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

AIMS Mathematics

Zuliang Lu¹  Fei Cai²  Fei Huang²  Xiankui Wu²  Shang Liu³  Yin Yang⁴ 

[1] 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;1. Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing, 404000, 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;

关键词: bilinear elliptic optimal control problem;    finite volume method;    a priori error estimates;    variational discretization;   

DOI  :  10.3934/math.2021498

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown