An h-p Version of the Continuous Petrov-Galerkin Method for Nonlinear Delay Differential Equations

We investigate an h-p version of the continuous Petrov-Galerkin time stepping method for nonlinear delay differential equations with vanishing delays. We derive a priori error estimates in the -, - and -norm that are completely explicit with respect to the local time steps, the local polynomial degrees, and the local regularity of the exact solution. Moreover, we show that the h-p version continuous Petrov-Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders achieves exponential rates of convergence for solutions with start-up singularities. The theoretical results are illustrated by some numerical experiments.