In this paper, we present case studies to illustrate the dependence of the rate of convergence of numerical schemes for singular neutral equations (SNFDEs) on the particular mesh employed in the computation. In Ito and Turi, a semigroup theoretical framework was used to show convergence of semi- and fully- discrete methods for a class of SNFDEs with weakly singular kernels. On the other hand, numerical experiments in Ito and Turi demonstrated a ``degradation" of the expected rate of convergence when uniform meshes were considered. In particular, it was numerically observed that the degradation of the rate of convergence was related to the strength of the singularity in the kernel of the SNFDE. Following the idea used for Volterra equations with weakly singular kernels, see, e.g., Brunner, we investigate graded meshes associated with the kernel of the SNFDE in attempting to restore convergence rates.
"Numerical solutions of a class of singular neutral functional differential equations on graded meshes." J. Integral Equations Applications 30 (3) 447 - 472, 2018. https://doi.org/10.1216/JIE-2018-30-3-447