A posteriori error estimates of hp spectral element method for parabolic optimal control problems

Zuliang Lu; Fei Cai; Ruixiang Xu; Chunjuan Hou; Xiankui Wu; Yin Yang; Zuliang Lu; Fei Cai; Ruixiang Xu; Chunjuan Hou; Xiankui Wu; Yin Yang

doi:10.3934/math.2022291

AIMS Mathematics

2022, Volume 7, Issue 4: 5220-5240. doi: 10.3934/math.2022291

Previous Article Next Article

Research article

A posteriori error estimates of hp spectral element method for parabolic optimal control problems

1.
Key Laboratory for Nonlinear Science and System Structure, Key Laboratory of Intelligent Information Processing and Control, Chongqing Three Gorges University, Chongqing 404000, China
2.
Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin 300222, China
3.
Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing 404000, China
4.
Guangzhou Huashang College, Guangzhou 511300, China
5.
School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105 Hunan, China

Received: 18 October 2021 Revised: 19 December 2021 Accepted: 20 December 2021 Published: 04 January 2022
MSC : 49J20, 65N30

In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $ L^2(H^1)-L^2(L^2) $ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $ L^2(L^2)-L^2(L^2) $ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.
- parabolic optimal control problems,
- hp spectral element method,
- a posteriori error estimates
Citation: Zuliang Lu, Fei Cai, Ruixiang Xu, Chunjuan Hou, Xiankui Wu, Yin Yang. A posteriori error estimates of hp spectral element method for parabolic optimal control problems[J]. AIMS Mathematics, 2022, 7(4): 5220-5240. doi: 10.3934/math.2022291

Related Papers:

Abstract

In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $ L^2(H^1)-L^2(L^2) $ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $ L^2(L^2)-L^2(L^2) $ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.

