arrow
Volume 9, Issue 2
A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems

Hongfei Fu, Hongxing Rui, Jiansong Zhang & Hui Guo

Numer. Math. Theor. Meth. Appl., 9 (2016), pp. 215-238.

Published online: 2016-09

Export citation
  • Abstract

In this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable $y$ and its flux $σ$ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables $z$ and $ω$, and also a variational inequality for the control variable $u$ is derived. As we can see the two resulting systems for the unknown state variable $y$ and its flux $σ$ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states $z$ and $ω$ are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable $y$ and its flux $σ$. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.

  • Keywords

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{NMTMA-9-215, author = {}, title = {A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2016}, volume = {9}, number = {2}, pages = {215--238}, abstract = {

In this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable $y$ and its flux $σ$ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables $z$ and $ω$, and also a variational inequality for the control variable $u$ is derived. As we can see the two resulting systems for the unknown state variable $y$ and its flux $σ$ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states $z$ and $ω$ are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable $y$ and its flux $σ$. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2016.m1409}, url = {http://global-sci.org/intro/article_detail/nmtma/12375.html} }
TY - JOUR T1 - A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 215 EP - 238 PY - 2016 DA - 2016/09 SN - 9 DO - http://doi.org/10.4208/nmtma.2016.m1409 UR - https://global-sci.org/intro/article_detail/nmtma/12375.html KW - AB -

In this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable $y$ and its flux $σ$ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables $z$ and $ω$, and also a variational inequality for the control variable $u$ is derived. As we can see the two resulting systems for the unknown state variable $y$ and its flux $σ$ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states $z$ and $ω$ are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable $y$ and its flux $σ$. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.

Hongfei Fu, Hongxing Rui, Jiansong Zhang & Hui Guo. (2020). A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems. Numerical Mathematics: Theory, Methods and Applications. 9 (2). 215-238. doi:10.4208/nmtma.2016.m1409
Copy to clipboard
The citation has been copied to your clipboard