Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems

Zuliang Lu; Fei Huang; Xiankui Wu; Lin Li; Shang Liu; Zuliang Lu; Fei Huang; Xiankui Wu; Lin Li; Shang Liu

doi:10.3934/era.2020077

Electronic Research Archive

2020, Volume 28, Issue 4: 1459-1486. doi: 10.3934/era.2020077

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Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems

Zuliang Lu ^{1,2
,
,},
Fei Huang ^{1
,},
Xiankui Wu ^{1
,},
Lin Li ^{3
,},
Shang Liu ^{4
,}

1.
Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing 404000, China
2.
Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin 300222, China
3.
College of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
4.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, Hunan, China

Received: 01 December 2019 Revised: 01 June 2020 Published: 31 July 2020
Primary: 49J20, 65N30; Secondary: 65N12

This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by $ L^2 $-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.
- Nonlinear optimal control problem,
- adaptive finite element method,
- convergence,
- quasi-optimality
Citation: Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems[J]. Electronic Research Archive, 2020, 28(4): 1459-1486. doi: 10.3934/era.2020077

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Abstract

This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by $ L^2 $-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.

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