Spectral element methods for boundary-value problems of functional differential equations

TL;DR

This paper proves that spectral element methods can efficiently approximate solutions to boundary-value problems in functional differential equations, achieving exponential convergence under certain regularity conditions.

Contribution

It establishes convergence and exponential accuracy of spectral element methods for functional differential equations with delays, including state-dependent delays.

Findings

Spectral element method converges with exponential rate for equations with fixed delays.

Convergence also holds for state-dependent delays under mild regularity.

The method achieves higher-than-finite order convergence even when solutions are not analytic.

Abstract

We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems for functional differential equations. In particular, we prove that the numerical collocation solution approximates the true solution with accuracy of order $\mathrm{e}^{-\eta m}$ for some $\eta>0$ and increasing degree $m$ of the polynomials for a case that is common in applications: differential equations where the right-hand side depends on a finite number of delayed arguments with parametric delays and real analytic coefficients. For state-dependent delays the spectral element method also converges under mild regularity assumptions, but the geometric convergence of the collocation solution depends on the properties of the true solution, which may in general not be real analytic even for analytic coefficients. However, in those cases the convergence rate is still higher than all finite orders.