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Towards a good characterization of spectrally correct finite element methods in electromagnetics. (English) Zbl 0862.65075
Summary: Spurious modes often appear in the computed spectrum when an electromagnetic eigenproblem is solved by the finite element method. The paper demonstrates that the inclusion condition, often claimed as the theoretical reason for the absence of (non-zero frequency) spurious modes, is a sufficient but not necessary condition for that. It does this by proving that edge elements, which are spectrally correct, do not satisfy the inclusion condition. As intermediate steps towards this result, it proves the equivalence of the inclusion condition to a less cryptic one and gives two more easily-checked necessary conditions for the latter.
The paper concludes that from this investigation, the inclusion condition seems too strong to be useful as a sufficient condition. It works out the present analysis in the framework of spectral approximation theory for non-compact operators, which emerges as a basic tool for a deeper understanding of the whole question of spurious modes.

MSC:
65Z05 Applications to the sciences
35P15 Estimates of eigenvalues in context of PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory, general
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