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In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain $\Omega$. To solve this problem numerically, it is usually necessary to approximate $\Omega$ by a (typically polygonal) new domain $\Omega_h$. The difference between the solutions of both infinite-dimensional control problems, one formulated in $\Omega$ and the second in $\Omega_h$, 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. Günther, and M. Hinze, SIAM J. Control Optim., 48 (2009), pp. 2798-2819], the numerical approximation of the problem defined in $\Omega$ was considered. The authors used a finite element method such that $\Omega_h$ was the polygon formed by the union of all triangles of the mesh of parameter $h$. They proved an error of order $O(h^{3/2})$ 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 $\Omega$ to $\Omega_h$.

A Paradox in the Approximation of Dirichlet Control Problems in Curved Domains.