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

Author:

Langer Ulrich1ORCID,Löscher Richard2,Steinbach Olaf2ORCID,Yang Huidong3

Affiliation:

1. Institute of Computational Mathematics , Johannes Kepler University Linz , Altenberger Straße 69, 4040 Linz , Austria

2. Institut für Angewandte Mathematik , Technische Universität Graz , Steyrergasse 30, 8010 Graz , Austria

3. Computational Science Center , Universität Wien , Oskar-Morgenstern-Platz 1, 1090 Wien , Austria

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 approximationuϱhu_{\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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Computational Mathematics,Numerical Analysis

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Recent Advances in Finite Element Methods;Computational Methods in Applied Mathematics;2023-07-25

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