Computational models of electromagnetic resonators: Analysis of edge element approximation

The purpose of this paper is to address some difficulties which arise in co mputing the eigenvalues of Maxwell's system by a finite element method. Dep ending on the method used, the spectrum may be polluted by spurious modes w hich are difficult to pick out among the approximations of the physically c orrect eigenvalues. Here we propose a criterion to establish whether or not a finite element scheme is well suited to approximate the eigensolutions a nd, in the positive case, we estimate the rate of convergence of the eigens olutions. This criterion involves some properties of the finite element spa ce and of a suitable Fortin operator. The lowest-order edge elements, under some regularity assumptions, give an example of space satisfying the requi red conditions. The construction of such a Fortin operator in very general geometries and for any order edge elements is still an open problem. Moreover, we give some justification for the spectral pollution which occur s when nodal elements are used. Results of numerical experiments confirming the theory are also reported.