Geodesic
Superconvergence and a~posteriori error estimates of RT1 mixed methods for elliptic control problems with an integral constraint
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 2, pp. 185-199

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In this paper, we investigate the superconvergence property and a posteriori error estimates of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by order $k=1$ Raviart–Thomas mixed finite element spaces, and the control variable is approximated by piecewise constant functions. Approximations of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order $h^2$. Moreover, we derive a posteriori error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results. 
@article{SJVM_2013_16_2_a7,
     author = {T. Hou},
     title = {Superconvergence and a~posteriori error estimates of {RT1} mixed methods for elliptic control problems with an integral constraint},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {185--199},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2013_16_2_a7/}
}
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T. Hou. Superconvergence and a~posteriori error estimates of RT1 mixed methods for elliptic control problems with an integral constraint. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 2, pp. 185-199. http://geodesic.mathdoc.fr/item/SJVM_2013_16_2_a7/