Finite Element Method and A Priori Error Estimates for Dirichlet Boundary Control Problems Governed by Parabolic PDEs

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• 作者：Wei Gong ; Michael Hinze ; Zhaojie Zhou
• 关键词：Optimal control problem ; Parabolic equation ; Finite element method ; A priori error estimate ; Dirichlet boundary control ; 49J20 ; 49K20 ; 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：941-967
• 全文大小：617 KB
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• 作者单位：Wei Gong (1)
  Michael Hinze (2)
  Zhaojie Zhou (3)
  
  1. LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
  2. Schwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany
  3. School of Mathematics Sciences, Shandong Normal University, Jinan, 250014, 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

Finite element approximations of Dirichlet boundary control problems governed by parabolic PDEs on convex polygonal domains are studied in this paper. The existence of a unique solution to optimal control problems is guaranteed based on very weak solution of the state equation and \(L^2(0,T;L^2(\varGamma ))\) as control space. For the numerical discretization of the state equation we use standard piecewise linear and continuous finite elements for the space discretization of the state, while a dG(0) scheme is used for time discretization. The Dirichlet boundary control is realized through a space–time \(L^2\)-projection. We consider both piecewise linear, continuous finite element approximation and variational discretization for the controls and derive a priori \(L^2\)-error bounds for controls and states. We finally present numerical examples to support our theoretical findings. Keywords Optimal control problem Parabolic equation Finite element method A priori error estimate Dirichlet boundary control