Error estimates in $ L^2 $ and $ L^\infty $ norms of finite volume method for the bilinear elliptic optimal control problem

Zuliang Lu; Xiankui Wu; Fei Cai; Fei Huang; Shang Liu; Yin Yang; Zuliang Lu; Xiankui Wu; Fei Cai; Fei Huang; Shang Liu; Yin Yang

doi:10.3934/math.2021498

AIMS Mathematics

2021, Volume 6, Issue 8: 8585-8599. doi: 10.3934/math.2021498

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Research article

Error estimates in $ L^2 $ and $ L^\infty $ norms of finite volume method for the bilinear elliptic 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.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, Hunan, China
4.
School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China

Received: 16 December 2020 Accepted: 24 May 2021 Published: 07 June 2021
MSC : 49J20, 65M08

This paper discusses some a priori error estimates of bilinear elliptic optimal control problems based on the finite volume element approximation. A case-based numerical example serves to discuss with optimal $ L^2 $-norm error estimates and $ L^{\infty} $-norm error estimates, and supports two key insights. First, the approximate orders for the state, costate and control variables are $ O(h^2) $ in the sense of $ L^{2} $-norm. Second, the approximate orders for the state, costate and control variables are $ O(h^2\sqrt{|lnh|}) $ in the sense of $ L^{\infty} $-norm.
- bilinear elliptic optimal control problem,
- finite volume method,
- a priori error estimates,
- variational discretization
Citation: Zuliang Lu, Xiankui Wu, Fei Cai, Fei Huang, Shang Liu, Yin Yang. Error estimates in $ L^2 $ and $ L^\infty $ norms of finite volume method for the bilinear elliptic optimal control problem[J]. AIMS Mathematics, 2021, 6(8): 8585-8599. doi: 10.3934/math.2021498

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Abstract

This paper discusses some a priori error estimates of bilinear elliptic optimal control problems based on the finite volume element approximation. A case-based numerical example serves to discuss with optimal $ L^2 $-norm error estimates and $ L^{\infty} $-norm error estimates, and supports two key insights. First, the approximate orders for the state, costate and control variables are $ O(h^2) $ in the sense of $ L^{2} $-norm. Second, the approximate orders for the state, costate and control variables are $ O(h^2\sqrt{|lnh|}) $ in the sense of $ L^{\infty} $-norm.

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