A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

Abstract

After discretization by the finite volume method, the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure. The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix. Moreover, we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations, which is unconditionally convergent for any positive constant. Meanwhile, the iteration methods introduce a new preconditioner for Krylov subspace methods. Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner, compared with the existing approaches.