In this article, a fully discrete finite element approximation is investigated for
constrained parabolic optimal control problems with time-dependent coefficients. The
spatial discretisation invokes finite elements, and the time discretisation a nonstandard
backward Euler method. On introducing some appropriate intermediate variables and
noting properties of the L
2 projection and the elliptic projection, we derive the superconvergence
for the control, the state and the adjoint state. Finally, we discuss some
numerical experiments that illustrate our theoretical results.