We give sufficient conditions to approximate the `nonessential' spectrum of a bounded operator acting on a Banach space by part of the spectra of a sequence of compact (or finite rank) operators , where is a suitable family of uniformly bounded operators which approach the identity. (By nonessential spectrum we mean here all the spectrum outside of the disc of radius equal to the essential spectral radius.) For this, we combine the formulae for the essential spectral radius with nonstandard perturbative results on the stability of the nonessential spectrum of quasicompact operators. We present concrete applications to transfer operators of smooth expanding maps using multiresolution analysis (large-scale approximation projections).