Error analysis of a collocation method on graded meshes for nonlocal diffusion problems with weakly singular kernels

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.