American Institute of Mathematical Sciences

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December  2020, 28(4): 1459-1486. doi: 10.3934/era.2020077

Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems

Zuliang Lu 1,2,, , Fei Huang 1, , Xiankui Wu 1, , Lin Li 3, and  Shang Liu 4,

1. 

Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Chongqing 404000, China
2. 

Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin 300222, China
3. 

College of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
4. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, Hunan, China

* Corresponding author: Zuliang Lu

Received  December 2019 Revised  June 2020 Published  December 2020 Early access  July 2020

Fund Project: The first author is supported by NSFC grant 11201510, NSSFC grant 19BGL190, Postdoctoral Science Foundation grant 2017T100155 and 2015M580197, Innovation Team Building at Institutions of Higher Education in Chongqing grant CXTDX201601035, Scientific and Technological Research Program of Chongqing Municipal Education Commission and Chongqing Research Program of Basic Research and Frontier Technology grant cstc2019jcyj-msxmX0280

This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by $ L^2 $-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.

Keywords: Nonlinear optimal control problem, adaptive finite element method, convergence, quasi-optimality.
Mathematics Subject Classification: Primary: 49J20, 65N30; Secondary: 65N12.
Citation: Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077
References:
[1]

M. Ainsworth and J. T. Oden, A posteriori error estimators in finite element analysis, Comput. Methods Appl. Mech. Engrg., 142 (1997), 1-88.  doi: 10.1016/S0045-7825(96)01107-3.  Google Scholar

[2]

I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite computations, SIAM J. Numer. Anal., 15 (1978), 736-754.  doi: 10.1137/0715049.  Google Scholar

[3]

R. Becker, S. Mao and Z. Shi, A convergent noncomforming adaptive finite element method with quasi-optimal complexity, SIAM J. Numer. Anal., 47 (2010), 4639-4659.  doi: 10.1137/070701479.  Google Scholar

[4]

P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), 219-268.  doi: 10.1007/s00211-003-0492-7.  Google Scholar

[5]

D. Braess, C. Carstensen and R. H. W. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math., 107 (2007), 455-471.  doi: 10.1007/s00211-007-0098-6.  Google Scholar

[6]

C. Carstensen and R. H. W. Hoppe, Error reduction and convergence for an adaptive mixed finite element method, Math. Comput., 75 (2006), 1033-1042.  doi: 10.1090/S0025-5718-06-01829-1.  Google Scholar

[7]

E. Casas, Error estimations for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM Control Optim. Calc. Var., 8 (2002), 345-374.  doi: 10.1051/cocv:2002049.  Google Scholar

[8]

J. M. Cascon, C. Kreuzer, R. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), 2524-2550.  doi: 10.1137/07069047X.  Google Scholar

[9]

Z. Chen and J. Feng, An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems, Math. Comput., 73 (2004), 1167-1193.  doi: 10.1090/S0025-5718-04-01634-5.  Google Scholar

[10]

H. Chen, X. Gong, L. He and A. Zhou, Adaptive finite element approximations for a class of nonlinear eigenvalue problems in quantum physics, Adv. Appl. Math. Mech., 3 (2011), 493-518.  doi: 10.4208/aamm.10-m1057.  Google Scholar

[11]

Y. Chen and Z. Lu, Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1415-1423.  doi: 10.1016/j.cma.2009.11.009.  Google Scholar

[12] Y. Chen and Z. Lu, High Efficient and Accuracy Numerical Methods for Optimal Control Problems, Science Press, Beijing, 2015.   Google Scholar
[13]

X. Dai, J. Xu and A. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. math., 110 (2008), 313-355.  doi: 10.1007/s00211-008-0169-3.  Google Scholar

[14]

A. Demlow and R. Stevenson, Convergence and quasi-optimality of an adaptive finite element method for controlling $L^2$ errors, Numer. Math., 117 (2011), 185-218.  doi: 10.1007/s00211-010-0349-9.  Google Scholar

[15]

W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054.  Google Scholar

[16]

K. Eriksson and C. Johnson, An adaptive finite element method for linear elliptic problems, Math. Comput., 50 (1988), 361-383.  doi: 10.1090/S0025-5718-1988-0929542-X.  Google Scholar

[17]

A. Gaevskaya, R. H. W. Hoppe, Y. llish and M. Kieweg, Convergence analysis of an adaptive finite element method for distributed control problems with control constrains, Inter. Ser. Numer. Math., 155 (2007), 47-68.  doi: 10.1007/978-3-7643-7721-2_3.  Google Scholar

[18]

L. Ge, W. Liu and D. Yang, $L^2$ norm equivalent a posteriori error for a constraint optimal control problem, Inter. J. Numer. Anal. Model., 6 (2009), 335-353.   Google Scholar

[19]

L. Ge, W. Liu and D. Yang, Adaptive finite element approximation for a constrained optimal control problem via multi-meshes, J. Sci. Comput., 41 (2009), 238-255.  doi: 10.1007/s10915-009-9296-y.  Google Scholar

