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(N2) and computational cost of O(N3) 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(N2) to O(N log N) per iteration and reduces the memory requirement from O(N2) 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.