Volume 33, Issue 2
A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems

Hongfei Fu & Hongxing Rui

J. Comp. Math., 33 (2015), pp. 113-127.

Published online: 2015-04

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  • 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.

  • Keywords

Optimal control Least-squares mixed finite element methods First-order elliptic system A priori error estimates

  • AMS Subject Headings

49K20 49M25 65N15 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hongfeifu@upc.edu.cn (Hongfei Fu)

hxrui@sdu.edu.cn (Hongxing Rui)

  • BibTex
  • RIS
  • TXT
@Article{JCM-33-113, author = {Fu , Hongfei and Rui , Hongxing }, title = {A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {2}, pages = {113--127}, 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1406-m4396}, url = {http://global-sci.org/intro/article_detail/jcm/9831.html} }
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://dor.org/10.4208/jcm.1406-m4396 UR - https://global-sci.org/intro/article_detail/jcm/9831.html KW - Optimal control KW - Least-squares mixed finite element methods KW - First-order elliptic system KW - A priori error estimates AB -

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.

Hongfei Fu & Hongxing Rui. (2020). A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems. Journal of Computational Mathematics. 33 (2). 113-127. doi:10.4208/jcm.1406-m4396
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