A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems

Hongbo Guan; Yong Yang; Huiqing Zhu

doi:10.3934/dcdsb.2020179

Article Contents

2021, Volume 26, Issue 3: 1711-1722. Doi: 10.3934/dcdsb.2020179

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A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems

1.
College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China
2.
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
3.
School of Mathematics and Natural Sciences, The University of Southern Mississippi, Hattiesburg, MS 39406, USA

^* Corresponding author: Huiqing Zhu
^* Corresponding author: Huiqing Zhu

Received: April 30, 2019

Revised: January 31, 2020

Early access: June 2020

Published: March 2021

The first author is supported by National Natural Science Foundation of China (No. 11501527)

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Abstract

In this paper, an anisotropic bilinear finite element method is constructed for the elliptic boundary layer optimal control problems. Supercloseness properties of the numerical state and numerical adjoint state in a $ \epsilon $-norm are established on anisotropic meshes. Moreover, an interpolation type post-processed solution is shown to be superconvergent of order $ O(N^{-2}) $, where the total number of nodes is of $ O(N^2) $. Finally, numerical results are provided to verify the theoretical analysis.

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Mathematics Subject Classification: Primary: 65M60, 49K20; Secondary: 65J10.

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Figure 1. Anisotropic mesh (left) and uniform mesh (right)

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Figure 2. Big element $ K_{0} $

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Figure 3. The profile of $ y_h $ (left plot) and $ p_h $ (right plot) on a uniform mesh with $ N = 8 $

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Figure 4. Pointwise errors of $ |y-y_h| $ (left plot) and $ |p-p_h| $ (right plot) on a uniform mesh with $ N = 8 $

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Figure 5. The profile of $ y_h $ (left plot) and $ p_h $ (right plot) on an anisotropic mesh with $ N = 8 $

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Figure 6. Pointwise errors of $ |y-y_h| $ (left plot) and $ |p-p_h| $ (right plot) on an anisotropic mesh with $ N = 8 $

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Table 1. Errors and convergence rates on uniform meshes

$N$	4	8	16	32	64
$\\|u-u_h\\|_{0}$	4.1470E-02	2.7610E-02	1.7622E-02	1.0661E-02	5.6118E-03
order	/	0.5869	0.6478	0.7250	0.9258
$\|\|\|\Pi_{h} y-y_h\|\|\|$	5.0583E-02	3.5215E-02	2.4487E-02	1.6591E-02	9.9351E-03
order	/	0.5225	0.5242	0.5616	0.7398
$\|\|\|\Pi_{h} p-p_h\|\|\|$	4.8403E-02	3.4837E-02	2.4420E-02	1.6569E-02	9.9076E-03
order	/	0.4745	0.5126	0.5596	0.7419
$\|\|\|y-\Pi_{2h}y_h\|\|\|$	1.7466E-01	9.7410E-02	5.7862E-02	3.5868E-02	2.1591E-02
order	/	0.8424	0.7515	0.6899	0.7323
$\|\|\|p-\Pi_{2h}p_h\|\|\|$	1.7207E-01	9.6955E-02	5.7777E-02	3.5844E-02	2.1562E-02
order	/	0.8276	0.7468	0.6888	0.7332

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Table 2. Errors and convergence rates on anisotropic meshes

$N$	4	8	16	32	64
$\\|u-u_h\\|_{0}$	1.1762E-02	3.0109E-03	7.5308E-04	1.8747E-04	4.6824E-05
order	/	1.9659	1.9993	2.0062	2.0013
$\|\|\|\Pi_{h} y-y_h\|\|\|$	8.5488E-03	2.9582E-03	8.8988E-04	2.3807E-04	6.2920E-05
order	/	1.5310	1.7330	1.9022	1.9198
$\|\|\|\Pi_{h} p-p_h\|\|\|$	6.3618E-03	2.5354E-03	7.8790E-04	2.1187E-04	5.8477E-05
order	/	1.3272	1.6861	1.8949	1.8572
$\|\|\|y-\Pi_{2h}y_h\|\|\|$	1.2562E-02	4.3649E-03	1.2406E-03	3.0455E-04	7.6278E-05
order	/	1.5250	1.8149	2.0263	1.9974
$\|\|\|p-\Pi_{2h}p_h\|\|\|$	1.0955E-02	4.0558E-03	1.1598E-03	2.8267E-04	7.3036E-05
order	/	1.4335	1.8061	2.0367	1.9524

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