An hp-spectral collocation method for nonlinear Volterra functional integro-differential equations with delays

In this paper, we propose a multistep Legendre–Gauss spectral collocation method for the nonlinear Volterra functional integro-differential equations (VFIDEs) with vanishing delays. This method is easy to implement and possesses the high-order accuracy. We also provide a rigorous convergence analysis of the hp  -version of the multistep spectral collocation method under H1H1-norm. Numerical results confirm the theoretical analysis.