Geodesic
Mixed methods for optimal control problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 3, pp. 333-343

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In this paper, we investigate a posteriori error estimates of a mixed finite element method for elliptic optimal control problems with an integral constraint. The gradient for our method belongs to the square integrable space instead of the classical $H(\mathrm{div};\Omega)$ space. The state and co-state are approximated by the $P^2_0$-$P_1$ (velocity-pressure) pair, and the control variable is approximated by piecewise constant functions. Using a duality argument method and an energy method, we derive residual a posteriori error estimates for all variables. 
@article{SJVM_2018_21_3_a6,
     author = {T. Hou},
     title = {Mixed methods for optimal control problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {333--343},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2018_21_3_a6/}
}
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T. Hou. Mixed methods for optimal control problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 3, pp. 333-343. http://geodesic.mathdoc.fr/item/SJVM_2018_21_3_a6/