An a priori error analysis of the hp-version of the C 0-continuous Petrov-Galerkin method for nonlinear second-order delay differential equations

Abstract

We study an hp-version of the -continuous Petrov-Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bound in the -norm that is fully explicit in the local time steps, in the local approximation degrees, and in the local regularity indexes of the exact solutions. Moreover, we prove that the -continuous Petrov-Galerkin based on special hp-version discretization can yield exponential rates of convergence for analytic solutions with initial singularities. Numerical experiments are provided to illustrate the theoretical results.