2024/01/31 by Minghua Chen, Chen, Minghua, Chao Min +5
Engineering · Mathematics · #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2401.17567
openalex publication_date 2024/01/31 · openalex created_date 2024/02/02 · openalex updated_date 2026/07/28
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.