  We examine the dual graph representation of simplicial manifolds in Causal
Dynamical Triangulations (CDT) as a mean to build observables, and propose a
new representation based on the Finite Element Methods (FEM). In particular,
with the application of FEM techniques, we extract the (low-lying) spectrum of
the Laplace-Beltrami (LB) operator on the Sobolev space $H^1$ of scalar
functions on piecewise flat manifolds, and compare them with corresponding
results obtained by using the dual graph representation. We show that, besides
for non-pathological cases in two dimensions, the dual graph spectrum and
spectral dimension do not generally agree, neither quantitatively nor
qualitatively, with the ones obtained from the LB operator on the continuous
space. We analyze the reasons of this discrepancy and discuss its possible
implications on the definition of generic observables built from the dual graph
representation.
