  Maximising dividends is one classical stability criterion in actuarial risk
theory. Motivated by the fact that dividends are paid periodically in real
life, $\textit{periodic}$ dividend strategies were recently introduced
(Albrecher, Gerber and Shiu, 2011). In this paper, we incorporate fixed
transaction costs into the model and study the optimal periodic dividend
strategy with fixed transaction costs for spectrally negative L\'evy processes.
  The value function of a periodic $(b_u,b_l)$ strategy is calculated by means
of exiting identities and It\^o's excusion when the surplus process is of
unbounded variation. We show that a sufficient condition for optimality is that
the L\'evy measure admits a density which is completely monotonic. Under such
assumptions, a periodic $(b_u,b_l)$ strategy is confirmed to be optimal.
  Results are illustrated.
