Volume 14, Issue 2
A Posteriori Error Estimates for $hp$ Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints

Adv. Appl. Math. Mech., 14 (2022), pp. 469-493.

Published online: 2022-01

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• Abstract

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state constraints. The control problem is approximated by the $hp$ spectral element method with high accuracy and geometric flexibility. Optimality conditions of the continuous and discrete optimal control problems are presented, respectively. The a posteriori error estimates both for the control and state variables are established in detail. In addition, illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.

• Keywords

Elliptic equations, optimal control, control-state constraints, a posteriori error estimates, $hp$ spectral element method.

35Q93, 49M25, 49M41

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@Article{AAMM-14-469, author = {Jinling and Zhang and and 22064 and and Jinling Zhang and Yanping and Chen and and 22065 and and Yanping Chen and Yunqing and Huang and and 22066 and and Yunqing Huang and Fenglin and Huang and and 22067 and and Fenglin Huang}, title = {A Posteriori Error Estimates for $hp$ Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {2}, pages = {469--493}, abstract = {

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state constraints. The control problem is approximated by the $hp$ spectral element method with high accuracy and geometric flexibility. Optimality conditions of the continuous and discrete optimal control problems are presented, respectively. The a posteriori error estimates both for the control and state variables are established in detail. In addition, illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0144}, url = {http://global-sci.org/intro/article_detail/aamm/20206.html} }
TY - JOUR T1 - A Posteriori Error Estimates for $hp$ Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints AU - Zhang , Jinling AU - Chen , Yanping AU - Huang , Yunqing AU - Huang , Fenglin JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 469 EP - 493 PY - 2022 DA - 2022/01 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0144 UR - https://global-sci.org/intro/article_detail/aamm/20206.html KW - Elliptic equations, optimal control, control-state constraints, a posteriori error estimates, $hp$ spectral element method. AB -

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state constraints. The control problem is approximated by the $hp$ spectral element method with high accuracy and geometric flexibility. Optimality conditions of the continuous and discrete optimal control problems are presented, respectively. The a posteriori error estimates both for the control and state variables are established in detail. In addition, illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.

Jinling Zhang, Yanping Chen, Yunqing Huang & Fenglin Huang. (2022). A Posteriori Error Estimates for $hp$ Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints. Advances in Applied Mathematics and Mechanics. 14 (2). 469-493. doi:10.4208/aamm.OA-2021-0144
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