Geodesic
Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 4, pp. 425-440

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In this paper, we investigate $L^\infty(L_2)$-error estimates and superconvergence of semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by order $k$ Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order $k$ ($k\ge0$). We derive error estimates for both the state and the control approximation. Moreover, we present superconvergence analysis for mixed finite element approximation of the optimal control problems. 
@article{SJVM_2012_15_4_a7,
     author = {T. Hou},
     title = {Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {425--440},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2012_15_4_a7/}
}
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T. Hou. Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 4, pp. 425-440. http://geodesic.mathdoc.fr/item/SJVM_2012_15_4_a7/