Preconditioners for Time-Harmonic Optimal Control Eddy-Current Problems

  • Owe Axelsson
  • Dalibor LukášEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10665)


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.


Preconditioning Krylov subspace methods Optimal control Eddy currents Time-harmonic 


  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Nédélec, J.C.: Mixed finite elements in \(\mathbb{R}^3\). Numer. Math. 35, 315–341 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Axelsson, O., Layton, W.: A two-level method for the discretization of nonlinear boundary value problems. SIAM J. Numer. Anal. 33, 2359–2374 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kollmann, M., Kolmbauer, M.: A preconditioned MinRes solver for time-periodic parabolic optimal control problems. Numer. Linear Algebra Appl. 20, 761–784 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Axelsson, O., Lukáš, D.: Preconditioning methods for eddy current optimally controlled time-harmonic electromagnetic problems. J. Numer. Math. (to appear)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute of GeonicsCzech Academy of SciencesOstrava-PorubaCzech Republic
  2. 2.VŠB-Technical University of OstravaOstrava-PorubaCzech Republic

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