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This paper explains how a popular commercially available software package for solving partial-differential-equations (PDEs), as based on the finite-element method, can be configured to efficiently calculate 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 nontrivial structures, i.e., 1) two different optical microcavities (one toroidal made of silica, the other an AlGaAs microdisk), 2) a third-order sapphire:air Bragg cavity, and 3) two different cryogenic sapphire WG-mode resonators; both 2) and 3) operate in the microwave X-band. By fitting one of 3) to a set of measured resonance frequencies, the dielectric constants of sapphire at liquid-helium temperature have been estimated.