On the finite element method for the time-space fractional advection dispersion equation

In this paper, we study the time-space fractional order (fractional for simplicity) advection dispersion equation, which can be an application as a model for anomalous diffusion or fractional diffusion. The fully discrete numerical approximation is analyzed where the Galerkin finite element method for the space Riemann-Liouville fractional derivative with order 1 + β ∈ [1; 2) and the finite difference scheme for the time Caputo derivative with order α ∈ (0, 1). Results on variational solution of the error estimates are presented. Numerical examples are included to confirm the theoretical estimates.