Abstract
Time-harmonic formulations enable solution of time-dependent PDEs without use of normally slow time-stepping methods. Two efficient preconditioners for the discretized parabolic and eddy current electromagnetic optimal control problems, one on block diagonal form and one utilizing the two by two block structure of the resulting matrix, are presented with simplified analysis and numerical illustrations. Both methods result in tight eigenvalue bounds for the preconditioned matrix and very few iterations that hold uniformly with respect to the mesh, problem and method parameters, with the exception of the dependence on reluctivity for the block diagonal preconditioner.
This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations excellence in science - LQ1602”. The second author was supported by the Czech Science Foundation under the project 17-22615S.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Kolmbauer, M., Langer, U.: A robust preconditioned MINRES solver for distributed time-periodic eddy current optimal control problems. SIAM J. Sci. Comput. 34, B785–B809 (2012)
Axelsson, O., Farouq, S., Neytcheva, M.: A preconditioner for optimal control problems constrained by Stokes equation with a time-harmonic control. J. Comp. Appl. Math. 310, 5–18 (2017)
Nédélec, J.C.: Mixed finite elements in \(\mathbb{R}^3\). Numer. Math. 35, 315–341 (1980)
Axelsson, O., Layton, W.: A two-level method for the discretization of nonlinear boundary value problems. SIAM J. Numer. Anal. 33, 2359–2374 (1996)
Kollmann, M., Kolmbauer, M.: A preconditioned MinRes solver for time-periodic parabolic optimal control problems. Numer. Linear Algebra Appl. 20, 761–784 (2013)
Kolmbauer, M.: The Multiharmonic finite element and boundary element method for simulation and control of eddy current problems. Ph.D. thesis, Johannes Kepler Universität, Linz, Austria (2012)
Axelsson, O., Lukáš, D.: Preconditioning methods for eddy current optimally controlled time-harmonic electromagnetic problems. J. Numer. Math. (to appear)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Axelsson, O., Lukáš, D. (2018). Preconditioners for Time-Harmonic Optimal Control Eddy-Current Problems. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science(), vol 10665. Springer, Cham. https://doi.org/10.1007/978-3-319-73441-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-73441-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-73440-8
Online ISBN: 978-3-319-73441-5
eBook Packages: Computer ScienceComputer Science (R0)