Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. (English) Zbl 0644.65087

The author presents some weak formulations for an eigenvalue problem in electromagnetism related to time-harmonic analysis. This study is intended as a theoretical basis for the three-dimensional investigation of such problems. Computations, however, are mainly restricted to two- dimensional models. Only the main ideas and results are presented, while proofs of theorems and technical details are omitted.
The author defines some function spaces which are not the usual Sobolev spaces and then develops what is known as the “mixed” and “penalty” formulations of the problem. These are applied to the finite element method and some numerical results are presented and discussed. Comparison is carried out with already existing methods. The author reaches the conclusion that the proposed method may be implemented as the standard “displacement method” with the Lagrangian multiplier eliminated.
Reviewer: A.Ghaleb


65Z05 Applications to the sciences
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
78A25 Electromagnetic theory (general)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35P15 Estimates of eigenvalues in context of PDEs
Full Text: DOI


[1] Bathe, K.-J.; Wilson, E. L., Numerical Methods in Finite Element Analysis (1976), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0387.65069
[2] Brezzi, F., On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO 8 Anal. Numér., 129-151 (1974) · Zbl 0338.90047
[3] Chatlin, F., Spectral Approximation of Linear Operators (1983), Academic Press: Academic Press New York · Zbl 0517.65036
[4] Duvaut, G.; Lions, J. L., Inequalities in Mechanics and Physics (1976), Springer: Springer Berlin · Zbl 0331.35002
[5] Gruber, R.; Rappaz, J., Finite Element Methods in Linear Ideal Magnetohydrodynamics (1985), Springer: Springer Berlin · Zbl 0573.76001
[6] Hara, M.; Wada, T.; Fukasawa, T.; Kikuchi, F., A three dimensional analysis of RF electromagnetic fields by the finite element method, IEEE Trans. Magn., 19, 2417-2420 (1983)
[7] Kikuchi, F., An isomorphic property of two Hilbert spaces appearing in electromagnetism-analysis by the mixed formulation, JJAM, 3, 53-58 (1986) · Zbl 0613.46040
[8] Kikuchi, F.; Hara, M.; Wada, T.; Nakahara, K., Calculation of RF electromagnetic field by finite element method (III), RIKEN (Institute of Physical and Chemical Research), Accelerator Prog. Rept., 17, 153-155 (1983)
[9] Leis, R., Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien, Math. Z., 106, 213-224 (1968)
[10] Nedelec, J. C., Mixed finite elements in \(R^3\), Numer. Math., 35, 315-341 (1980) · Zbl 0419.65069
[11] Strang, G.; Fix, G. J., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[12] Stratton, J. A., Electromagnetic Theory (1941), McGraw-Hill: McGraw-Hill New York · Zbl 0022.09303
[13] Weck, N., Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries, J. Math. Anal. Appl., 46, 410-437 (1974) · Zbl 0281.35022
[14] Weiland, T., A dicretization method for the solution of Maxwell’s equations and applications for sixcomponent fields, AEU, 31, 116-120 (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.