Bu et al. discretize the fractional diffusion equation in time using a modified compound trapezoid formula, and the Caputo fractional derivative in space is done using the L1 method, thus obtaining an unconditionally stable semi-discrete variational formulation that is proven through assuming an exact solution satisfying the aforementioned formulation. The fully discrete finite element scheme is defined through the discrete form of a projection operator satisfying the variational form in terms of the discrete Caputo fractional derivative (up to low-order terms), leading to a discretization in space.

Since the above scheme was found to converge slowly, it was improved assuming the initial condition to be zero, and smoothness of the function u(X,t). The Caputo fractional derivative is then implemented through the WSGD formula, using a Fourier transform and the Riemann-Liouville fractional derivative, thus resulting in a weighted sum where the weights are a function of gamma functions, and in turn a function of the fractional order of the derivative. Again, an unconditionally stable semi-discrete scheme is obtained, whereby a projection operator for the finite elements in space is defined satisfying the variational form. The Caputo fractional derivative was later improved with Lagrange functions and Richardson extrapolation. Finally, the numerical experiments indeed show a superlinear convergence rate on average.

I strongly recommend this paper to specialized researchers and PhD students of numerical analysis.