  Can graded meshes yield more accurate numerical solution than uniform meshes?
A time-dependent nonlocal diffusion problem with a weakly singular kernel is
considered using collocation method. For its steady-state counterpart, under
the sufficiently smooth solution, we first clarify that the standard graded
meshes are worse than uniform meshes and may even lead to divergence; instead,
an optimal convergence rate arises in so-called anomalous graded meshes.
Furthermore, under low regularity solutions, it may suffer from a severe order
reduction in (Chen, Qi, Shi and Wu, IMA J. Numer. Anal., 41 (2021) 3145--3174).
In this case, conversely, a sharp error estimates appears in standard graded
meshes, but offering far less than first-order accuracy. For the time-dependent
case, however, second-order convergence can be achieved on graded meshes. The
related analysis are easily extended for certain multidimensional problems.
Numerical results are provided that confirm the sharpness of the error
estimates.
