  In this work, we introduce a Variational Multi-Scale (VMS) method for the
numerical approximation of parabolic problems, where sub-grid scales are
approximated from the eigenpairs of associated elliptic operator. The abstract
method is particularized to the one-dimensional advection-diffusion equations,
for which the sub-grid components are exactly calculated in terms of a spectral
expansion when the advection velocity is approximated by piecewise constant
velocities on the grid elements.
  We prove error estimates that in particular imply that when Lagrange finite
element discretisations in space are used, the spectral VMS method coincides
with the exact solution of the implicit Euler semi-discretisation of the
advection-diffusion problem at the Lagrange interpolation nodes. We also build
a feasible method to solve the evolutive advection-diffusion problems by means
of an offline/online strategy with reduced computational complexity.
  We perform some numerical tests in good agreement with the theoretical
expectations, that show an improved accuracy with respect to several stabilised
methods.
