A paradox in the approximation of dirichlet control problems in curved domains

Abstract

In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain O. To solve this problem numerically, it is usually necessary to approximate O by a (typically polygonal) new domain Oh. The difference between the solutions of both infinite-dimensional control problems, one formulated in O and the second in Oh, 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 O was considered. The authors used a finite element method such that Oh 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 O to Oh.