Block preconditioners for elliptic PDE-constrained optimization problems

For the structured systems of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems, we construct block-counter-diagonal and block-counter-tridiagonal preconditioning matrices to precondition the Krylov subspace methods such as GMRES. We derive explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices. Numerical implementations show that these structured preconditioners may lead to satisfactory experimental results of the preconditioned GMRES methods when the regularization parameter is suitably small.