Finite difference approximation of space-fractional diffusion problems: The matrix transformation method

A mathematical analysis is presented to establish the convergence of the matrix transformation (or matrix transfer) method for the finite difference approximation of space-fractional diffusion problems. Combined this with an implicit Euler time discretization, the optimal order convergence is proved with respect to the discrete L2 and the maximum norm. The analysis is performed on general two and three-dimensional domains with homogeneous boundary conditions. The corresponding error estimates are illustrated with some numerical experiments.