Published by De Gruyter February 18, 2023

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

Ulrich Langer , Richard Löscher , Olaf Steinbach and Huidong Yang

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2022-0138

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Abstract

We consider standard tracking-type, distributed elliptic optimal control problems with L 2 regularization, and their finite element discretization. We are investigating the L 2 error between the finite element approximation u ϱ ⁢ h of the state u ϱ and the desired state (target) u ¯ in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice ϱ = 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.

Keywords: Elliptic Optimal Control Problems; Finite Element Discretization; Robust Error Estimates; Robust Solvers

MSC 2010: 49J20; 49M05; 35J05; 65M60; 65M15; 65N22

Acknowledgements

The authors would like to acknowledge the computing support of the supercomputer MACH-2 (https://www3.risc.jku.at/projects/mach2/) from Johannes Kepler Universität Linz and of the high performance computing cluster Radon1 (https://www.oeaw.ac.at/ricam/hpc) from Johann Radon Institute for Computational and Applied Mathematics (RICAM) on which the numerical examples are performed.

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Received: 2022-07-09

Revised: 2022-12-08

Accepted: 2023-01-24

Published Online: 2023-02-18

Published in Print: 2023-10-01

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

Abstract

Acknowledgements

References

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