What library can one use to compute efficiently the lowest eigenvalues of the Laplace operator in a polyhedral domain in $R^3$? For the application I have in mind one has to consider very acute polyhedra (which moreover are non-convex) so it would probably be necessary to use a smart enough algorithm that can deal well with what happens near the vertices.
Any of the finite element libraries will provide a generic way to do this. You can take a look at this example program of deal.II to see how this can be done. (Disclaimer: I am one of the authors of deal.II.)
There may be techniques that deal specifically with the situation of polyhedral domains, for example by decomposing them into triangles and doing something special with them. However, I am unaware of such methods.