An h-p version of the continuous Petrov-Galerkin time stepping method for nonlinear second-order delay differential equations

Abstract

We propose and analyze an h-p version of the continuous Petrov-Galerkin time stepping method for nonlinear second-order delay differential equations with vanishing delays. We establish a priori error estimates in the L2- and L∞-norm that are completely explicit in the local time steps, in the local polynomial degrees, and in the local regularity of the exact solutions. In particular, it is proved that, for analytic solutions with start-up singularities exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Numerical examples are given illustrating the theoretical results.