  We present a full space-time numerical solution of the advection-diffusion
equation using a continuous Galerkin finite element method. The
Galerkin/least-square method is employed to ensure stability of the discrete
variational problem. In the full space-time formulation, time is considered
another dimension, and the time derivative is interpreted as an additional
advection term of the field variable. We derive a priori error estimates and
illustrate spatio-temporal convergence with several numerical examples. We also
derive a posteriori error estimates, which coupled with adaptive space-time
mesh refinement provide efficient and accurate solutions. The accuracy of the
space-time solutions is illustrated against analytical solutions as well as
against numerical solutions using a conventional time-marching algorithm.
