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

• Published in: Journal of computational methods in applied mathematics Vol. 23; no. 4; pp. 989 - 1005
• Main Authors: Langer, Ulrich, Löscher, Richard, Steinbach, Olaf, Yang, Huidong
• Format: Journal Article
• Language: English
• Published: Minsk De Gruyter 01.10.2023; Walter de Gruyter GmbH
• Subjects: 35J05; 49J20; 49M05; 65M15; 65M60; 65N22; Discretization; Elliptic Optimal Control Problems; Error analysis; Finite Element Discretization; Mathematical analysis; Optimal control; Optimization; Parameters; Regularization; Robust Error Estimates; Robust Solvers
• ISSN: 1609-4840; 1609-9389
• DOI: 10.1515/cmam-2022-0138

We consider standard tracking-type, distributed elliptic optimal control problems with regularization, and their finite element discretization. We are investigating the error between the finite element approximation of the state and the desired state (target) in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice . 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.