On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems

The convergence of Galerkin finite element approximations of electromagneti c eigenproblems modelling cavity resonators is studied. Since the operator involved is noncompact, the rst part of the analysis is carried out in term s of the specific definition of convergence that is known to be appropriate for this case. Then, a slightly stronger definition of convergence is prop osed, which is tuned to the features a practitioner of the numerical simula tion of electromagnetic devices requires for a good computational model of a resonant cavity. or 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 fea ture of the newly defined convergence, the previously developed theory is a pplied to generalize the convergence proof for the lowest order edge elemen t approximations to the case of anisotropic, inhomogeneous and discontinuou s material properties. Results clarifying the relationships among the vario us conditions occurring in our analysis and examples showing what may happe n when not all the conditions for convergence hold true are also reported a nd contribute to a clear picture about the origin and the behavior of spuri ous modes.