A circulant preconditioner for fractional diffusion equations ?
The implicit finite difference scheme with the shifted Grnwald formula, which is unconditionally stable, is employed to discretize fractional diffusion equations. The resulting systems are Toeplitz-like and then the fast Fourier transform can be used to reduce the computational cost of the matrixvector multiplication. The preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems. The spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant; hence the convergence rate of the proposed iterative algorithm is superlinear. Numerical experiments are carried out to demonstrate that our circulant preconditioner works very well, even though for cases of variable diffusion coefficients.
circulant preconditioner fractional diffusion equations fast fourier transform shifted grnwald discretization toeplitz cgnr method