Convergence and quasi-optimality based on an adaptive finite element method for the bilinear optimal control problem

• Published in: AIMS mathematics Vol. 6; no. 9; pp. 9510 - 9535
• Main Authors: Lu, Zuliang, Wu, Xiankui, Huang, Fei, Cai, Fei, Hou, Chunjuan, Yang, Yin
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: adaptive finite element method; bilinear optimal control problem; convergence; quasi-optimality
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021553

This paper investigates the adaptive finite element method for an optimal control problem governed by a bilinear elliptic equation. We establish the finite element discrete scheme for the bilinear optimal control problem and use a dual argument, linearization method, bubble function, and new bubble function to obtain a posteriori error estimates. To prove the convergence and the quasi-optimality for adaptive finite element methods, we introduce the adaptive finite element algorithm, local perturbation, error reduction, discrete local upper bound, Dörfler property, dual argument method, and quasi orthogonality. A few numerical examples are given at the end of the paper to demonstrate our theoretical analysis.