Finite element methods for fractional diffusion problems: smooth and nonsmooth initial data

Abstract: 

The Galerkin finite element method is applied to approximate the solution of a time fractional diffusion equation with variable diffusivity. By a delicate energy analysis, a priori error bounds are derived for both smooth and nonsmooth initial data. The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.