An improved fast approximation to two-sided variable-order space-fractional diffusion equation and its preconditioning

Abstract

For two-sided variable-order space-fractional diffusion equation, due to the impact of variable fractional order, the discretized stiffness matrix no longer holds Toeplitz-like structure, which brings great challenge to develop efficient solvers. To overcome the difficulty, a fast approximation scheme was proposed in Jia et al. (2021). The main aim of this paper is to propose an improved fast scheme by approximating the stiffness matrix via Chebyshev interpolation technique. Moreover, a block diagonal approximate inverse preconditioner is developed for the proposed scheme to accelerate the convergence of Krylov subspace iteration method. Both theoretical and numerical results demonstrate that the new fast scheme can attain desired solution accuracy with much fewer involved Toeplitz-like approximation terms and hence is evidently more efficient. The effectiveness of the developed preconditioner is also validated.