@article{10902/1638,
year = {2011},
url = {http://hdl.handle.net/10902/1638},
abstract = {In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain Ω. To solve this problem numerically, it is usually necessary to approximate Ω by a (typically polygonal) new domain Ωh. The difference between the solutions of both infinite-dimensional control problems, one formulated in Ω and the second in Ωh, was studied in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746–3780], where an error of order O(h) was proved. In [K. Deckelnick, A. G¨unther, and M. Hinze, SIAM J. Control Optim., 48 (2009), pp. 2798–2819], the numerical approximation of the problem defined in Ω was considered. The authors used a finite element method such that Ωh was the polygon formed by the union of all triangles of the mesh of parameter h. They proved an error of order O(h3/2) for the difference between continuous and discrete optimal controls. Here we show that the estimate obtained in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746–3780] cannot be improved, which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from Ω to Ωh.},
publisher = {Society for Industrial and Applied Mathematics},
publisher = {Siam Journal on Control and Optimization, 2011, 49(5), 1998-2007},
title = {A paradox in the approximation of Dirichlet control problems in curved domains},
author = {Casas Rentería, Eduardo and Günther, Andreas and Mateos Alberdi, Mariano},
}