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

Abstract

The purpose of this article is to investigate the effects of the use of mass‐lumping in the finite element discretization with mesh size of the reduced first‐order optimality system arising from a standard tracking‐type, distributed elliptic optimal control problem with regularization, involving a regularization (cost) parameter on which the solution depends. We show that mass‐lumping will not affect the error between the desired state and the computed finite element 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. While the proposed approach is applicable independently of the regularity of the given target, our particular interest is in discontinuous desired states that do not belong to the state space. However, the corresponding control belongs to whereas the cost as . This motivates to use in order to balance the error and the maximal costs we are willing to accept. This can be embedded into a nested iteration process on a sequence of refined finite element meshes in order to control the error and the cost simultaneously.