[20]

W. Gong and N. Yan, Adaptive finite element method for elliptic optimal control problems: Convergence and optimality, Numer. Math., 135 (2017), 1121-1170.  doi: 10.1007/s00211-016-0827-9.  Google Scholar

[21]

W. Gong, N. Yan and Z. Zhou, Convergence of $L^2$-norm based adaptive finite element method for elliptic optimality control problem, arXiv: 1608.08699. Google Scholar

[22]

K. Kohls, K. Siebert and A. Rösch, Convergence of adaptive finite elements for optimal control problems with control constraints, Inter. Ser. Numer. Math., 165 (2014), 403-419.  doi: 10.1007/978-3-319-05083-6_25.  Google Scholar

[23]

H. Leng and Y. Chen, Convergence and quasi-optimality of an adaptive finite element method for optimal control problems on $L^2$-errors, J. Sci. Comput., 73 (2017), 438-458.  doi: 10.1007/s10915-017-0425-8.  Google Scholar

[24]

H. Leng and Y. Chen, Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint, Adv. Comput. Math., 44 (2018), 367-394.  doi: 10.1007/s10444-017-9546-8.  Google Scholar

[25]

R. Li, W. Liu, H. Ma and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim., 41 (2002), 1321-1349.  doi: 10.1137/S0363012901389342.  Google Scholar

[26]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.  Google Scholar

[27] W. Liu and N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008.   Google Scholar
[28]

W. Liu and N. Yan, A posteriori error estimates for control problems governed by nonlinear elliptic equation, Appl. Numer. Math., 47 (2003), 173-187.  doi: 10.1016/S0168-9274(03)00054-0.  Google Scholar

[29]

K. Mekchay and R. H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal., 43 (2005), 1803-1827.  doi: 10.1137/04060929X.  Google Scholar

[30]

P. Morin, R. H. Nochetto and K. G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), 466-488.  doi: 10.1137/S0036142999360044.  Google Scholar

[31]

P. Morin, R. H. Nochetto and K. G. Siebert, Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), 631-658.  doi: 10.1137/S0036144502409093.  Google Scholar

[32]

R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), 245-269.  doi: 10.1007/s10208-005-0183-0.  Google Scholar

show all references

References:
[1]

M. Ainsworth and J. T. Oden, A posteriori error estimators in finite element analysis, Comput. Methods Appl. Mech. Engrg., 142 (1997), 1-88.  doi: 10.1016/S0045-7825(96)01107-3.  Google Scholar

[2]

I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite computations, SIAM J. Numer. Anal., 15 (1978), 736-754.  doi: 10.1137/0715049.  Google Scholar

[3]

R. Becker, S. Mao and Z. Shi, A convergent noncomforming adaptive finite element method with quasi-optimal complexity, SIAM J. Numer. Anal., 47 (2010), 4639-4659.  doi: 10.1137/070701479.  Google Scholar

[4]

P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), 219-268.  doi: 10.1007/s00211-003-0492-7.  Google Scholar

[5]

D. Braess, C. Carstensen and R. H. W. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math., 107 (2007), 455-471.  doi: 10.1007/s00211-007-0098-6.  Google Scholar

[6]

C. Carstensen and R. H. W. Hoppe, Error reduction and convergence for an adaptive mixed finite element method, Math. Comput., 75 (2006), 1033-1042.  doi: 10.1090/S0025-5718-06-01829-1.  Google Scholar

[7]

E. Casas, Error estimations for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM Control Optim. Calc. Var., 8 (2002), 345-374.  doi: 10.1051/cocv:2002049.  Google Scholar

[8]

J. M. Cascon, C. Kreuzer, R. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), 2524-2550.  doi: 10.1137/07069047X.  Google Scholar

[9]

Z. Chen and J. Feng, An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems, Math. Comput., 73 (2004), 1167-1193.  doi: 10.1090/S0025-5718-04-01634-5.  Google Scholar

[10]

H. Chen, X. Gong, L. He and A. Zhou, Adaptive finite element approximations for a class of nonlinear eigenvalue problems in quantum physics, Adv. Appl. Math. Mech., 3 (2011), 493-518.  doi: 10.4208/aamm.10-m1057.  Google Scholar

[11]

Y. Chen and Z. Lu, Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1415-1423.  doi: 10.1016/j.cma.2009.11.009.  Google Scholar

[12] Y. Chen and Z. Lu, High Efficient and Accuracy Numerical Methods for Optimal Control Problems, Science Press, Beijing, 2015.   Google Scholar
[13]

X. Dai, J. Xu and A. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. math., 110 (2008), 313-355.  doi: 10.1007/s00211-008-0169-3.  Google Scholar

[14]

A. Demlow and R. Stevenson, Convergence and quasi-optimality of an adaptive finite element method for controlling $L^2$ errors, Numer. Math., 117 (2011), 185-218.  doi: 10.1007/s00211-010-0349-9.  Google Scholar

