Electron. J. Differential Equations, Vol. 2017 (2017), No. 267, pp. 1-15.

A priori error estimates of finite volume methods for general elliptic optimal control problems

Yuming Feng, Zuliang Lu, Longzhou Cao, Lin Li, Shuhua Zhang

Abstract:
In this article, we establish a priori error estimates for the finite volume approximation of general elliptic optimal control problems. We use finite volume methods to discretize the state and adjoint equation of the optimal control problems. For the variational inequality, we use the variational discretization methods to discretize the control. We show the existence and the uniqueness of the solution for discrete optimality conditions. Under some reasonable assumptions, we obtain some optimal order error estimates for the state, costate and control variables. On one hand, the convergence rate for the state, costate and control variables is $O(h^2)$ or $O(h^2 \sqrt{|\log(\frac{1}{h})|})$ in the sense of $L^2$ norm or $L^{\infty}$ norm. On the other hand, the convergence rate for the state and costate variables is $O(h)$ or $O(h{|\log (\frac{1}{h})|})$ in the sense of $H^1$ norm or $W^{1,\infty}$ norm.

Submitted August 9, 2017. Published October 27, 2017.
Math Subject Classifications: 49J20, 65N30.
Key Words: A priori error estimates; general elliptic optimal control problems; finite volume methods; optimal-order.

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Yuming Feng
Key Laboratory of Intelligent Information Processing and Control
Chongqing Three Gorges University
Wanzhou, Chongqing, 404100, China
email: yumingfeng25928@163.com
Zuliang Lu
Key Laboratory for Nonlinear Science and System Structure
Chongqing Three Gorges University, Chongqing 404100, China
email: zulianglux@126.com
Longzhou Cao
Key Laboratory for Nonlinear Science and System Structure
Chongqing Three Gorges University
Chongqing 404100, China
email: caolongzhou@126.com
Lin Li
Key Laboratory for Nonlinear Science and System Structure
Chongqing Three Gorges University
Chongqing 404100, China
email: linligx@126.com
Shuhua Zhang
Research Center for Mathematics and Economics
Tianjin University of Finance and Economics
Tianjin 300222, China
email: szhang@tjufe.edu.cn

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