Volume 12, Issue 2
The Convergence and Superconvergence of a MFEM for Elliptic Optimal Control Problems

Adv. Appl. Math. Mech., 12 (2020), pp. 527-544.

Published online: 2020-01

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• Abstract

In this paper, we investigate a mixed finite element method (MFEM) for the elliptic optimal control problems (OCPs) with a distributive control. The state variable and adjoint state variable are approximated by the conforming rectangular $Q_{11}+Q_{01}\times Q_{10}$ elements pair. The discrete B-B condition is satisfied automatically, which is usually considered to be the key point of the MFEM. The control is then obtained by the orthogonal projection through the adjoint state. Optimal orders of convergence are derived for the above mentioned variables. Furthermore, supercloseness and superconvergence results are also established under certain reasonable regularity assumptions. Some numerical results are provided to verify the theoretical analysis. At last, the proposed method is extended to some other low order conforming and nonconforming elements.

• Keywords

MFEMs, OCPs, optimal order error estimates, supercloseness and superconvergence.

65N30, 65N15

Huiqing.Zhu@usm.edu (Huiqing Zhu)

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@Article{AAMM-12-527, author = {Hongbo and Guan and and 6495 and College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China and Hongbo Guan and Yong and Yang and and 6496 and College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China and Yong Yang and Huiqing and Zhu and Huiqing.Zhu@usm.edu and 6497 and School of Mathematics and Natural Sciences, The University of Southern Mississippi, Hattiesburg, MS 39406, USA and Huiqing Zhu}, title = {The Convergence and Superconvergence of a MFEM for Elliptic Optimal Control Problems}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {2}, pages = {527--544}, abstract = {

In this paper, we investigate a mixed finite element method (MFEM) for the elliptic optimal control problems (OCPs) with a distributive control. The state variable and adjoint state variable are approximated by the conforming rectangular $Q_{11}+Q_{01}\times Q_{10}$ elements pair. The discrete B-B condition is satisfied automatically, which is usually considered to be the key point of the MFEM. The control is then obtained by the orthogonal projection through the adjoint state. Optimal orders of convergence are derived for the above mentioned variables. Furthermore, supercloseness and superconvergence results are also established under certain reasonable regularity assumptions. Some numerical results are provided to verify the theoretical analysis. At last, the proposed method is extended to some other low order conforming and nonconforming elements.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0019}, url = {http://global-sci.org/intro/article_detail/aamm/13632.html} }
TY - JOUR T1 - The Convergence and Superconvergence of a MFEM for Elliptic Optimal Control Problems AU - Guan , Hongbo AU - Yang , Yong AU - Zhu , Huiqing JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 527 EP - 544 PY - 2020 DA - 2020/01 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0019 UR - https://global-sci.org/intro/article_detail/aamm/13632.html KW - MFEMs, OCPs, optimal order error estimates, supercloseness and superconvergence. AB -

In this paper, we investigate a mixed finite element method (MFEM) for the elliptic optimal control problems (OCPs) with a distributive control. The state variable and adjoint state variable are approximated by the conforming rectangular $Q_{11}+Q_{01}\times Q_{10}$ elements pair. The discrete B-B condition is satisfied automatically, which is usually considered to be the key point of the MFEM. The control is then obtained by the orthogonal projection through the adjoint state. Optimal orders of convergence are derived for the above mentioned variables. Furthermore, supercloseness and superconvergence results are also established under certain reasonable regularity assumptions. Some numerical results are provided to verify the theoretical analysis. At last, the proposed method is extended to some other low order conforming and nonconforming elements.

Hongbo Guan, Yong Yang & Huiqing Zhu. (2020). The Convergence and Superconvergence of a MFEM for Elliptic Optimal Control Problems. Advances in Applied Mathematics and Mechanics. 12 (2). 527-544. doi:10.4208/aamm.OA-2019-0019
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