A preconditioned fast quadratic spline collocation method for two-sided space-fractional partial differential equations

Abstract

Abstract Quadratic spline collocation methods for the one and two-dimensional fractional diffusion equations are proposed. By carefully exploring the mathematical structure of the coefficient matrix, we propose a matrix-free fast Krylov subspace iterative solver for the corresponding quadratic spline collocation scheme. And then, preconditioning technique is applied to further accelerate the convergence of the fast Krylov subspace iterative method. It shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from O ( N 2 ) to O ( N log N ) per iteration and memory requirement from O ( N 2 ) to O ( N ) , while still well approximates the fractional diffusion equations without any accuracy lost. Numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method.