Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements. (English) Zbl 1180.78048

This paper deals with the numerical analysis of a class of time-harmonic Maxwell equations in \(3D\) polyhedral domains. The main contribution in this study is that the authors propose a constrained formulation which is obtained by adding a constraint on the divergence of the field. In the first part of the paper the authors recall the time-harmonic Maxwell equations, which are expressed as a set of second-order partial differential equations. Next, it is introduced the functional framework and it is developed the continuous variational formulation of the problem. The authors also prove the convergence of the discretized eigenmodes towards the exact eigenmodes. In the last section of the paper there are proposed some numerical examples to illustrate the behavior of the method.


78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
35Q60 PDEs in connection with optics and electromagnetic theory
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
Full Text: DOI


[1] Amrouche C., Bernardi C., Dauge M., Girault V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998) · Zbl 0914.35094
[2] Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1999) · Zbl 0934.65121
[3] Assous F., Ciarlet P. Jr, Garcia E., Segré J.: Time-dependent Maxwell’s equations with charges in singular geometries. Comput. Methods Appl. Mech. Eng. 196, 665–681 (2006) · Zbl 1121.78305
[4] Assous F., Ciarlet P. Jr, Segré J.: Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method. J. Comput. Phys. 161, 218–249 (2000) · Zbl 1007.78014
[5] Assous F., Ciarlet P. Jr, Sonnendrücker E.: Resolution of the Maxwell equations in a domain with reentrant corners. Modél. Math. Anal. Numér. 32, 359–389 (1998) · Zbl 0924.65111
[6] Assous F., Degond P., Heintzé E., Raviart P.-A., Segré J.: On a finite element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109, 222–237 (1993) · Zbl 0795.65087
[7] Babuska, I., Osborn, J.E.: Eigenvalue problems. In: Handbook of Numerical Analysis, vol. II, pp. 641–787. North Holland, Amsterdam (1991) · Zbl 0875.65087
[8] Boffi D.: Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34, 664–670 (1997) · Zbl 0874.76032
[9] Boffi, D.: Compatible discretizations for eigenvalue problems. In: Compatible Spatial Discretizations, IMA Volumes in Mathematics and its Applications, vol. 142, pp. 121–142. Springer, Berlin (2006) · Zbl 1110.65104
[10] Boffi D., Brezzi F., Gastaldi L.: On the convergence of eigenvalues for mixed formulations. Annali Sc. Norm. Sup. Pisa Cl. Sci. 25, 131–154 (1997) · Zbl 1003.65052
[11] Brenner S., Li F., Sung L.-Y.: A locally divergence-free interior penalty method for two dimensional curl–curl problems. SIAM J. Numer. Anal. 46, 1190–1211 (2008) · Zbl 1168.65068
[12] Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, vol. 15. Springer, Berlin (1991) · Zbl 0788.73002
[13] Ciarlet P. Jr: Augmented formulations for solving Maxwell equations. Comput. Methods Appl. Mech. Eng. 194, 559–586 (2005) · Zbl 1063.78018
[14] Ciarlet P. Jr, Garcia E., Zou J.: Solving Maxwell equations in 3D prismatic domains. C. R. Acad. Sci. Paris, Ser. I 339, 721–726 (2004) · Zbl 1068.78001
[15] Ciarlet P. Jr, Girault V.: Inf-sup condition for the 3D, P 2so 1 Taylor–Hood finite element; application to Maxwell equations. C. R. Acad. Sci. Paris, Ser. I 335, 827–832 (2002) · Zbl 1021.78009
[16] Ciarlet, P. Jr., Hechme, G.: Mixed, augmented variational formulations for Maxwell’s equations: numerical analysis via the macroelement technique. Numer. Math. (submitted)
[17] Ciarlet P. Jr, Hechme G.: Computing electromagnetic eigenmodes with continuous Galerkin approximations. Comput. Methods Appl. Mech. Eng. 198, 358–365 (2008) · Zbl 1194.78053
[18] Costabel M.: A coercive bilinear form for Maxwell’s equations. J. Math. An. Appl. 157, 527–541 (1991) · Zbl 0738.35095
[19] Costabel M., Dauge M.: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93, 239–277 (2002) · Zbl 1019.78009
[20] Costabel, M., Dauge, M.: Computation of resonance frequencies for Maxwell equations in non smooth domains. In: Topics in Computational Wave Propagation. Lecture Notes in Computational Science and Engineering, vol. 31, pp. 125–161. Springer, Berlin (2003) · Zbl 1116.78002
[21] Costabel M., Dauge M., Schwab C.: Exponential convergence of hp-FEM for Maxwell’s equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci. 15, 575–622 (2005) · Zbl 1078.65089
[22] Dauge, M.: Benchmark computations for Maxwell equations for the approximation of highly singular solutions (2004). See Monique Dauge’s personal web page at the location http://perso.univ-rennes1.fr/monique.dauge/core/index.html
[23] Garcia, E.: Solution to the instationary Maxwell equations with charges in non-convex domains (in French). Ph.D. Thesis, Université Paris VI, France (2002)
[24] Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986) · Zbl 0585.65077
[25] Hazard C., Lohrengel S.: A singular field method for Maxwell’s equations: numerical aspects for 2D magnetostatics. SIAM J. Appl. Math. 40, 1021–1040 (2002) · Zbl 1055.78011
[26] Heintzé, E.: Solution to the 3D instationary Maxwell equations with conforming finite elements (in French). Ph.D. Thesis, Université Paris VI, France (1992)
[27] Jamelot E.: Éléments finis nodaux pour les équations de Maxwell. C. R. Acad. Sci. Paris, Sér. I 339, 809–814 (2004) · Zbl 1076.78013
[28] Jamelot, E.: Solution to Maxwell equations with continuous Galerkin finite elements (in French). Ph.D. Thesis, École Polytechnique, Palaiseau, France (2005) · Zbl 1185.65006
[29] Labrunie, S.: The Fourier singular complement method for Maxwell equations in axisymmetric domains (in French). Technical Report 2004-42, Institut Elie Cartan, Nancy I University, Vandoeuvre-lès-Nancy, France (2004)
[30] Scott L.R., Zhang S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990) · Zbl 0696.65007
[31] Sorokina T., Worsey A.J.: A multivariate Powell–Sabin interpolant. Adv. Comput. Math. 29, 71–89 (2008) · Zbl 1154.65009
[32] Weber C.: A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci. 2, 12–25 (1980) · Zbl 0432.35032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.