References

[1]	R. Ghanem, H. Sissaoui, A posteriori error estimate by a spectral method of an elliptic optimal control problem, J. Comput. Math. Optim., 2 (2006), 111–125.
[2]	Y. Chen, Superconvergence of optimal control problems by rectangular mixed finite element methods, Math. Comput., 77 (2008), 1269–1291. https://doi.org/10.1090/S0025-5718-08-02104-2 doi: 10.1090/S0025-5718-08-02104-2
[3]	Y. Chen, W. Liu, Error estimates and superconvergence of mixed finite element for quadratic optimal control, Int. J. Numer. Anal. Mod., 3 (2006), 311–321. https://doi.org/10.1080/00207160601117354 doi: 10.1080/00207160601117354
[4]	Y. Chen, W. Liu, A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comput. Appl. Math., 211 (2008), 76–89. https://doi.org/10.1016/j.cam.2006.11.015 doi: 10.1016/j.cam.2006.11.015
[5]	Y. Chen, Z. Lu, High efficient and accuracy numerical methods for optimal control problems, Science Press, Beijing, 2015.
[6]	Y. Chen, Z. Lin, A posteriori error estimates of semidiscrete mixed finite element methods for parabolic optimal control problems, E. Asian J. Appl. Math., 5 (2015), 957–965. https://doi.org/10.4208/eajam.010314.110115a doi: 10.4208/eajam.010314.110115a
[7]	A. Kröner, B. Vexler, A priori error estimates for elliptic optimal control problems with a bilinear state equation, Comput. Math. Appl., 2 (2009), 781–802. https://doi.org/10.1016/j.cam.2009.01.023 doi: 10.1016/j.cam.2009.01.023
[8]	Y. Chen, N. Yi, W. Liu, A Legendre Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46 (2008), 2254–2275. https://doi.org/10.1137/070679703 doi: 10.1137/070679703
[9]	L. Li, Z. Lu, W. Zhang, F. Huang, Y. Yang, A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem, J. Inequal. Appl., 1 (2018), 1–23. https://doi.org/10.1186/s13660-018-1729-4 doi: 10.1186/s13660-018-1729-4
[10]	R. Li, W. Liu, H. Ma, T. Tang, Adaptive finite element approximation of elliptic optimal control, SIAM J. Control Optim., 41 (2002), 1321–1349. https://doi.org/10.1137/S0363012901389342 doi: 10.1137/S0363012901389342
[11]	J. L. Lions, Optimal control of systems governed by partial differential equations, Springer-Verlag, Berlin, 1971.
[12]	J. L. Lions, E. Magenes, Non homogeneous boundary value problems and applications, Springer-Verlag, Berlin, 1972.
[13]	W. Liu, J. Barrett, Error bounds for the finite element approximation some degenerate quasilinear parabolic equations and variational inequalities, Adv. Comput. Math., 1 (1993), 223–239.
[14]	W. Liu, D. Tiba, Error estimates for the finite element approximation of nonlinear optimal control problems, J. Numer. Func. Optim., 22 (2001), 953–972.
[15]	W. Liu, N. Yan, A posteriori error analysis for convex distributed optimal control problems, Adv. Comp. Math., 15 (2001), 285–309. https://doi.org/10.1023/A:1014239012739 doi: 10.1023/A:1014239012739
[16]	W. Liu, N. Yan, A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 93 (2003), 497–521. https://doi.org/10.1007/s002110100380 doi: 10.1007/s002110100380
[17]	W. Liu, N. Yan, A posteriori error estimates for optimal control of stokes flows, SIAM J. Numer. Anal., 40 (2003), 1805–1869.
[18]	Y. Tang, Y. Chen, Recovery type a posteriori error estimates of fully discrete finite element methods for general convex parabolic optimal control problems, Numer. Math.-Theory Me., 4 (2012), 573–591. https://doi.org/10.1017/S1004897900001069 doi: 10.1017/S1004897900001069
[19]	Z. Lu, S. Zhang, $L^\infty$-error estimates of rectangular mixed finite element methods for bilinear optimal control problem, Appl. Math. Comp., 300 (2017), 79–94. https://doi.org/10.1016/j.amc.2016.12.006 doi: 10.1016/j.amc.2016.12.006
[20]	J. M. Melenk, hp-interpolation of non-smooth functions, SIAM J. Numer. Anal., 43 (2005), 127–155. https://doi.org/10.1137/S0036142903432930 doi: 10.1137/S0036142903432930
[21]	A. T. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys., 54 (1984), 468–488. https://doi.org/10.1016/0021-9991(84)90128-1 doi: 10.1016/0021-9991(84)90128-1
[22]	L. R. Scott, S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), 483–493. https://doi.org/10.1090/S0025-5718-1990-1011446-7 doi: 10.1090/S0025-5718-1990-1011446-7
[23]	X. Xing, Y. Chen, $L^{\infty}$-error estimates for general optimal control problem by mixed finite element methods, Int. J. Numer. Anal. Mod., 5 (2008), 441–456. https://doi.org/10.1007/s11424-010-8015-y doi: 10.1007/s11424-010-8015-y
[24]	X. Xing, Y. Chen, Error estimates of mixed methods for optimal control problems governed by parabolic equations, Int. J. Numer. Meth. Eng., 75 (2010), 735–754. https://doi.org/10.1002/nme.2289 doi: 10.1002/nme.2289
[25]	S. Boulaaras, Some new properties of asynchronous algorithms of theta scheme combined with finite elements methods for an evolutionary implicit 2-sided obstacle problem, Math. Meth. App., 40 (2017), 7231–7239. https://doi.org/10.1002/mma.4525 doi: 10.1002/mma.4525
[26]	S. Boulaaras, Polynomial decay rate for a new class of viscoelastic Kirchhoff equation related with Balakrishnan-Taylor dissipation and logarithmic source terms, Alex. Eng. J., 4 (2020), 1059–1071. https://doi.org/10.1016/j.aej.2019.12.013 doi: 10.1016/j.aej.2019.12.013
[27]	S. Boulaaras, M. S. Touati Brahim, S. Bouzenada, A. Zarai, An asymptotic behavior and a posteriori error estimates for the generalized Schwartz method of advection-diffusion equation, Acta Math. Sci., 4 (2018), 1227–1244. https://doi.org/10.1016/S0252-9602(18)30810-5 doi: 10.1016/S0252-9602(18)30810-5
[28]	S. Boulaaras, M. Haiour, The finite element approximation of evolutionary Hamilton-Jacobi-Bellman equations with nonlinear source terms, Indagat. Math., 24 (2013), 161–173. https://doi.org/10.1016/j.indag.2012.07.005 doi: 10.1016/j.indag.2012.07.005
[29]	L. Bonifacius, K. Pieper, B. Vexler, A priori error estimates for space-time finite element discretization of parabolic time-optimal control problems, Numer. Math., 120 (2018), 345–386. https://doi.org/10.1007/s00211-011-0409-9 doi: 10.1007/s00211-011-0409-9
[30]	Z. Lu, X. Huang, A priori error estimates of mixed finite element methods for general linear hyperbolic convex optimal control problems, Abst. Appl. Anal., 7 (2014), 1–10. https://doi.org/10.1155/2014/547490 doi: 10.1155/2014/547490

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)