Nonlocal convection–diffusion problems and finite element approximations

Abstract Nonlocal models are becoming increasingly popular in numerical simulations of important application problems, ranging from anomalous diffusion in heterogeneous environment to crack and fracture in composites. In this paper, we propose and compare some nonlocal models for convection–diffusion problems and study their finite element approximations by piecewise constant and piecewise bilinear basis functions. Our focus is on the convection-dominated case. Through analysis and numerical experiments, we demonstrate the advantage of the upwind models which are more consistent with the physical nature and allow more stable and robust numerical approximations. On the other hand, the central model may produce unphysical oscillations.