  This paper explains how a popular, commercially-available software package
for solving partial-differential-equations (PDEs), as based on the
finite-element method (FEM), can be configured to calculate, efficiently, the
frequencies and fields of the whispering-gallery (WG) modes of axisymmetric
dielectric resonators. The approach is traceable; it exploits the PDE-solver's
ability to accept the definition of solutions to Maxwell's equations in
so-called `weak form'. Associated expressions and methods for estimating a WG
mode's volume, filling factor(s) and, in the case of closed(open) resonators,
its wall(radiation) loss, are provided. As no transverse approximation is
imposed, the approach remains accurate even for quasi-transverse
magnetic/electric modes of low, finite azimuthal mode order. The approach's
generality and utility are demonstrated by modeling several non-trivial
structures: (i) two different optical microcavities [one toroidal made of
silica, the other an AlGaAs microdisk]; (ii) a 3rd-order sapphire:air Bragg
cavity; (iii) two different cryogenic sapphire WG-mode resonators; both (ii)
and (iii) operate in the microwave X-band. By fitting one of (iii) to a set of
measured resonance frequencies, the dielectric constants of sapphire at
liquid-helium temperature have been estimated.
