A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the corresponding numerical methods have full coefficient matrices which require storage of O(N-2) and computational cost of O(N-3) for a problem of size N. We develop a superfast-preconditioned conjugate gradient squared method for the efficient solution of steady-state space-fractional diffusion equations. The method reduces the computational work from O(N-2) to O(N log N) per iteration and reduces the memory requirement from O(N-2) to O(N). Furthermore, the method significantly reduces the number of iterations to be mesh size independent. Preliminary numerical experiments for a one-dimensional steady-state diffusion equation with 213 nodes show that the fast method reduces the overall CPU time from 3 h and 27 min for the Gaussian elimination to 0.39 s for the fast method while retaining the accuracy of Gaussian elimination. In contrast, the regular conjugate gradient squared method diverges after 2 days of simulations and more than 20,000 iterations. (C) 2012 Elsevier Inc. All rights reserved.