of a nonconforming finite element method for a linear quadratic elliptic distributed optimal control problem with two different sets of constraints, namely (i) integral state constraint and integral control constraint; (ii) integral state constraint and pointwise control constraints. In the analysis, we have taken the approach of reducing the state-control constrained minimization problem into a state minimization problem obtained by eliminating the control variable. The reliability and efficiency of a posteriori error estimator are discussed. Numerical results are reported to illustrate the behavior of the error estimator.

Key words: Elliptic optimal control problem, Fourth-order variational inequality, Integral state constraints, Adaptive finite element method

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