A PRIORI ERROR ESTIMATES FOR LEAST-SQUARES MIXED FINITE ELEMENT APPROXIMATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS

• Published in: Journal of computational mathematics Vol. 33; no. 2; pp. 113 - 127
• Main Authors: Fu, Hongfei, Rui, Hongxing
• Format: Journal Article
• Language: English
• Published: Chinese Academy of Mathematices and System Sciences (AMSS) Chinese Academy of Sciences 01.03.2015
• Subjects: 先验误差估计; 最优控制问题; 最小二乘; 有限元逼近; 椭圆型; 混合有限元方法; 状态变量; 预条件共轭梯度
• ISSN: 0254-9409; 1991-7139
• DOI: 10.4208/jcm.1406-m4396

In this paper, a constrained distributed optimal control problem governed by a first- order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient （PCG） and algebraic multi-grid （AMG） can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in L2 （Ω）-norm, for the original state and adjoint state in H1 （Ω）-norm, and for the flux state and adjoint flux state in H（div; Ω）-norm. Finally, we use one numerical example to validate the theoretical findings.