Solving large systems arising from fractional models by preconditioned methods

This study develops and analyzes preconditioned Krylov subspace methods for solving discretization of the time-independent space-fractional models. First we apply a shifted Grnwald formulas to obtain a stable finite difference approximation to fractional advection-diffusion equations. Then, we apply two preconditioned iterative methods, namely, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient for normal residual(preconditioned CGN) method, to solve the corresponding discritized systems. We make comparisons between the preconditioners commonly used in the parallelization of the preconditioned Krylov subspace methods. The results suggest that preconditioning technique is a promising candidate for solving large-scale linear systems arising from fractional models.