Journal of Optimization Theory and Applications

, Volume 154, Issue 3, pp 879–903 | Cite as

Finite Element Analysis of an Optimal Control Problem in the Coefficients of Time-Harmonic Eddy Current Equations

  • Irwin YouseptEmail author


This paper is concerned with an optimal control problem governed by time-harmonic eddy current equations on a Lipschitz polyhedral domain. The controls are given by scalar functions entering in the coefficients of the curl-curl differential operator in the state equation. We present a mathematical analysis of the optimal control problem, including sensitivity analysis, regularity results, existence of an optimal control, and optimality conditions. Based on these results, we study the finite element analysis of the optimal control problem. Here, the state is discretized by the lowest order edge elements of Nédélec’s first family, and the control is discretized by continuous piecewise linear elements. Our main findings are convergence results of the finite element discretization (without a rate).


Optimal control with PDEs Eddy current equations Control in coefficients Nédélec’s curl-conforming edge elements 


  1. 1.
    Albanese, R., Monk, P.: The inverse source problem for Maxwell’s equations. Inverse Probl. 22, 1023–1035 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput. 68, 607–631 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alonso, A., Valli, A.: Eddy Current Approximation of Maxwell Equations: Theory, Algorithms and Applications. Springer, New York (2010) zbMATHGoogle Scholar
  4. 4.
    Ammari, H., Buffa, A., Nédélec, J.-C.: A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60, 1805–1823 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Beck, R., Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B.: Residual based a posteriori error estimators for eddy current computation. Modél. Math. Anal. Numér. 34, 159–182 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bossavit, A.: Computational Electromagnetism. Academic Press, San Diego (1998) zbMATHGoogle Scholar
  7. 7.
    Chen, Z., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37, 1542–1570 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ciarlet, P. Jr., Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82, 193–219 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin (1998) zbMATHGoogle Scholar
  10. 10.
    Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. Modél. Math. Anal. Numér. 33, 627–649 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Costabel, M., Dauge, M., Nicaise, S.: Singularities of eddy current problems. Modél. Math. Anal. Numér. 37, 807–831 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hiptmair, R.: Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36, 204–225 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hoppe, R.H.W.: Adaptive multigrid and domain decomposition methods in the computation of electromagnetic fields. J. Comput. Appl. Math. 168, 245–254 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hoppe, R.H.W., Schöberl, J.: Convergence of adaptive edge element methods for the 3D eddy currents equations. J. Comput. Math. 27, 657–676 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Monk, P.: Analysis of a finite element method for Maxwell’s equations. SIAM J. Numer. Anal. 29, 714–729 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon, Oxford (2003) zbMATHCrossRefGoogle Scholar
  18. 18.
    Nédélec, J.-C.: Mixed finite elements in ℝ3. Numer. Math. 35, 315–341 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Nédélec, J.-C.: A new family of mixed finite elements in ℝ3. Numer. Math. 50, 57–81 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Nédélec, J.-C.: Acoustic and Electromagnetic Equations. Springer, New York (2001) zbMATHGoogle Scholar
  21. 21.
    Neittaanmäki, P., Picard, R.: Error estimates for the finite element approximation to a Maxwell-type boundary value problem. Numer. Funct. Anal. Optim. 2, 267–285 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Neittaanmäki, P., Rudnicki, M., Savini, A.: Inverse Problems and Optimal Design in Electricity and Magnetism. Oxford University Press, Oxford (1996) Google Scholar
  23. 23.
    Romanov, V.G., Kabanikhin, S.I.: Inverse Problems for Maxwell’s Equations. VSP International Science Publishers, Utrecht (1994) zbMATHGoogle Scholar
  24. 24.
    Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Birkhäuser Boston, Cambridge (1989) zbMATHCrossRefGoogle Scholar
  25. 25.
    Chen, Z., Zou, J.: An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems. SIAM J. Control Optim. 37, 892–910 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Falk, R.S.: Error estimates for the numerical identification of a variable coefficient. Math. Comput. 40, 537–546 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Ito, K., Kunisch, K.: The augmented Lagrangian method for parameter estimation in elliptic systems. SIAM J. Control Optim. 28, 113–136 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008) zbMATHCrossRefGoogle Scholar
  29. 29.
    Rannacher, R., Vexler, B.: A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements. SIAM J. Control Optim. 44, 1844–1863 (2005) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Casas, E.: Optimal control in coefficients of elliptic equations with state constraints. Appl. Math. Optim. 26, 21–37 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Druet, P.-E., Klein, O., Sprekels, J., Tröltzsch, F., Yousept, I.: Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects. SIAM J. Control Optim. 49, 1707–1736 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Tröltzsch, F., Yousept, I.: PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages. ESAIM: Math. Model. Numer. Anal. 46, 709–729 (2012) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Yousept, I.: Optimal control of a nonlinear coupled electromagnetic induction heating system with pointwise state constraints. Ann. Acad. Romanian Sci. Ser. Math. Appl. 2, 45–77 (2010) MathSciNetzbMATHGoogle Scholar
  34. 34.
    Yousept, I.: Optimal control of Maxwell’s equations with regularized state constraints. Comput. Optim. Appl. (2011). doi: 10.1007/s10589-011-9422-2 zbMATHGoogle Scholar
  35. 35.
    Kolmbauer, M., Langer, U.: A robust preconditioned-MinRes-solver for distributed time-periodic eddy current optimal control problems. DK-report No. 2011-07, JKU, Linz (2011) Google Scholar
  36. 36.
    Wellander, N., Kristensson, G.: Homogenization of the Maxwell equations at fixed frequency. SIAM J. Appl. Math. 64, 170–195 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Adams, R.A.: Sobolev Spaces. Academic Press, Boston (1978) Google Scholar
  38. 38.
    Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Am. Math. Soc., Providence (2010) zbMATHGoogle Scholar
  40. 40.
    Carstensen, C.: Quasi-interpolation and a posteriori error analysis in finite element method. Modél. Math. Anal. Numér. 33, 1187–1202 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    De Los Reyes, J.C., Meyer, C., Vexler, B.: Finite element error analysis for state-constrained optimal control of the stokes equations. Control Cybern. 37, 251–284 (2008) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations