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 coef-ficient 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 approx-imations uh(x)of u(x)by plugging vh(x)into the representation of u(x).Optimal-order con-vergence estimates of u(x)-uh(x)are proved in both L2 and Hα∕2 norms.Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.