Let M be a compact Riemannian manifold. The work of Dodziuk and Patodi shows that the Hodge Laplacian on the differential forms of M can be approximated using a sequence of discretizations — Hilbert spaces built using iterative refinements of triangulations of M. In a computational setting, however, one seldom has access to M and desires an iterative approximation scheme that exhibits convergence of steps. I will describe a framework in which a given sequence of discretizations equipped with self-adjoint non-negative operators can be evaluated for convergence and show that the work of Dodziuk and Patodi constitutes an example.