An Indirect Finite Element Method for Variable-Coefficient Space-Fractional Diffusion Equations and Its Optimal-Order Error Estimates

We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions u(x) to the proposed variable coefficient models in terms of v(x), the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations vh(x) to v(x) and then obtain the approximations uh(x) of u(x) by plugging vh(x) into the representation of u(x). Optimal-order convergence estimates of u(x)-uh(x) are proved in both L2 and H alpha /2 norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.