The exact computation of an operator's spectrum is only possible in rare cases and its high susceptibility to perturbations poses a substantial challenge for approximations. This thesis delves into various approaches aimed at addressing this problem, utilizing spectral supersets such as the pseudospectrum and the quadratic numerical range as key tools. A new method for enclosing the pseudospectrum of a linear operator via the computable intersection of sets which are expressed in terms of the numerical ranges of shifted inverses of approximating matrices is introduced. The result is illustrated by means of the advection-diffusion, the Hain-Lüst and a Stokes-type operator. Moreover, a novel algorithm for the computation of the quadratic numerical range that is based on the maximization of a purposefully tailored objective function is presented. Numerous examples showcase that this computational technique outperforms the prevailing random vector sampling approach in both the quality of the produced images and computational speed.