  We analyze the convergence of piecewise collocation methods for computing
periodic solutions of general retarded functional differential equations under
the abstract framework recently developed in [S. Maset, Numer. Math. (2016)
133(3):525-555], [S. Maset, SIAM J. Numer. Anal. (2015) 53(6):2771--2793] and
[S. Maset, SIAM J. Numer. Anal. (2015) 53(6):2794--2821]. We rigorously show
that a reformulation as a boundary value problem requires a proper
infinite-dimensional boundary periodic condition in order to be amenable of
such analysis. In this regard, we also highlight the role of the period acting
as an unknown parameter, which is critical since it is directly linked to the
course of time. Finally, we prove that the finite element method is convergent,
while we limit ourselves to commenting on the infeasibility of this approach as
far as the spectral element method is concerned.

Numerical Analysis;Numerical Analysis