Alternating direction implicit methods for two-dimensional fractional diffusion problems

In this project, we propose and study two different types of alternating direction implicit (ADI) methods for solving numerically a two-dimensional fractional sub-diffusion problem with a Reimann-Liouville fractional derivative of order (0,1). The methods use first-order backward difference and Crank-Nicolson ADI schemes for the time discretization, in combination with a spatial discretization by the standard Galerkin finite elements. A theoretical analysis of the convergence properties of the methods will be carried out. Moreover, simulations and numerical computations for different classes of fractional diffusion models will be delivered to support our theoretical predictions,and alsoto demonstrate the wide applicability of the proposed numerical tools.