2018/02/28 by Hao Tian, Jing Zhang, Tian, Hao +1
Engineering · Mathematics · #65M12 #65M70 #74S25 #76R50 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1802.10545
openalex publication_date 2018/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Nonlocal diffusion model provides an appropriate description of the diffusion process of solute in the complex medium, which cannot be described properly by classical theory of PDE. However, the operators in the nonlocal diffusion models are nonlocal, so the resulting numerical methods generate dense or full stiffness matrices. This imposes significant computational and memory challenge for a nonlocal diffusion model. In this paper, we develop a spectral collocation method for the nonlocal diffusion model and provide a rigorous error analysis which theoretically justifies the spectral rate of convergence provided that the kernel functions and the source functions are sufficiently smooth. Compared to finite difference methods and finite element methods, because of the high order convergence rates, the numerical cost of spectral collocation methods will be greatly decreased. Numerical results confirm the exponential rate of convergence.