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**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.

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

### MSC:

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 |

### Keywords:

electromagnetic waves in resonators of particle accelerators; penalty formulations; eigenvalue problem; electromagnetism; time-harmonic analysis; finite element method; Comparison; displacement method
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\textit{F. Kikuchi}, Comput. Methods Appl. Mech. Eng. 64, 509--521 (1987; Zbl 0644.65087)

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### References:

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