Robust Parameter-Free Multilevel Methods for Neumann Boundary Control Problems

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 Hessian 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.