Hostname: page-component-7b7567c8c-mnklg Total loading time: 0 Render date: 2026-11-12T21:03:31.668Z Has data issue: false hasContentIssue false
  • English
  • Français

Edge finite elements for the approximation of Maxwell resolventoperator

Published online by Cambridge University Press:  15 May 2002

Daniele Boffi and
Lucia Gastaldi
Daniele Boffi
Affiliation:
Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy. boffi@dimat.unipv.it.
Lucia Gastaldi
Affiliation:
Dipartimento di Matematica, Università di Brescia, 25133 Brescia, Italy. gastaldi@bsing.ing.unibs.it.
Get access

Abstract

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

Keywords

Edge finite elementstime-harmonic Maxwell's equationsmixed finite elements.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 2 , March 2002, pp. 293 - 305
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potential in three-dimensional nonsmooth domains. Math Methods Appl. Sci. 21 (1998) 823-864. 3.0.CO;2-B>CrossRef
Bermúdez, A., Durán, R., Muschietti, A., Rodríguez, R. and Solomin, J., Finite element vibration analysis of fluid-solid systems without spurious modes. SIAM J. Numer. Anal. 32 (1995) 1280-1295. CrossRef
Boffi, D., Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229-246. CrossRef
Boffi, D., A note on the de Rham complex and a discrete compactness property. Appl. Math. Lett. 14 (2001) 33-38. CrossRef
Boffi, D., Brezzi, F. and Gastaldi, L., On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000) 121-140. CrossRef
Boffi, D., Fernandes, P., Gastaldi, L. and Perugia, I., Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36 (1998) 1264-1290. CrossRef
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991).
Brezzi, F., Rappaz, J. and Raviart, P.A., Finite dimensional approximation of nonlinear problems. Part i: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1-25. CrossRef
Caorsi, S., Fernandes, P. and Raffetto, M., On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems. SIAM J. Numer. Anal. 38 (2000) 580-607. CrossRef
Demkowicz, L., Monk, P., Vardapetyan, L. and Rachowicz, W., de Rham diagram for hp finite element spaces. Comput. Math. Appl. 39 (2000) 29-38. CrossRef
Demkowicz, L. and Vardapetyan, L., 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). CrossRef
Descloux, J., Nassif, N. and Rappaz, J., On spectral approximation. I. The problem of convergence. RAIRO Anal. Numér. 12 (1978) 97-112. CrossRef
Fernandes, P and Gilardi, G., Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 (1997) 957-991. CrossRef
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.
F. Kikuchi, On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci., Univ. Tokyo, Sect. I A 36 (1989) 479-490.
Monk, P., A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63 (1992) 243-261. CrossRef
Monk, P. and Demkowicz, L., Discrete compactness and the approximation of Maxwell's equations in ${\mathbb{R}}\sp 3$ . Math. Comp. 70 (2001) 507-523. CrossRef
Nédélec, J.-C., Mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 35 (1980) 315-341. CrossRef
Nédélec, J.-C., A new family of mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 50 (1986) 57-81. CrossRef
J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Preprint ISC-01-10-MATH, Texas A&M University, 2001.
Vardapetyan, L. and Demkowicz, L., hp-adaptive finite elements in electromagnetics. Comput. Methods Appl. Mech. Engrg. 169 (1999) 331-344. CrossRef