TY - JOUR
T1 - A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems
AU - Fu , Hongfei
AU - Rui , Hongxing
JO - Journal of Computational Mathematics
VL - 2
SP - 113
EP - 127
PY - 2015
DA - 2015/04
SN - 33
DO - http://doi.org/10.4208/jcm.1406-m4396
UR - https://global-sci.org/intro/article_detail/jcm/9831.html
KW - Optimal control, Least-squares mixed finite element methods, First-order elliptic system, A priori error estimates.
AB - <p style="text-align: justify;">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 $L^2(Ω)$-norm, for the original state and adjoint state in $H^1(Ω)$-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.</p>