The finite element method approximates the spectrum of
an operator $S$ by computing the spectra of a sequence of operators
$S_N$ defined in terms of the finite element spaces. For the case that
$S$ is compact, convergence of the approximate spectra follows from
the convergence of $S_N$ to $S$
in the operator norm. We consider the case that $S$ is non-compact, in
which case such operator norm convergence cannot take place, and the
approximations may be polluted by spurious eigenvalues. Pollution-free
convergence of the eigenvalues can, however, be
guaranteed outside the {\em essential numerical range} of $S$, which is
related to the essential spectrum of $S$. We present  
results  for estimating this essential numerical range and apply them
to an algorithm for the buckling of
three-dimensional bodies (that gives rise to a non-compact $S$).   
Our results show, for instance, that for the example of a circular disc,
 the algorithm will
be free of spurious eigenvalues provided the body is thin enough. The
case that singularities in the stresses can lead to
non-physical spectral values being approximated is also investigated.