Mass-lumping discretization and solvers for distributed elliptic optimal control problems

The purpose of this paper is to investigate the effects of the use of mass-lumping in the finite element discretization of the reduced first-order optimality system arising from a standard tracking-type, distributed elliptic optimal control problem with $L_2$ regularization. We show that mass-lumping will not affect the $L_2$ error between the desired state and the computed state, but will lead to a Schur-complement system that allows for a fast matrix-by-vector multiplication. We show that the use of the Schur-Complement Preconditioned Conjugate Gradient method in a nested iteration setting leads to an asymptotically optimal solver with respect to the complexity.