A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. (Q2839905)

The authors consider the optimal control subject to partial differential equations (PDE) constraints. Their focus is on the solution of the discretized first-order conditions which are of saddle point form. The PDE-constraint considered is given by the Poisson equation. The authors introduce various general preconditioners well-suited to precondition linear systems in saddle point form. They then introduce a new Schur-complement approximation as an extension of the work presented in an article by \textit{T. Rees} et al. [SIAM J. Sci. Comput. 32, No. 1, 271--298 (2010; Zbl 1208.49035)]. The authors construct their approximation in such a way that more terms of the original Schur-complement are matched in contrast to previously existing results. Then, using some basic facts about finite element matrices, they prove that the eigenvalues of the preconditioned Schur-complement are bounded in a tight interval independently of the mesh and regularization parameters. The authors proceed to perform numerical experiments indicating that the new approximation is more robust with respect to the regularization parameter.