Multigrid method for fractional diffusion equations

The fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted Grünwald formula. The scheme is unconditionally stable and the coefficient matrix possesses the Toeplitz-like structure. A multigrid method is proposed to solve the resulting system. Meanwhile, the fast Toeplitz matrix–vector multiplication is utilized to lower the computational cost with only O ( N log N ) complexity, where N is the number of the grid points. Numerical experiments are given to demonstrate the efficiency of the method.