A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems
Hongfei Fu 1, Hongxing Rui 21 College of Science, China University of Petroleum, Qingdao 266580, China
2 School of Mathematics, Shandong University, Jinan 250100, China
Received 2013-6-2 Accepted 2014-9-26
Available online 2015-3-13
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.
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