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

Spectral Variational Multi-Scale method for parabolic problems. Application to 1D transient advection-diffusion equations