  In this work, we consider an extension to parabolic problems of the
variational multiscale method with spectral approximation of the sub-scales. We
first discretize in time using a finite difference scheme and second, apply the
generalization of the spectral variational multi-scale method. To obtain error
estimations in convection-dominated flows, we find a helpful link between the
stabilized term expressed in terms of Green's functions and in terms of
spectral functions. Finally, we present some numerical tests to show the
reliability of the method. We consider the stationary one-dimensional
advection-diffusion-reaction equation and the evolutive one-dimensional
advection-diffusion equation.

Numerical Analysis