Summary
The time harmonic Maxwell’s equations for a lossless medium are neither elliptic or definite. Hence the analysis of numerical schemes for these equations presents some unusual difficulties. In this paper we give a simple proof, based on the use of duality, for the convergence of edge finite element methods applied to the cavity problem for Maxwell’s equations. The cavity is assumed to be a general Lipschitz polyhedron, and the mesh is assumed to be regular but not quasi-uniform.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Mathematics of Computations, 68 (1999), pp. 607–631.
C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional nonsmooth domains, Math. Meth. Appl. Sci., 21 (1998), pp. 823–864.
D. Arnold, R. Falk, and R. Winthur, Multigrid in H(div) and H (curl), Numerische Mathematik, 85 (2000), pp. 197–217.
D. Boffi, Fortin operators and discrete compactness for edge elements, Numer. Math., 87 (2000), pp. 229–246.
D. Boffi —, A note on the de Rham complex and a discrete compactness property, Applied Mathematics Letters, 14 (2001), pp. 33–38.
D. Boffi and L. Gastaldi, Edge finite elements for the approximation of Maxwell resolvent operators. Preprint.
S. Caorsi, P. Fernandes, and M. Raffetto, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM. J. Numer. Anal., 38 (2000), pp. 580–607.
P. Ciarlet, The Finite Element Method for Elliptic Problems, vol. 4 of Studies In Mathematics and It’s Applications, Elsevier North-Holland, New York, 1978.
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, no. 93 in Applied Mathematical Sciences, Springer-Verlag, New York, second ed., 1998.
M. Costabel, A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains, Math. Meth. Appl. Sci., 12 (1990), pp. 365–368.
M. Costabel and M. Dauge, Maxwell and Lamé eigenvalues on polyhedra, Mathematical Methods in the Applied Sciences, 22 (1999), pp. 243–258.
M. Dauge, M. Costabel, and D. Martin, Numerical investigation of a boundary penalization method for Maxwell equations. Report available at http://www.maths.univ-rennesl.fr/~dauge/, 1999.
_M. Dauge, M. Costabel, and D. Martin —, Weighted regularisation of Maxwell equations in polyhedral domains. Report available at http://www.maths.univ-rennesl.fr/~dauge/, 2001.
L. Demkowicz and P. Monk, Discrete compactness and the approximation of Maxwell’s equations in ℝ3, Math. Comp., 70 (2001), pp. 507–523.
L. Demkowicz, P. Monk, and L. Vardapetyan, de Rham diagram for hp finite element spaces, Comput. Math. Appl., 39 (2000), pp. 29–38.
L. Demkowicz and L. Vardapetyan, Modelling electromagnetic absorbtion/scattering problems using hp-adaptive finite elements, Compu. Methods Appl. Mech. Engrg., 152 (1998), pp. 103–124.
J. Descloux, N. Nassif, and J. Rappaz, On spectral approximation. I. The problem, of convergence, RAIRO Anal. Númer., 12 (1978), pp. 97–112.
V. Girault, Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in ℝ3, Math. Comp., 51 (1988), pp. 53-58.
V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, vol. 749 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1979.
J. Gopalakrishnan and J. Pasciak, Overlapping Schwarz preconditioners for indefinite Maxwell equations. To appear in Math. Comp.
F. Kikuchi, On a discrete compactness property for the Nedelec finite elements, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math, 36 (1989), pp. 479–490.
P. Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math., 63 (1992), pp. 243–261.
J. NÉdÉlec, Mixed finite elements in ℝ3, Numer. Math., 35 (1980), pp. 315–341.
J. Nédélec —, A new family of mixed finite elements in ℝ3, Numer. Math., 50 (1986), pp. 57–81.
A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp., 28 (1974), pp. 959–962.
A. Schatz and J. Wang, Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions, Mathematics of Computation, 65 (1996), p. 19.
L. Vardapetyan and L. Demkowicz, hp-Adaptive finite elements in electromagnetics, Computers Methods in Applied Mechanics and Engineering, 169 (1999), pp. 331–344.
C. Weber, A local compactness theorem for Maxwell’s equations, Math. Meth. in the Appl. Sci., 2 (1980), pp. 12–25.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Monk, P. (2003). A Simple Proof of Convergence for an Edge Element Discretization of Maxwell’s Equations. In: Monk, P., Carstensen, C., Funken, S., Hackbusch, W., Hoppe, R.H.W. (eds) Computational Electromagnetics. Lecture Notes in Computational Science and Engineering, vol 28. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55745-3_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-55745-3_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44392-6
Online ISBN: 978-3-642-55745-3
eBook Packages: Springer Book Archive