θθ schemes for finite element discretization of the space–time fractional diffusion equations

The numerical solution of a space–time fractional diffusion equation used to model the anomalous diffusion is considered. Spatial discretization is effected using a finite element method whereas the θθ-scheme is used for temporal discretization. The fully discrete scheme is analyzed for all 0≤θ≤10≤θ≤1 to determine conditional and unconditional stability regimes for the scheme and also to obtain error estimates for the approximate solution. The analysis is facilitated by making use of a variational formulation of the equations that is based on a recently developed nonlocal calculus. One-dimensional numerical examples are provided that illustrate the theoretical stability and convergence results.