Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations

TL;DR

This paper rigorously analyzes the convergence of collocation methods for finding periodic solutions of retarded functional differential equations, emphasizing the importance of boundary conditions and the role of the period as an unknown parameter.

Contribution

It provides a new convergence analysis framework for collocation methods applied to these equations, highlighting the necessity of an infinite-dimensional boundary condition reformulation.

Findings

Finite element method converges for the problem.

Spectral element method's infeasibility is discussed.

Proper boundary conditions are crucial for analysis.

Abstract

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.