A Fast Finite Element Method for Space-Fractional Dispersion Equations on Bounded Domains in $\mathbb{R}^2$

Abstract

We develop a fast and accurate finite element method for space-fractional dispersion equations in two space dimensions, which are expressed in terms of fractional directional derivatives in all the directions that are integrated with respect to a probability measure on the unit circle. The fast method significantly reduces the computational work of solving the discrete linear algebraic systems from $O(N^3)$ by a direct solver to $O(N \log N)$ per iteration and a memory requirement from $O(N^2)$ to $O(N)$. Furthermore, the fast method reduces the evaluation of the stiffness matrix, which often constitutes a large portion of the CPU time, from $O(N^2)$ to $O(N)$. The developed preconditioned fast Krylov subspace iterative solver significantly reduces the number of iterations in a Krylov subspace iterative method and may improve the convergence behavior of the solver. Numerical results show the utility of the method.