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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.

international journal of computer mathematics

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