  A new method to enclose the pseudospectrum via the numerical range of the
inverse of a matrix or linear operator is presented. The method is applied to
finite-dimensional discretizations of an operator on an infinite-dimensional
Hilbert space, and convergence results for different approximation schemes are
obtained, including finite element methods. We show that the pseudospectrum of
the full operator is contained in an intersection of sets which are expressed
in terms of the numerical ranges of shifted inverses of the approximating
matrices. The results are illustrated by means of two examples: the
advection-diffusion operator and the Hain-L\"ust operator.
