Geodesic
Finite-difference approximation of dirichlet observation problems for weak solutions to the wave equation subject to Robin boundary conditions
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 8, pp. 1323-1339

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For the wave equation with variable coefficients subject to Neumann and Robin boundary conditions, two mutually dual problems are considered: the Dirichlet observation problem with weak generalized solutions and the control problem with strong generalized solutions. Both problems are approximated by finite differences preserving the duality relation. The convergence of the approximate solutions is established in the norms of the corresponding dual spaces. 
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     author = {M. M. Potapov},
     title = {Finite-difference approximation of dirichlet observation problems for weak solutions to the wave equation subject to {Robin} boundary conditions},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1323--1339},
     publisher = {mathdoc},
     volume = {47},
     number = {8},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_8_a4/}
}
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M. M. Potapov. Finite-difference approximation of dirichlet observation problems for weak solutions to the wave equation subject to Robin boundary conditions. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 8, pp. 1323-1339. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_8_a4/