标题：A PRIORI ERROR ESTIMATES FOR LEAST-SQUARES MIXED FINITE ELEMENT APPROXIMATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS
作者：Fu, Hongfei; Rui, Hongxing
作者机构：[Fu, H] College of Science, China University of Petroleum, Qingdao, 266580, China;[ Rui, H] School of Mathematics, Shandong University, Jinan, 250100, 更多
通讯作者地址：[Fu, HF]China Univ Petr, Coll Sci, Qingdao 266580, Peoples R China.
关键词：Optimal control; Least-squares mixed finite element methods; First-order; elliptic system; A priori error estimates
摘要：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 L-2(Omega)-norm, for the original state and adjoint state in H-1(Omega)-norm, and for the flux state and adjoint flux state in H(div;Omega)-norm. Finally, we use one numerical example to validate the theoretical findings.