Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 2, p. 331-354

By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec's edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proof is new, instead, and shows how families of spurious-free elements can be systematically constructed. The tools here developed are used to define a new family of spurious-free edge elements which, in some sense, are complementary to those defined in 1986 by Nedelec.

Classification:  65N25,  65N30,  65N12
Keywords: electromagnetic eigenproblems, new families of edge elements, Galerkin finite element approximations, convergence, spurious modes, discontinuous material properties, symmetry exploitation, mixed boundary conditions, discrete compactness
@article{M2AN_2001__35_2_331_0,
     author = {Caorsi, Salvatore and Fernandes, Paolo and Raffetto, Mirco},
     title = {Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {2},
     year = {2001},
     pages = {331-354},
     zbl = {0993.78016},
     mrnumber = {1825702},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_2_331_0}
}
Caorsi, Salvatore; Fernandes, Paolo; Raffetto, Mirco. Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 2, pp. 331-354. http://www.numdam.org/item/M2AN_2001__35_2_331_0/

[1] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potential in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21 (1998) 823-864. | Zbl 0914.35094

[2] M.L. Barton and Z.J. Cendes, New vector finite elements for three-dimensional magnetic field computation. J. Appl. Phys. 61 (1987) 3919-3921.

[3] A. Bermudez and D.G. Pedreira, Mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides. Numer. Math. 61 (1992) 39-57. | Zbl 0741.65095

[4] D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math. 86 (2000). DOI 10.1007/s002110000182. | MR 1804657 | Zbl 0967.65106

[5] D. Boffi, A note on the discrete compactness property and the de Rham complex. Technical Report AM188, Department of Mathematics, Penn State University, 1999. Appl. Math. Lett. 14 (2001) 33-38. | Zbl 0983.65125

[6] D. Boffi, P. Fernandes, L. Gastaldi and I. Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36 (1999) 1264-1290. | Zbl 1025.78014

[7] A. Bossavit, Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEE Proceedings, part A 135 (1988) 493-500.

[8] A. Bossavit, A rationale for ‘edge-elements' in 3-D fields computations. IEEE Trans. Magnet. 24 (1988) 74-79.

[9] A. Bossavit, Solving maxwell's equations in a closed cavity, and the question of ‘spurious modes'. IEEE Trans. Magnet. 26 (1990) 702-705.

[10] S. Caorsi, P. Fernandes and M. Raffetto, Edge elements and the inclusion condition. IEEE Microwave Guided Wave Lett. 5 (1995) 222-223.

[11] S. Caorsi, P. Fernandes and M. Raffetto, Towards a good characterization of spectrally correct finite element methods in electromagnetics. COMPEL 15 (1996) 21-35. | Zbl 0862.65075

[12] S. Caorsi, P. Fernandes and M. Raffetto, Do covariant projection elements really satisfy the inclusion condition? IEEE Trans. Microwave Theory Tech. 45 (1997) 1643-1644.

[13] S. Caorsi, P. Fernandes and M. Raffetto, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems. SIAM J. Numer. Anal. 38 (2000) 580-607. | Zbl 1005.78012

[14] S. Caorsi, P. Fernandes and M. Raffetto, Characteristic conditions for spurious-free finite element approximations of electromagnetic eigenproblems, in Proceedings of ECCOMAS 2000, Barcelona, Spain (2000) 1-13.

[15] Z.J. Cendes and P.P. Silvester, Numerical solution of dielectric loaded waveguides: I-finite-element analysis. IEEE Trans. Microwave Theory Tech. 18 (1970) 1124-1131.

[16] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[17] C.W. Crowley, P.P. Silvester and H. Hurwitz Jr., Covariant projection elements for 3d vector field problems. IEEE Trans. Magnet. 24 (1988) 397-400.

[18] J.B. Davies, F.A. Fernandez and G.Y. Philippou, Finite element analysis of all modes in cavities with circular symmetry. IEEE Trans. Microwave Theory Tech. 30 (1982) 1975-1980.

[19] P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 (1997) 957-991. | Zbl 0910.35123

[20] V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin (1986). | Zbl 0585.65077

[21] M. Hano, Vector finite element solution of anisotropic waveguides using novel triangular elements. Electron. Com. Japan, Part 2, 71 (1988) 71-80.

[22] M. Hara, T. Wada, T. Fukasawa and F. Kikuchi, A three dimensional analysis of rf electromagnetic fields by the finite element method. IEEE Trans. Magnet. 19 (1983) 2417-2420.

[23] H.C. Hoyt, Numerical studies of the shapes of drift tubes and Linac cavities. IEEE Trans. Nucl. Sci. 12 (1965) 153-155.

[24] H.C. Hoyt, D.D. Simmonds and W.F. Rich, Computer designed 805 MHz proton Linac cavities. The Review of Scientific Instruments 37 (1966) 755-762.

[25] F. Kikuchi, On a discrete compactness property for the Nedelec finite elements. J. Fac. Sci., Univ. Tokyo 36 (1989) 479-490. | Zbl 0698.65067

[26] F. Kikuchi, Theoretical analysis of Nedelec's edge elements, in Proceedings of Computational Engineering Conference, Tokyo, Japan, May 26-28 (1999).

[27] R. Miniowitz and J.P. Webb, Covariant-projection quadrilateral elements for the analysis of waveguides with sharp edges. IEEE Trans. Microwave Theory Tech. 39 (1991) 501-505.

[28] P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in 3 . Math. Comput. 70 (2001) 507-523. | Zbl 1035.65131

[29] J.C. Nedelec, Mixed finite elements in 3 . Numer. Math. 35 (1980) 315-341. | Zbl 0419.65069

[30] J.C. Nedelec, A new family of mixed finite elements in 3 . Numer. Math. 50 (1986) 57-81. | Zbl 0625.65107

[31] R. Parodi, A. Stella and P. Fernandes, Rf tests of a band overlap free daw accelerating structure, in Proceedings of the IEEE 1991 Particle Accelerator Conference, San Francisco, USA (1991) 3026-3028.

[32] J.S. Wang and N. Ida, Curvilinear and higher order ‘edge' finite elements in electromagnetic field computation. IEEE Trans. Magnet. 29 (1993) 1491-1494.

[33] S.H. Wong and Z.J. Cendes, Combined finite element-modal solution of three-dimensional eddy current problems. IEEE Trans. Magnet. 24 (1988) 2685-2687.

[34] J.P. Webb, Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements. IEEE Trans. Antennas Propagation 47 (1999) 1244-1253. | Zbl 0955.78014

[35] J.P. Webb and R. Miniowitz, Analysis of 3-D microwave resonators using covariant-projection elements. IEEE Trans. Microwave Theory Tech. 39 (1991) 1895-1899.