A priori error estimates of finite volume element method for bilinear parabolic optimal control problem

Zuliang Lu; Ruixiang Xu; Chunjuan Hou; Lu Xing; Zuliang Lu; Ruixiang Xu; Chunjuan Hou; Lu Xing

doi:10.3934/math.2023988

AIMS Mathematics

2023, Volume 8, Issue 8: 19374-19390. doi: 10.3934/math.2023988

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Research article

A priori error estimates of finite volume element method for bilinear parabolic optimal control problem

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.
Department of Data Science, Guangzhou Huashang College, Guangzhou 511300, China

Received: 29 March 2023 Revised: 10 May 2023 Accepted: 16 May 2023 Published: 08 June 2023
MSC : 49J20, 65N30

In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $ O(h^{3/2}+\triangle t) $ in the sense of $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. Finally, a numerical experiment is presented to test these theoretical results.
- bilinear parabolic optimal control problems,
- finite volume element method
Citation: Zuliang Lu, Ruixiang Xu, Chunjuan Hou, Lu Xing. A priori error estimates of finite volume element method for bilinear parabolic optimal control problem[J]. AIMS Mathematics, 2023, 8(8): 19374-19390. doi: 10.3934/math.2023988

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Abstract

In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $ O(h^{3/2}+\triangle t) $ in the sense of $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. Finally, a numerical experiment is presented to test these theoretical results.

