Numerical approximation of a one-dimensional space fractional advection-dispersion equation with boundary layer

Finite element computations for singularly perturbed convection-diffusion equations have long been an attractive theme for numerical analysis. In this article, we consider the singularly perturbed fractional advection-dispersion equation (FADE) with boundary layer behavior. We derive a theoretical estimate which shows that the under-resolved case corresponds to ǫ < hα−1, where α is the order of the diffusion operator. We also present a numerical method for solving such an FADE in which the boundary layer is incorporated into the finite element basis, and provide numerical experiments which show that knowledge of the boundary layer (either analytically or through a direct numerical simulation) can be used to greatly enhance the efficiency of the finite element method.