Edge finite elements for the approximation of Maxwell resolvent operator
ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 2, pp. 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 : 10.1051/m2an:2002013
Classification : 65N25, 65N30
Mots clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {293--305},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {2},
     year = {2002},
     doi = {10.1051/m2an:2002013},
     mrnumber = {1906819},
     zbl = {1042.65087},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2002013/}
}
TY  - JOUR
AU  - Boffi, Daniele
AU  - Gastaldi, Lucia
TI  - Edge finite elements for the approximation of Maxwell resolvent operator
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2002
SP  - 293
EP  - 305
VL  - 36
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2002013/
DO  - 10.1051/m2an:2002013
LA  - en
ID  - M2AN_2002__36_2_293_0
ER  - 
%0 Journal Article
%A Boffi, Daniele
%A Gastaldi, Lucia
%T Edge finite elements for the approximation of Maxwell resolvent operator
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2002
%P 293-305
%V 36
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2002013/
%R 10.1051/m2an:2002013
%G en
%F M2AN_2002__36_2_293_0
Boffi, Daniele; Gastaldi, Lucia. Edge finite elements for the approximation of Maxwell resolvent operator. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 2, pp. 293-305. doi : 10.1051/m2an:2002013. http://archive.numdam.org/articles/10.1051/m2an:2002013/

[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

[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

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

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

[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

[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

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

[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

[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

[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

[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

[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

[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

[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

[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

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

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

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

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

[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

Cité par Sources :