In this article, we present a numerical scheme based on a finite element method in order
to solve a time-dependent convection-diffusion equation problem and satisfy some
conservation properties. In particular, our scheme is able to conserve the total energy
for a heat equation or the total mass of a solute in a fluid for a concentration equation,
even if the approximation of the velocity field is not completely divergence-free. We
establish a priori errror estimates for this scheme and we give some numerical examples
which show the efficiency of the method.