On the Convergence of Galerkin Finite Element Approximations of Electromagnetic Eigenproblems

The convergence of Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators is studied. Since the operator involved is noncompact, the first part of the analysis is carried out in terms of the specific definition of convergence that is known to be appropriate for this case. Then, a slightly stronger definition of convergence is proposed, which is tuned to the features a practitioner of the numerical simulation of electromagnetic devices requires for a good computational model of a resonant cavity. For both definitions, necessary and sufficient conditions are introduced and discussed. Moreover, it is proved that the convergence of an approximation in the stronger sense is unaffected by the presence of different materials filling the cavity resonator. Exploiting this basic feature of the newly defined convergence, the previously developed theory is applied to generalize the convergence proof for the lowest order edge element approximations to the case of anisotropic, inhomogeneous and discontinuous material properties. Results clarifying the relationships among the various conditions occurring in our analysis and examples showing what may happen when not all the conditions for convergence hold true are also reported and contribute to a clear picture about the origin and the behavior of spurious modes.

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