Geometry has been, in recent times, a great inspiration for mathematical problems. It is therefore useful to consider numerical methods in effort to visualize and explore properties of the solutions to the partial differential equations arising from these problems. We examine a modern framework, Hilbert complexes, which abstracts many of the essential details relevant for partial differential equations, such as exterior derivatives and coderivatives, Laplacians, and Poincaré inequalities. This viewpoint also proves useful for approximation via the finite element method. We prove some abstract error estimates and apply these results to the case of numerically computing parabolic equations on Riemannian hypersurfaces.