We introduce and compare two approaches to slow manifolds for infinite-dimensional evolution equations: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions, allowing us to change between different characterizations of slow invariant manifolds. This gives a suitable framework to study extensions of slow manifolds around non-hyperblic singularities in reaction-diffusion PDEs, using the geometric blow-up method in the respective Galerkin ODE systems. If time allows, I will also explain birefly some results on arbitrarily strong stabilization along slow manifolds by time discretization effects.