A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

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• 作者：Hongfei Fu ; Hongxing Rui ; Jian Hou ; Haihong Li
• 关键词：Optimal control ; Stabilized mixed finite element ; LBB condition ; A priori error estimates ; Numerical experiments ; 49K20 ; 49M25 ; 65N15 ; 65N30
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：66
• 期：3
• 页码：968-986
• 全文大小：4,324 KB
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• 作者单位：Hongfei Fu (1)
  Hongxing Rui (2)
  Jian Hou (3)
  Haihong Li (1)
  
  1. College of Science, China University of Petroleum, Qingdao, 266580, China
  2. School of Mathematics, Shandong University, Jinan, 250100, China
  3. College of Petroleum Engineering, China University of Petroleum, Qingdao, 266580, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

In this paper, we propose a new mixed finite element method, called stabilized mixed finite element method, for the approximation of optimal control problems constrained by a first-order elliptic system. This method is obtained by adding suitable elementwise least-squares residual terms for the primal state variable y and its flux \(\sigma \). We prove the coercive and continuous properties for the new mixed bilinear formulation at both continuous and discrete levels. Therefore, the finite element function spaces do not require to satisfy the Ladyzhenkaya–Babuska–Brezzi consistency condition. Furthermore, the state and flux state variables can be approximated by the standard Lagrange finite element. We derive optimality conditions for such optimal control problems under the concept of Discretization-then-Optimization, and then a priori error estimates in a weighted norm are discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the stabilized method. Keywords Optimal control Stabilized mixed finite element LBB condition A priori error estimates Numerical experiments