An efficient multigrid method with preconditioned smoother for two-dimensional anisotropic space-fractional diffusion equations

The anisotropic space-fractional diffusion equations in two dimensions are discretized by the Crank-Nicolson difference scheme with the weighted and shifted Grunwald formula, which is unconditionally stable and second -order convergence. The coefficient matrix of the discretized linear system possesses a two-level Toeplitz-like structure. Due to the anisotropy, the standard multigrid method converges slowly. By utilizing the GMRES method with a newly proposed tridiagonal preconditioner as a smoother, the convergence rate of the multigrid method can be accelerated significantly. The proposed tridiagonal preconditioner is shown to be invertible and a numerical experiment is given to demonstrate the efficiency of the proposed multigrid method with preconditioned smoother.