Many continuation-based software packages have been developed for the stability and bifurcation analysis of ordinary differential equations. Moreover, thanks to software such as DDE-Biftool [3] it is also possible to analyze the dynamics of functional differential equations defined by an arbitrary number of (possibly state-dependent) discrete delays. All these tools are based on the collocation method in [4]. Since the method is based on collocation with piecewise polynomials, two main strategies can be used to meet higher accuracy requirements. The first one consists in improving resolution uniformly in the whole domain by decreasing the size of the subdomains (thus increasing their number). Alternatively, one can keep the subdomains fixed, while increasing the degree of the polynomial used in each subdomain. The former corresponds to the finite element method, for which the convergence analysis has been carried out in [1, 2]. The latter constitutes the basis of the spectral element methods, which is the focus of the present contribution. Specifically, we prove that if the exact solution of the periodic boundary-value problem has an analytic extension then the collocation solution converges geometrically. If the exact solution has a finite order of continuous differentiability then the collocation solution converges with this order. [1] Andò A. and Breda D. Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations. SIAM Journal on Numerical Analysis (2020) 58(5):3010–3039. [2] Andò A. Sieber J. Boundary-value problems of functional differential equations with state-dependent delays. (submitted 2024). https://arxiv.org/abs/2410.07375. [3] Engelborghs K., Luzyanina T. and Roose D. Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Transactions on Mathematical Software (2022) 28(1):1–21. [4] Engelborghs K., Luzyanina T., in’t Hout K., and Roose D. Collocation methods for the computation of periodic solutions of delay differential equations. SIAM Journal on Scientific Computing (2001) 22(5):1593–1609.