We present new numerical methods for the porous media equation (PME), a non-linear parabolic PDE used to model a variety of diffusive processes. It is well-known that, unlike the archetypal linear parabolic heat equation, the PME evolves compactly supported initial data to compactly supported solutions for all time. Compactness of solutions gives rise to the "free boundary" of the support set, which itself exposes computational concerns. Additionally, it is well-known that solutions of the PME tend to a self-similar Barenblatt-Pattle solution as time tends to infinity. We introduce new spectral Galerkin (sG) methods for this problem: solutions that are the result of forcing truncated series expansions in bases of functions to satisfy a finite-dimensional, weak formulation of the PDE. We prove that by carefully tracking the free boundary and adding the Barenblatt-Pattle solution to our bases of functions our numerical solutions preserve the correct asymptotic behavior as time tends to infinity. Our method is preferable for long-time numerical simulations because it preserves this useful property whereas previous methods do not. Numerical experiments suggest convergence to the true solution for all time as the number of basis functions tends to infinity. Next, we investigate sG methods for an equation of unsteady filtration (EUF), of which the PME is a special case. We show that the asymptotic computational cost of our approach is better than that exhibited by prior methods based on finite-difference discretizations.