Geodesic
A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 63-76

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In this paper, we discuss a priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems. The state variables and co-state variables are approximated by a $P_0^2-P_1$ (velocity-pressure) pair and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, the state variables and the co-state variables. Then we obtain a superconvergence result for the control variable by using postprocessing projection operator. 
@article{SJVM_2021_24_1_a5,
     author = {C. Xu},
     title = {A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {63--76},
     publisher = {mathdoc},
     volume = {24},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a5/}
}
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C. Xu. A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 63-76. http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a5/