Spectrally correct finite element operators for electromagnetic vector fields

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Abstract
A single finite element formulation using the magnetic (H) field vector directly is proposed for analysis of electromagnetic fields throughout the frequency spectrum. Results for waveguide and cavity analysis, as well as recent solutions to benchmark low-frequency eddy current examples such as the ‘‘Bath cube,’’ demonstrate the flexibility of the formulation. Applying earlier finite element methods to vector Helmholtz or diffusion equation problems, various workers have obtained incorrect solutions because the eigenmode spectra of the discrete (finite element) operators for such problems may contain eigenvalues and eigenmodes which do not correspond to modes of the underlying continuum (physical) problem. Such ‘‘spurious’’ modes have long been documented in high-frequency modal analysis. They are clearly identified as the cause for error in deterministic problems. Error is avoided by employing finite element operators whose spectra contain no spurious components. Application of the formulation may be limited by computer round-off at matrix assembly which affects solenoidality of magnetic fields in the solutions. Furthermore, the extreme values encountered in low-frequency eddy current analyses lead to ill conditioning and information loss and subsequent unreliability of the solution. These numerical instabilities may be overcome by parametric adjustment of permittivities, or by increased computer word length.