Superconvergence in collocation methods on quasi-graded meshes for functional differential equations with vanishing delays

We study the optimal orders of (global and local) superconvergence in piecewise polynomial collocation on quasi-graded meshes for functional differential equations with (nonlinear) delays vanishing at t=0. It is shown that while for linear delays (e.g. proportional delays qt with 0<q<1) and certain nonlinear delays the classical optimal order results still hold, high degree of tangency with the identity function at t=0 leads not only to a reduction in the order of superconvergence but also to very serious difficulties in the actual computation of numerical approximations.