References

[1]	R. A. Adams, J. J. F. Fournier, Sobolev spaces, 2 Eds., New York: Academic Press, 2003.
[2]	N. Arada, E. Casas, F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl., 23 (2002), 201–229. http://dx.doi.org/10.1023/A:1020576801966 doi: 10.1023/A:1020576801966
[3]	R. E. Bank, D. J. Rose, Some error estimates for the box method, SIAM J. Numer. Anal., 24 (1987), 777–787. http://dx.doi.org/10.1137/0724050 doi: 10.1137/0724050
[4]	C. M. Bollo, C. M. Gariboldi, D. A. Tarzia, Simultaneous distributed and Neumann boundary optimal control problems for elliptic hemivariational inequalities, J. Nonlinear Var. Anal., 6 (2022), 535–549. http://dx.doi.org/10.23952/jnva.6.2022.5.07 doi: 10.23952/jnva.6.2022.5.07
[5]	S. C. Brenner, L. R. Scott, The mathematical theory of finite elementMethods, 3 Eds., New York: Springer, 2008. http://dx.doi.org/10.1007/978-0-387-75934-0
[6]	P. Chatzipantelidis, R. Lazarov, V. Thomée, Error estimate for a finite volume element method for parabolic equations in convex polygonal domains, Numer. Meth. Part. D. E., 20 (2004), 650–674. http://dx.doi.org/10.1002/num.20006 doi: 10.1002/num.20006
[7]	Y. Chen, Y. Huang, W. Liu, N. Yan, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput., 42 (2010), 382–403. https://dx.doi.org/10.1007/s10915-009-9327-8 doi: 10.1007/s10915-009-9327-8
[8]	Y. Chen, Z. Lu, High efficient and accuracy numerical methods for optimal control problems, 1 Eds., Beijing: Science Press, 2015.
[9]	Y. Chen, Z. Lu, Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems, Comput. Method. Appl. M., 199 (2010), 1415–1423. http://dx.doi.org/10.1016/j.cma.2009.11.009 doi: 10.1016/j.cma.2009.11.009
[10]	Y. Chen, Z. Lu, R. Guo, Error estimates of triangularmixed finite element methods for quasilinear optimal controlproblems, Front. Math. China, 7 (2012), 397–413. http://dx.doi.org/10.1007/s11464-012-0179-4 doi: 10.1007/s11464-012-0179-4
[11]	Y. Chen, Z. Lu, Y. Huang, Superconvergence of triangular Raviart-Thomas mixed finite element methods for bilinear constrained optimal control problem, Comput. Math. Appl., 66 (2013), 1498–1513. http://dx.doi.org/10.1016/j.camwa.2013.08.019 doi: 10.1016/j.camwa.2013.08.019
[12]	Y. Chen, N. Yi, W. Li, A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46 (2008), 2254–2275. https://dx.doi.org/10.1137/070679703 doi: 10.1137/070679703
[13]	S. Chou, Q. Li, Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: a unified approach, Math. Comput., 69 (2000), 103–120. https://dx.doi.org/10.1090/S0025-5718-99-01192-8 doi: 10.1090/S0025-5718-99-01192-8
[14]	S. Chou, X. Ye, Unified analysis of finite volume methods for second order elliptic problems, SIAM J. Numer. Anal., 45 (2007), 1639–1653. http://dx.doi.org/10.1137/050643994 doi: 10.1137/050643994
[15]	R. E. Ewing, T. Lin, Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal., 39 (2002), 1865–1888. http://dx.doi.org/10.1137/S0036142900368873 doi: 10.1137/S0036142900368873
[16]	Y. Feng, Z. Lu, S. Zhang, L. Cao, L. Li, A priori error estimates of finite volume methods for general elliptic optimal control problems, Electron. J. Differ. Eq., 267 (2017), 1–15.
[17]	M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Applic., 30 (2005), 45–61. http://dx.doi.org/10.1007/s10589-005-4559-5 doi: 10.1007/s10589-005-4559-5
[18]	B. T. Kien, X. Qin, C. F. Wen, J. C. Yao, Second-order optimality conditions for multiobjective optimal control problems with mixed pointwise constraints and free right end point, SIAM J. Control Optim., 58 (2020), 2658–2677. http://dx.doi.org/10.1137/19M1281770 doi: 10.1137/19M1281770
[19]	R. Li, W. Liu, H. Ma, T. Tang, Adaptive finite element approximation for distributed convexoptimal control problems, SIAM J. Control Optim., 41 (2002), 1321–1349. http://dx.doi.org/10.1137/S0363012901389342 doi: 10.1137/S0363012901389342
[20]	W. Liu, T. Zhao, K. Ito, Z. Zhang, Error estimates of Fourier finite volume element method for parabolic Dirichlet boundary optimal control problems on complex connected domains, Appl. Numer. Math., 186 (2023), 164–201. http://dx.doi.org/10.1016/j.apnum.2023.01.007 doi: 10.1016/j.apnum.2023.01.007
[21]	Z. Lu, $L^{\infty}$-estimates of rectangular mixed methods for nonlinear constrained optimal control problem, Bull. Malays. Math. Sci. Soc., 37 (2014), 271–284. http://dx.doi.org/228721511
[22]	Z. Lu, L. Li, L. Cao, C. Hou, A priori error estimates offinite volume method for nonlinear optimal control problem, Numer. Analys. Appl., 10 (2017), 224–236. http://dx.doi.org/10.1134/S1995423917030041 doi: 10.1134/S1995423917030041
[23]	Z. Lu, S. Zhang, $L^{\infty}$-error estimates of rectangular mixed finitec element methods for bilinear optimal control problem, Appl. Math. Comput., 300 (2017), 79–94. http://dx.doi.org/10.1016/j.amc.2016.12.006 doi: 10.1016/j.amc.2016.12.006
[24]	Q. Li, Z. Liu, Finite volume element methods for nonlinear parabolic problems, J. KSIAM, 6 (2002), 85–97.
[25]	J. L. Lions, Optimal control of systems governed by partial differential equations, 1 Eds., Berlin: Springer, 1971.
[26]	W. Liu, N. Yan, Adaptive finite element methods for optimal control governed by PDEs, 1 Eds., Beijing: Springer, 2008.
[27]	X. Luo, Y. Chen, Y. Huang, T. Hou, Some error estimates offinite volume element method for parabolic optimal control problems, Optim. Contr. Appl. Met., 35 (2014), 145–165. http://dx.doi.org/10.1002/oca.2059 doi: 10.1002/oca.2059
[28]	D. Yang, Y. Chang, W. Liu, A priori error estimate and superconvergence analysis for an optimal control problem of bilinear type, J. Comput. Math., 26 (2008), 471–487.
[29]	T. Zhang, H. Zhong, J. Zhao, A full discrete two-gridb finite-volume method for a nonlinear parabolic problem, Int. J. Comput. Math., 88 (2011), 1644–1663. http://dx.doi.org/abs/10.1080/00207160.2010.521550

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