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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.
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Abstract

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 L2 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.

Keywords

Edge finite elementstime-harmonic Maxwell's equationsmixed finite elements.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 2 , March 2002 , pp. 293 - 305
Copyright
© EDP Sciences, SMAI, 2002

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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