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We consider a linear-quadratic elliptic control problem (LQECP) where 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 system (the discrete Hessian system) resulting from the application of the Schur complement method to the discrete LQECP. In order to derive robust preconditioners with respect to stabilization parameters, we first show that the continuous reduced Hessian operator and the corresponding discrete Hessian matrix are associated to a linear combination of fractional negative Sobolev norms. Then we propose a preconditioner based on multilevel methods, including cases where the stabilization parameters are set equal to zero. We also present numerical experiments which agree with the theoretical results.

Analysis of Robust Parameter-Free Multilevel Methods for Neumann Boundary Control Problems