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 being it directly linked to the course of time. Finally, we prove that the finite element method is convergent, while limit ourselves to comment on the unfeasibility of this approach as far as the spectral element method is concerned.

中文翻译：

---

计算滞后泛函微分方程周期解的搭配方法的收敛性分析

我们分析了在最近开发的抽象框架下用于计算一般延迟泛函微分方程周期解的分段搭配方法的收敛性。马塞特，数字。数学。(2016) 133(3):525-555], [S. Maset, SIAM J. 数字。肛门。(2015) 53(6):2771--2793] 和 [S. Maset, SIAM J. 数字。肛门。(2015) 53(6):2794--2821]。我们严格地表明，作为边界值问题的重新表述需要适当的无限维边界周期条件才能进行此类分析。在这方面，我们还强调了作为未知参数的时期的作用，这一点至关重要，因为它与时间进程直接相关。最后，我们证明有限元方法是收敛的，