  The method of choice for describing attractive quantum systems is
Hartree-Fock-Bogoliubov (HFB) theory. This is a nonlinear model which allows
for the description of pairing effects, the main explanation for the
superconductivity of certain materials at very low temperature. This paper is
the first study of Hartree-Fock-Bogoliubov theory from the point of view of
numerical analysis. We start by discussing its proper discretization and then
analyze the convergence of the simple fixed point (Roothaan) algorithm.
Following works by Canc\`es, Le Bris and Levitt for electrons in atoms and
molecules, we show that this algorithm either converges to a solution of the
equation, or oscillates between two states, none of them being a solution to
the HFB equations. We also adapt the Optimal Damping Algorithm of Canc\`es and
Le Bris to the HFB setting and we analyze it. The last part of the paper is
devoted to numerical experiments. We consider a purely gravitational system and
numerically discover that pairing always occurs. We then examine a simplified
model for nucleons, with an effective interaction similar to what is often used
in nuclear physics. In both cases we discuss the importance of using a damping
algorithm.
