A spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions

Abstract

The nonlocal diffusion model describes the diffusion process of solutes in complex media properly, while the classical theory of partial differential equations can not provide an appropriate description. The purpose of this paper is to provide illustrations from both theoretical and numerical perspectives of the computation of nonlocal diffusion models by Legendre collocation methods. Compared to local numerical methods, Legendre collocation methods can achieve a fixed accuracy with much fewer unknowns whenever the computational domain is regular and the solutions are sufficiently smooth. This paper is a groundwork towards efficient high order methods including spectral and spectral element methods for nonlocal diffusion equations with volume constrained boundary conditions.