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(NlogN)O(NlogN) complexity, where N is the number of the grid points. Numerical experiments are given to demonstrate the efficiency of the method.

► The coefficient matrix of the fractional diffusion equation is Toeplitz-like.
► A multigrid method is proposed to solve the resulting Toeplitz-like system.
► The smoothing operator is chosen as the damped-Jacobi method.
► The coarse grid operator is constructed to retain the Toeplitz-like structure.
► Numerical results show the robustness and efficiency of the multigrid method.