We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a nonintrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap postprocessing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence toward the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.


  1. Maxwell's equations
  2. time-harmonic scattering
  3. exact controllability
  4. discontinuous Galerkin

MSC codes

  1. 65N30
  2. 78M10

Get full access to this article

View all available purchase options and get full access to this article.


R. Adams and J. Fournier, Sobolev Spaces, Academic Press, New York, 2003.
P. R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. Y. L'Excellent, and C. Weisbecker, Improving multifrontal methods by means of block low-rank reresentations, SIAM J. Sci. Comput., 37 (2015), pp. A1451--A1474.
P. R. Amestoy, I. S. Duff, and J. Y. L'Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Engrg., 184 (2000), pp. 501--520.
D. Appelö, F. Garcia, and O. Runborg, WaveHoltz: Iterative solution of the Helmholtz equation via the wave equation, SIAM J. Sci. Comput., 42 (2020), pp. A1950--A1983.
F. Assous, P. Ciarlet, and S. Labrunie, Mathematical Foundations of Computational Electromagnetism, Springer, New York, 2018.
C. Bardos and J. Rauch, Variational algorithms for the Helmholtz equation using time evolution and artificial boundaries, Asymptot. Anal., 9 (1994), pp. 101--117.
M. Bonazzoli, V. Dolean, I. G. Graham, E. A. Spence, and P. H. Tournier, Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption, Math. Comp., 88 (2019), pp. 2559--2604.
M. O. Bristeau, R. Glowinski, and J. Periaux, Controllability methods for the computation of time-periodic solutions; application to scattering, J. Comput. Phys., 147 (1998), pp. 265--292.
M. O. Bristeau, R. Glowinski, and J. Periaux, On the numerical solution of the Helmholtz equation at large wave numbers using exact controllability methods. Application to scattering, Contemp. Math., 157 (1994), pp. 399--419.
M.-O. Bristeau, R. Glowinski, J. Périaux, and T. Rossi, 3D Harmonic Maxwell Solutions on Vector and Parallel Computers Using Controllability and Finite Element Methods, Research Report RR-3607, INRIA, 1999, https://hal.inria.fr/inria-00073072.
T. Chaumont-Frelet and S. Nicaise, Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems, IMA J. Numer. Anal., 40 (2020), pp. 1503--1543.
P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge, 1989.
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 2012.
O. G. Ernst and M.J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, in Numerical Analysis of Multiscale Problems, Springer, New York, 2012, pp. 325--363.
P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Methods Appl. Sci., 47 (1997), pp. 2872--2896.
L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, ESAIM Math. Model. Numer. Anal., 39 (2005), pp. 1149--1176.
M. J. Gander, I. G. Graham, and E. A. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed?, Numer. Math., 131 (2015), pp. 567--614.
C. Geuzaine and J. F. Remacle, Gmsh: A 3D finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79 (2009), pp. 1309--1331.
V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, 1986.
I. G. Graham, O. R. Pembery, and E. A. Spence, The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness and resonances, J. Differential Equations, 266 (2019), pp. 2869--2923.
D. J. Griffiths, Introduction to Electrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1999.
M. J. Grote, M. Mehlin, and T. Mitkova, Runge--Kutta-based explicit local time-stepping methods for wave propagation, SIAM J. Sci. Comput., 37 (2015), pp. A747--A775.
M. J. Grote, F. Nataf, J. H. Tang, and P. H. Tournier, Parallel controllability methods for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 362 (2020), 112846.
M. J. Grote, A. Schneebeli, and D. Schötzau, Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm erro estimates, J. Comput. Appl. Math., 204 (2007), pp. 375--386.
M. J. Grote and J. H. Tang, On controllability methods for the Helmholtz equation, J. Comput. Appl. Math., 358 (2019), pp. 306--326.
E. Heikkola, S. Mönkölä, A. Pennanen, and T. Rossi, Controllability method for acoustic scattering with spectral elements, J. Comput. Appl. Math., 204 (2007), pp. 344--355.
E. Heikkola, S. Mönkölä, A. Pennanen, and T. Rossi, Controllability method for the Helmholtz equation with higher-order discretizations, J. Comput. Phys., 225 (2007), pp. 1553--1576.
J. S. Hesthaven and T. Warburton, Nodal high-order methods on unstructured grids. Part I. Time-domain solution of Maxwell's equations, J. Comput. Phys., 181 (2002), pp. 1266--1288.
R. Hiptmair, A. Moiola, and I. Perugia, Stability results for the time-harmonic Maxwell equations with impedance boundary conditions, Math. Methods Appl. Sci., 21 (2010), pp. 2263--2287.
S. Kähkönen, R. Glowinski, T. Rossi, and R. A. Mäkinen, Solution of time-periodic wave equation using mixed finite elements and controllability techniques, J. Comput. Acoust., 19 (2011), pp. 335--352.
L. Li, S. Lanteri, and R. Perrussel, A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3d time-harmonic Maxwell's equations, J. Comput. Phys., 256 (2014), pp. 563--581.
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), pp. 1--68.
J. M. Melenk and S. Sauter, Wavenumber explicit $hp$-FEM analysis of Maxwell's equations with transparent boundary conditions, Found. Comput. Math., 49 (2020), pp. 1210--1243.
A. Moiola and E. A. Spence, Electromagnetic transmission problems: Wavenumber-explicit bounds, presented at MAFELAP, 2019.
P. Monk, Finite Element Methods for Maxwell's Equations, Oxford Science Publications, Oxford, UK, 2003.
C. S. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math., 15 (1962), pp. 349--361.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
Z. Peng and D. Appelö, EM-WaveHoltz: A Flexible Frequency-Domain Method Built from Time-Domain Solvers, arXiv [math.NA] 2103.14789, 2021.
J. Räbinä, S. Mönkölä, and T. Rossi, Efficient time integration of Maxwell's equations with generalized finite differences, SIAM J. Sci. Comput., 37 (2015), pp. B834--B854.
A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite-Difference Time-Domain Method, Artech House, London, 2005.
J. H. Tang, Solving Forward and Inverse Helmholtz Equations via Controllability Methods, Ph.D. thesis, Universität Basel, 2020.
P. Tsuji, B. Engquist, and L. Ying, A sweeping preconditioner for time-harmonic Maxwell's equations with finite elements, J. Comput. Phys., 231 (2012), pp. 3770--3783.
K. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propag., 16 (1966), pp. 302--307.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A3700 - A3727
ISSN (online): 1095-7197


Submitted: 3 June 2021
Accepted: 21 June 2022
Published online: 8 December 2022


  1. Maxwell's equations
  2. time-harmonic scattering
  3. exact controllability
  4. discontinuous Galerkin

MSC codes

  1. 65N30
  2. 78M10



Funding Information

Centre National de la Recherche Scientifique https://doi.org/10.13039/501100004794 : UMR CNRS 6621

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.