  We prove a general Mosco convergence theorem for bounded Euclidean domains
satisfying a set of mild geometric hypotheses. For bounded domains, this notion
implies norm-resolvent convergence for the Dirichlet Laplacian which in turn
ensures spectral convergence. A key element of the proof is the development of
a novel, explicit Poincar\'e-type inequality. These results allow us to
construct a universal algorithm capable of computing the eigenvalues of the
Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal
boundaries, such as the Koch snowflake and certain filled Julia sets, are
included among this class. Conversely, we construct a counter example showing
that there does not exist a universal algorithm of the same type capable of
computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded
domain.