[15]

W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054.  Google Scholar

[16]

K. Eriksson and C. Johnson, An adaptive finite element method for linear elliptic problems, Math. Comput., 50 (1988), 361-383.  doi: 10.1090/S0025-5718-1988-0929542-X.  Google Scholar

[17]

A. Gaevskaya, R. H. W. Hoppe, Y. llish and M. Kieweg, Convergence analysis of an adaptive finite element method for distributed control problems with control constrains, Inter. Ser. Numer. Math., 155 (2007), 47-68.  doi: 10.1007/978-3-7643-7721-2_3.  Google Scholar

[18]

L. Ge, W. Liu and D. Yang, $L^2$ norm equivalent a posteriori error for a constraint optimal control problem, Inter. J. Numer. Anal. Model., 6 (2009), 335-353.   Google Scholar

[19]

L. Ge, W. Liu and D. Yang, Adaptive finite element approximation for a constrained optimal control problem via multi-meshes, J. Sci. Comput., 41 (2009), 238-255.  doi: 10.1007/s10915-009-9296-y.  Google Scholar

[20]

W. Gong and N. Yan, Adaptive finite element method for elliptic optimal control problems: Convergence and optimality, Numer. Math., 135 (2017), 1121-1170.  doi: 10.1007/s00211-016-0827-9.  Google Scholar

[21]

W. Gong, N. Yan and Z. Zhou, Convergence of $L^2$-norm based adaptive finite element method for elliptic optimality control problem, arXiv: 1608.08699. Google Scholar

[22]

K. Kohls, K. Siebert and A. Rösch, Convergence of adaptive finite elements for optimal control problems with control constraints, Inter. Ser. Numer. Math., 165 (2014), 403-419.  doi: 10.1007/978-3-319-05083-6_25.  Google Scholar

[23]

H. Leng and Y. Chen, Convergence and quasi-optimality of an adaptive finite element method for optimal control problems on $L^2$-errors, J. Sci. Comput., 73 (2017), 438-458.  doi: 10.1007/s10915-017-0425-8.  Google Scholar

[24]

H. Leng and Y. Chen, Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint, Adv. Comput. Math., 44 (2018), 367-394.  doi: 10.1007/s10444-017-9546-8.  Google Scholar

[25]

R. Li, W. Liu, H. Ma and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim., 41 (2002), 1321-1349.  doi: 10.1137/S0363012901389342.  Google Scholar

[26]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.  Google Scholar

[27] W. Liu and N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008.   Google Scholar
[28]

W. Liu and N. Yan, A posteriori error estimates for control problems governed by nonlinear elliptic equation, Appl. Numer. Math., 47 (2003), 173-187.  doi: 10.1016/S0168-9274(03)00054-0.  Google Scholar

[29]

K. Mekchay and R. H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal., 43 (2005), 1803-1827.  doi: 10.1137/04060929X.  Google Scholar

[30]

P. Morin, R. H. Nochetto and K. G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), 466-488.  doi: 10.1137/S0036142999360044.  Google Scholar

[31]

P. Morin, R. H. Nochetto and K. G. Siebert, Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), 631-658.  doi: 10.1137/S0036144502409093.  Google Scholar

[32]

R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), 245-269.  doi: 10.1007/s10208-005-0183-0.  Google Scholar

Figure 1.  The profiles of the exact state and the numerical state on adaptively refined grids with $ \theta = 0.3 $ and 15 adaptive loops for Example 1
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Figure 2.  The profiles of the exact state, the numerical state, the exact co-state variables and the co-state variables on adaptively refined grids with $ \theta = 0.3 $ and 15 adaptive loops for Example 1
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Figure 3.  The adaptive girds after 5 steps and 13 steps with $ \theta = 0.3 $ and 15 adaptive loops for Example 1 generated by Algorithm 3.1
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Figure 4.  The profiles of the numerical state and the co-state variables on uniformly refined grids ($ \theta = 1 $) and 15 adaptive loops for Example 1
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Figure 5.  The adaptive girds after 5 steps with $ \theta = 0.3 $ and the uniform refinement ($ \theta = 1 $) after 5 steps for Example 1 generated by Algorithm 3.1
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Figure 6.  The error estimations of adaptively refined grids with $ \theta = 0.3 $, $ \theta = 0.4 $, and the error estimates of uniformly refined grids for Example 1
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Figure 7.  The profiles of the numerical state and the co-state on adaptively refined grids with $ \theta = 0.3 $ and 27 adaptive loops for Example 2
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Figure 8.  The adaptive girds after 15 steps and 25 steps with $ \theta = 0.3 $ for Example 2 generated by Algorithm 3.1
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Figure 9.  The error estimations of adaptively refined grids with $ \theta = 0.3 $ and the error estimations of uniformly refined grids for Example 2
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