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A robust adaptive grid method for singularly perturbed Burger-Huxley equations
Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
W. Dörfler,
A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.
doi: 10.1137/0733054. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
W. Dörfler,
A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.
doi: 10.1137/0733054. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. |
[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. |
[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. |
[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. |
[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. |
[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. |









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