Time-harmonic problems arise in many important applications, such as eddy current optimally controlled electromagnetic problems. Eddy current modelling can also be used in non-destructive testings of conducting materials. Using a truncated Fourier series to approximate the solution, for linear problems the equation for different frequencies separate, so it suffices to study solution methods for the problem for a single frequency.
The arising discretized system takes a two-by-two or four-by-four block matrix form. Since the problems are in general three-dimensional in space and hence of very large scale, one must use an iterative solution method. It is then crucial to construct efficient preconditioners.
It is shown that an earlier used preconditioner for optimal control problems is applicable here also and leads to very tight eigenvalue bounds and hence very fast convergence such as for a Krylov subspace iterative solution method. A comparison is done with an earlier used block diagonal preconditioner.
The authors gratefully acknowledge a comment by professor Yuri Kuznetsov which improved the presentation of this paper. We also acknowledge helpful comments by two reviewers.
Funding The first author was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations of excellence in science – LQ1602”. The second author was supported by the Czech Science Foundation under the project 17-22615S. In this work we used the IT4Innovations infrastructure which is supported by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center – LM2015070”.
 O. Axelsson and W. Layton. A two-level method for the discretization of nonlinear boundary value problems. SIAM J. Numer. Anal. 33 (1996), 2359–2374.10.1137/S0036142993247104 Search in Google Scholar
 O. Axelsson, S. Farouq, and M. Neytcheva, Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Poisson and convection–diffusion control. Numerical Algorithms, 73 (2016), 631–663.10.1007/s11075-016-0111-1 Search in Google Scholar
 O. Axelsson, S. Farouq, and M. Neytcheva. Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Stokes control. Numerical Algorithms, 74 (2017), 19–37.10.1007/s11075-016-0136-5 Search in Google Scholar
 O. Axelsson, S. Farouq, and M. Neytcheva. A preconditioner for optimal control problems constrained by Stokes equation with a time-harmonic control. J. Comp. Appl. Math. 310 (2017), 5–18.10.1016/j.cam.2016.05.029 Search in Google Scholar
 O. Axelsson and D. Lukáš, Preconditioners for time-harmonic optimal control eddy-current problems. In: Large-Scale Scientific Computing (Eds. I. Lirkov and S. Margenov), Lecture Notes in Computer Science, Vol. 10665. Springer, Cham, 2018, pp. 47–54. Search in Google Scholar
 O. Axelsson and A. Kucherov, Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7 (2000), 197–218.10.1002/1099-1506(200005)7:4<197::AID-NLA194>3.0.CO;2-S Search in Google Scholar
 O. Axelsson, M. Neytcheva, and B. Ahmad, A comparison of iterative methods to solve complex valued linear algebraic systems. Numerical Algorithms, 66 (2014), 811–841.10.1007/s11075-013-9764-1 Search in Google Scholar
 F. Bachinger, U. Langer, and J. Schöberl, Efficient solvers for nonlinear time-periodic eddy current problems. Comput. Vis. Sci. 9 (2006), 197–207.10.1007/s00791-006-0023-z Search in Google Scholar
 N. Ida, Numerical Modeling for Electromagnetic Non-destructive Evaluation. Engineering NDE. Chapman & Halls, London, 1995. Search in Google Scholar
 M. Kollmann and M. Kolmbauer, A preconditioned MinRes solver for time-periodic parabolic optimal control problems. Numer. Linear Algebra Appl. 20 (2013), 761–784.10.1002/nla.1842 Search in Google Scholar
 M. Kolmbauer and U. Langer, A robust preconditioned MINRES solver for distributed time-periodic eddy current optimal control problems. SIAM J. Sci. Comput. 34 (2012), B785–B809.10.1137/110842533 Search in Google Scholar
 M. Kolmbauer, 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. Search in Google Scholar
 Y. Notay, An aggregation-based algebraic multigrid method. Electron. Trans. Numer. Anal. 37 (2010), 123–146. Search in Google Scholar
 F. Tröltzsch, Optimal control of partial differential equations: theory, methods and applications. Grad. Stud. Math. Vol. 112, American Mathematical Society, Providence, RI, 2010. Search in Google Scholar
 P. S. Vassilevski, Multilevel Block Factorization Preconditioners. Matrix-based Analysis and Algorithms for Solving Finite Element Equations. Springer, New York, 2008. Search in Google Scholar
 S. Yamada and K. Bessho, Harmonic field calculation by the combination of finite element analysis and harmonic balance method. IEEE Trans. Magnetics, 24 (1988), 2588–2590.10.1109/20.92182 Search in Google Scholar
 I. Yousept, Optimal control of quasilinear H(curl)-elliptic partial differential equations in magnetostatic field problems. SIAM J. Control Optim. 51 (2013), 3624–3651.10.1137/120904299 Search in Google Scholar
 I. Yousept, Optimal control of non-smooth hyperbolic evolution Maxwell equations in type-II superconductivity. SIAM J. Control Optim. 55 (2017), 2305–2332.10.1137/16M1074229 Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston