Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems

Abstract

AbstractWe consider standard tracking-type, distributed elliptic optimal control problems withL2L^{2}regularization, and their finite element discretization. We are investigating theL2L^{2}error between the finite element approximationuϱ⁢hu_{\varrho h}of the stateuϱu_{\varrho}and the desired state (target)u¯\overline{u}in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choiceϱ=h4\varrho=h^{4}. It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.