  We describe and analyze a finite element numerical scheme for the
parabolic-parabolic Keller-Segel model. The scalar auxiliary variable method is
used to retrieve the monotonic decay of the energy associated with the system
at the discrete level. This method relies on the interpretation of the
Keller-Segel model as a gradient flow. The resulting numerical scheme is
efficient and easy to implement. We show the existence of a unique non-negative
solution and that a modified discrete energy is obtained due to the use of the
SAV method. We also prove the convergence of the discrete solutions to the ones
of the weak form of the continuous Keller-Segel model.
