Edge finite elements for the approximation of Maxwell resolvent operator
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 2, p. 293-305

In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the L 2 norm for the sequence of discrete operators. These results, together with a general theory introduced by Brezzi, Rappaz and Raviart [8], allow an immediate proof of convergence for the finite element approximation of the time-harmonic Maxwell system.

DOI : https://doi.org/10.1051/m2an:2002013
Classification:  65N25,  65N30
Keywords: edge finite elements, time-harmonic Maxwell's equations, mixed finite elements
@article{M2AN_2002__36_2_293_0,
     author = {Boffi, Daniele and Gastaldi, Lucia},
     title = {Edge finite elements for the approximation of Maxwell resolvent operator},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {2},
     year = {2002},
     pages = {293-305},
     doi = {10.1051/m2an:2002013},
     zbl = {1042.65087},
     mrnumber = {1906819},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2002__36_2_293_0}
}
Boffi, Daniele; Gastaldi, Lucia. Edge finite elements for the approximation of Maxwell resolvent operator. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 2, pp. 293-305. doi : 10.1051/m2an:2002013. http://www.numdam.org/item/M2AN_2002__36_2_293_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] A. Bermúdez, R. Durán, A. Muschietti, R. Rodríguez and J. Solomin, Finite element vibration analysis of fluid-solid systems without spurious modes. SIAM J. Numer. Anal. 32 (1995) 1280-1295. | Zbl 0833.73050

[3] D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229-246. | Zbl 0967.65106

[4] D. Boffi, A note on the de Rham complex and a discrete compactness property. Appl. Math. Lett. 14 (2001) 33-38. | Zbl 0983.65125

[5] D. Boffi, F. Brezzi and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000) 121-140. | Zbl 0938.65126

[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 (1998) 1264-1290. | Zbl 1025.78014

[7] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[8] F. Brezzi, J. Rappaz and P.A. Raviart, Finite dimensional approximation of nonlinear problems. Part i: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1-25. | Zbl 0488.65021

[9] 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

[10] L. Demkowicz, P. Monk, L. Vardapetyan and W. Rachowicz, de Rham diagram for hp finite element spaces. Comput. Math. Appl. 39 (2000) 29-38. | Zbl 0955.65084

[11] L. Demkowicz and L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements. Comput. Methods Appl. Mech. Engrg. 152 (1998) 103-124. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997). | Zbl 0994.78011

[12] J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. I. The problem of convergence. RAIRO Anal. Numér. 12 (1978) 97-112. | Numdam | Zbl 0393.65024

[13] 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

[14] F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. In Proceedings of the first world congress on computational mechanics (Austin, Tex., 1986), Vol. 64, pages 509-521, 1987. | Zbl 0644.65087

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

[16] P. Monk, A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63 (1992) 243-261. | Zbl 0757.65126

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

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

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

[20] J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Preprint ISC-01-10-MATH, Texas A&M University, 2001.

[21] L. Vardapetyan and L. Demkowicz, hp-adaptive finite elements in electromagnetics. Comput. Methods Appl. Mech. Engrg. 169 (1999) 331-344. | Zbl 0956.78013