The $h$-$p$ Version of the Continuous Petrov-Galerkin Method for Nonlinear Volterra Functional Integro-Differential Equations with Vanishing Delays

Abstract

We investigate an $h$-$p$ version of the continuous Petrov-Galerkin method for the nonlinear Volterra functional integro-differential equations with vanishing delays. We derive $h$-$p$ version a priori error estimates in the $L^2$-, $H^1$- and $L^∞$-norms, which are completely explicit in the local discretization and regularity parameters. Numerical computations supporting the theoretical results are also presented.