Description:
We consider a linear-quadratic elliptic control problem (LQECP). For the problem we consider here, the control variable corresponds to the Neumann data on the boundary of a convex polygonal domain. The optimal control unknown is the one for which the harmonic extension approximates best a specified target in the interior of the domain. We propose multilevel preconditioners for the reduced Hes- sian resulting from the application of the Schur complement method to the discrete LQECP. In order to derive robust stabilization parameters-free preconditioners, we first show that the Schur complement matrix is associated to a linear combination of negative Sobolev norms and then propose preconditioner based on multilevel methods.We also present numerical experiments which agree with the theoretical results.