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In this paper we review some basic facts concerning the numerical approximation of partial differential equations. In particular, we apply the finite element method to the approximations of eigenvalues and eigenfunctions of elliptic operators defined in parametric domains. We consider two applications and compute using the specialized software FreeFem++. The first domain is parameterized by an angle θ and consists of a unit disc from which a sector of angle θ its been cut-off; the value of θ runs from zero to ninety degrees. We obtain a quadratic dependence of the first two eigenvalues with respect to the angle θ. For the second example we consider an integer-valued parameter n and the domain consists of a unit disc with n circular holes. In this example we observe localization of eigenvalues. [ABSTRACT FROM AUTHOR]