Convergence and quasi-optimality based on an adaptive finite element method for the bilinear optimal control problem

Zuliang Lu; Xiankui Wu; Fei Huang; Fei Cai; Chunjuan Hou; Yin Yang; Zuliang Lu; Xiankui Wu; Fei Huang; Fei Cai; Chunjuan Hou; Yin Yang

doi:10.3934/math.2021553

AIMS Mathematics

2021, Volume 6, Issue 9: 9510-9535. doi: 10.3934/math.2021553

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Research article

Convergence and quasi-optimality based on an adaptive finite element method for the bilinear optimal control problem

1.
Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, 404000, China
2.
Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin, 300222, China
3.
Guangzhou Huashang College, Guangzhou 511300, China
4.
School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China

Received: 05 March 2021 Accepted: 24 May 2021 Published: 24 June 2021
MSC : 49J20, 65M08

This paper investigates the adaptive finite element method for an optimal control problem governed by a bilinear elliptic equation. We establish the finite element discrete scheme for the bilinear optimal control problem and use a dual argument, linearization method, bubble function, and new bubble function to obtain a posteriori error estimates. To prove the convergence and the quasi-optimality for adaptive finite element methods, we introduce the adaptive finite element algorithm, local perturbation, error reduction, discrete local upper bound, Dörfler property, dual argument method, and quasi orthogonality. A few numerical examples are given at the end of the paper to demonstrate our theoretical analysis.
- bilinear optimal control problem,
- adaptive finite element method,
- convergence,
- quasi-optimality
Citation: Zuliang Lu, Xiankui Wu, Fei Huang, Fei Cai, Chunjuan Hou, Yin Yang. Convergence and quasi-optimality based on an adaptive finite element method for the bilinear optimal control problem[J]. AIMS Mathematics, 2021, 6(9): 9510-9535. doi: 10.3934/math.2021553

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Abstract

This paper investigates the adaptive finite element method for an optimal control problem governed by a bilinear elliptic equation. We establish the finite element discrete scheme for the bilinear optimal control problem and use a dual argument, linearization method, bubble function, and new bubble function to obtain a posteriori error estimates. To prove the convergence and the quasi-optimality for adaptive finite element methods, we introduce the adaptive finite element algorithm, local perturbation, error reduction, discrete local upper bound, Dörfler property, dual argument method, and quasi orthogonality. A few numerical examples are given at the end of the paper to demonstrate our theoretical analysis.

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