J. Comp. Math., 33 (2015), pp. 113-127. |
A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems Hongfei Fu ^{1}, Hongxing Rui ^{2} 1 College of Science, China University of Petroleum, Qingdao 266580, China2 School of Mathematics, Shandong University, Jinan 250100, China Received 2013-6-2 Accepted 2014-9-26 Available online 2015-3-13 doi:10.4208/jcm.1406-m4396 Abstract In this paper, a constrained distributed optimal control problem governed by a firstorder 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. Key words: Optimal control, Least-squares mixed finite element methods, First-order elliptic system, A priori error estimates. AMS subject classifications: 49K20, 49M25, 65N15, 65N30. Email: hongfeifu@upc.edu.cn, hxrui@sdu.edu.cn |