The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of linear systems with a block structure. Correspondingly, if the solution method is a non-overlapping domain decomposition method, we need to solve interface problems which exhibit a block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners.
In this talk we describe a technique which employs a preconditioner block structure based on the fractional Sobolev norms corresponding to the domains of the boundary operators arising in the matrix interface problem, some of which may include a dependence on the control regularization parameter. We illustrate our approach on standard elliptic control problems. We present analysis which shows that the resulting iterative method converges independently of the size of the problem. We include numerical results which indicate that performance is also independent of the control regularization parameter and exhibits only a mild dependence on the number of the subdomains.