θ 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 θ-scheme is used for temporal discretization. The fully discrete scheme is analyzed for all 0 ≤ 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.