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A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems

Part of: Optimality conditions Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  24 May 2016

Hongfei Fu ,
Hongxing Rui ,
Jiansong Zhang  and
Hui Guo
Hongfei Fu*
Affiliation:
College of Science, China University of Petroleum, Qingdao, 266580, China
Hongxing Rui*
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China
Jiansong Zhang*
Affiliation:
College of Science, China University of Petroleum, Qingdao, 266580, China
Hui Guo*
Affiliation:
College of Science, China University of Petroleum, Qingdao, 266580, China
*
*Corresponding author. Email addresses:hongfeifu@upc.edu.cn(H. Fu), hxrui@sdu.edu.cn(H. Rui), jszhang@upc.edu.cn(J. Zhang), sdugh@163.com(H. Guo)
*Corresponding author. Email addresses:hongfeifu@upc.edu.cn(H. Fu), hxrui@sdu.edu.cn(H. Rui), jszhang@upc.edu.cn(J. Zhang), sdugh@163.com(H. Guo)
*Corresponding author. Email addresses:hongfeifu@upc.edu.cn(H. Fu), hxrui@sdu.edu.cn(H. Rui), jszhang@upc.edu.cn(J. Zhang), sdugh@163.com(H. Guo)
*Corresponding author. Email addresses:hongfeifu@upc.edu.cn(H. Fu), hxrui@sdu.edu.cn(H. Rui), jszhang@upc.edu.cn(J. Zhang), sdugh@163.com(H. Guo)
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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

Parabolic optimal controlsplitting mixed finite element methodpositive definitea priori error estimatesnumerical experiments

MSC classification

Secondary: 49K20: Problems involving partial differential equations 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 2 , May 2016 , pp. 215 - 238
Copyright
Copyright © Global-Science Press 2016 

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