  We establish stable finite element (FE) approximations of
convection-diffusion initial boundary value problems using the automatic
variationally stable finite element (AVS-FE) method. The transient
convection-diffusion problem leads to issues in classical FE methods as the
differential operator can be considered singular perturbation in both space and
time. The unconditional stability of the AVS-FE method, regardless of the
underlying differential operator, allows us significant flexibility in the
construction of FE approximations. We take two distinct approaches to the FE
discretization of the convection-diffusion problem: i) considering a space-time
approach in which the temporal discretization is established using finite
elements, and ii) a method of lines approach in which we employ the AVS-FE
method in space whereas the temporal domain is discretized using the
generalized-alpha method. In the generalized-alpha method, we discretize the
temporal domain into finite sized time-steps and adopt the generalized-alpha
method as time integrator. Then, we derive a corresponding norm for the
obtained operator to guarantee the temporal stability of the method.
  We present numerical verifications for both approaches, including numerical
asymptotic convergence studies highlighting optimal convergence properties.
Furthermore, in the spirit of the discontinuous Petrov-Galerkin method by
Demkowicz and Gopalakrishnan, the AVS-FE method also leads to readily available
a posteriori error estimates through a Riesz representer of the residual of the
AVS-FE approximations. Hence, the norm of the resulting local restrictions of
these estimates serve as error indicators in both space and time for which we
present multiple numerical verifications adaptive strategies.
